Intrinsic Geometry of Varifolds in Riemannian Manifolds: Monotonicity and Poincare-Sobolev Inequalities

Julio Cesar Correa Hoyos

Tese apresentada ao Instituto de Matemática e Estatística da Universidade de São Paulo para obtenção do título de Doutor em Ciências

Programa: Doutorado em Matemática Orientador: Prof. Dr. Stefano Nardulli

Durante o desenvolvimento deste trabalho o autor recebeu auxílio ﬁnanceiro da CAPES e CNPq

São Paulo, Agosto de 2020 Intrinsic Geometry of Varifolds in Riemannian Manifolds: Monotonicity and Poincare-Sobolev Inequalities

Esta é a versão original da tese elaborada pelo candidato Julio Cesar Correa Hoyos, tal como submetida à Comissão Julgadora. Resumo

CORREA HOYOS, J.C. Geometría Intrínsica de Varifolds em Variedades Riemannianas: Monotonia e Desigualdades do Tipo Poincaré-Sobolev . 2010. 120 f. Tese (Doutorado) - Instituto de Matemática e Estatística, Universidade de São Paulo, São Paulo, 2020.

São provadas desigualdades do tipo Poincaré e Sobolev para funções com suporte compacto definidas em uma varifold k-rectificavel V definida em uma variedade Riemanniana com raio de injetividade positivo e curvatura secional limitada por cima. As técnica usadas permitem consid- erar variedades Riemannianas (M n, g) com métrica g de classe C2 ou mais regular, evitando o uso do mergulho isométrico de Nash. Dito análise permite re-fazer algums fragmentos importantes da teoría geométrica da medida no caso das variedades Riemannianas com métrica C2, que não é Ck+α, com k + α > 2. A classe de varifolds consideradas, são aquelas em que sua primeira variação δV p esta em um espaço de Labesgue L com respeito à sua medida de massa kV k com expoente p ∈ R satisfazendo p > k.

Palavras-chave: Geometría métrica, cálculo das variações, teoría geomética da medida, análise em variedades, primeira variação de uma varifold, desigualdade de Michael-Simon.

iii iv Abstract

CORREA HOYOS, J. C. Intrinsic Geometry of Varifolds in Riemannian Manifolds: Mono- tonicity and Poincare-Sobolev Inequalities. 2010. 120 f. Tese (Doutorado) - Instituto de Matemática e Estatística, Universidade de São Paulo, São Paulo, 2020.

We prove a Poincaré, and a general Sobolev type inequalities for functions with compact support defined on a k-rectifiable varifold V defined on a complete Riemannian manifold with positive injectivity radius and sectional curvature bounded above. Our techniques allow us to consider Rie- mannian manifolds (M n, g) with g of class C2 or more regular, avoiding the use of Nash’s isometric embedding theorem. Our analysis permits to do some quite important fragments of geometric measure theory also for those Riemannian manifolds carrying a C2 metric g, that is not Ck+α with k +α > 2. The class of varifolds we consider are those which first variation δV lies in an appropriate p Lebesgue space L with respect to its weight measure kV k with the exponent p ∈ R satisfying p > k.

Keywords: Metric geometry, calculus of variations, geometric measure theory, analysis on manifolds, ﬁrst variation of a varifold, Michael-Simon inequality.

v vi Contents

1 Introduction 1 1.1 A quick overview...... 2 1.2 Monotonicity Inequality...... 3 1.3 Rectiﬁable varifolds...... 4 1.4 Poincaré and Sobolev inequalities...... 4 1.5 Structure of this thesis...... 5

2 Preliminaries 7 2.1 Basic Geometric Measure Theory...... 7 2.1.1 Radon Measures...... 7 2.1.2 Convergence of Radon measures...... 8 2.1.3 Covering Theorems...... 9 2.1.4 Densities and Diﬀerentiation of measures...... 12 2.1.5 Grassmanian manifold...... 13 2.2 Riemannian Geometry...... 14 2.2.1 Lie Derivatives and the Metric...... 17 2.2.2 Jacobi Fields and Rauch Comparison Theorem...... 18 2.2.3 Geometric Applications of Jacobi Fields Estimates...... 20

3 Intrinsic Riemannian Theory of General Varifolds 25 3.1 General Varifolds...... 25 3.2 The First Variation of a Varifold...... 26

4 Monotonicity and Consequences 31 4.1 Weighted Monotonicity Formulae for Abstrac Varifolds...... 31 4.2 Lp Monotonicity Formula...... 38

5 Poincaré and Sobolev Type Inequalities for Intrinsic Varifolds 45 5.1 Poincaré type inequality...... 45 5.2 Sobolev type Inequality...... 51

Bibliography 57

vii viii CONTENTS Chapter 1

Introduction

In the theory of PDE’s the Sobolev spaces is the setting in which to apply ideas of functional analysis to glean information regarding various specific partial differential equations, in order to recast them abstractly as operators acting on appropriate linear spaces, where the operator encodes the structure of the partial differential equations, including possibly boundary conditions, etc. The great advantage is that once a PDE problem has been appropriately interpreted in this way, the general and elegant principles of functional analysis to study the solvability of various equations involving an abstract operator encoding the properties of the problem. The really hard work is not so much the invocation of functional analysis, but rather finding the "right" spaces and the "right" abstract operators. Sobolev spaces are designed precisely to make all this work out properly, and so these are usually the proper choices for. Once this kind of spaces are at hand, an important questions is: whether a function on a Sobolev space belongs to a certain space? The answer depends on the appropriate choice of certain parame- ters as dimension and desired summability, and discover embeddings of various Sobolev spaces into others involves the uses of crucial analytic tools so-called "Sobolev type inequalities".

Among this "Sobolev type inequalities" there is the Gagliardo-Nirenberg-Sobolev inequality, which let the geometry enter at scene, relating them with the broadly studied isoperimetric inequality for domains in the euclidean space as follows. Theorem 1.0.1 (c.f. [Eva10] sec. 5.6.1.). Assume 2 ≤ p < n. There exists a constant C, depending only on p and n, such that

kuk p∗ n ≤ Ck∇uk p n , L (R ) L (R ) 1 n ∗ np for all u ∈ Cc (R ), where p = n−p . In the special case in which p = 1, and u = χΩ (the characteristic function of Ω), with Ω ⊂ n 1 R open set with boundary of class C , we can proof via integration by parts and a standard regularisation argument that: n n−1 n (L (Ω)) n ≤ CL (∂Ω) , n n where L is the n-dimensional measure of R .

So, the study of this kind of "analytic inequalities" is in fact related with important geometric concepts, in fact in 1960, in his seminal work, [FF60] Federer and Fleming proves that once we can establish an isoperimetric inequality, there exists an associated Sobolev type inequality.

This is why Sobolev inequality has been studied for many years in the Euclidean case, as well as in the Riemannian case, and its prominent role in the theory of partial differential equations and geometry is well known. In [Mir67] Miranda obtained a Sobolev inequality for functions defined on minimal graphs, The Mirandas proof follows from the Isoperimetric inequality proved in [FF60] for integer currents and a procedure introduced by De Giorgi in [BDGM69a]. Bombieri used a refined version of this new inequality, De Giorgi and Miranda (employing the isoperimetric inequality of [FF60]), to derive gradient bounds for solutions to the minimal surface equation (see [BDGM69b]).

1 2 INTRODUCTION 1.1

In [MS73], Michael and Simon prove a general Sobolev type inequality, which proof follows an argument which is, in some aspects, evocative of the potential theory. That inequality is obtained on what might be termed a generalized manifold and in particular, the classical Sobolev inequality, a Sobolev inequality on graphs of weak solutions to the mean curvature equation, and a Sobolev 2 n inequality on arbitrary C submanifolds of R (of arbitrary co-dimension) are derived.

On the other hand, Allard in a pioneering work [All72] proved a Sobolev type inequality for non-negative, compact supported functions defined on a varifold V , whose first variation δV lies in an appropriate Lebesgue space endowed with the measure kδV k, by generalization of the Isoperi- metric inequality for integer currents in [FF60]. The proof of this inequality given by Allards follows from the monotonicity formula derived from the computation of the first variation of a varifold for a suitable perturbation of a radial vector field.

Following the ideas of Michael and Simon, Hoﬀman and Spruk proved in [HS74] a general Sobolev inequality for submanifolds N of a Riemannian manifold M, satisfying geometric restric- tions involving the volume of M, the sectional curvatures and the injectivity radius of M. Their proof is inspired in the Michael and Simon work, therefore, is an extrinsic perspective.

Since this general Sobolev inequality has been largely studied in diﬀerent contexts from an extrinsic point of view, a natural question is whether or not this kind of inequality remains valid from an intrinsic point o view. For functions with compact support on a varifold V whose ﬁrst variation δV lies in an appropriate Lebesgue space with respect to kδV k.

In this thesis, we show an intrinsic Riemannian analog to the Allard result, considering a k- dimensional varifold V defined in an n-dimensional Riemannian manifold (M n, g) (with 1 ≤ k ≤ n) defined intrinsically. We achieve this goal by recovering a monotonicity inequality (instead of monotonicity equality) in this intrinsic Riemannian context, which takes into account the bounds on the geometry of M, this procedure envolves ideas of the classical geometric measure theory and analysis on Riemannian manifolds. Then we follow the ideas of Simon and Michael in [MS73] and [Sim83], to get a local version of the desired inequality. Finally, The Sobolev type inequality is then obtained by a standard covering argument. 1.1 A quick overview We present a natural extension of Allard’s work. In fact, instead of define a general varifold on a Riemannian manifold via an isometric embedding (i.e., as a Radon measure on Gk(i(M)) where n N i : M → U ⊂ R is an isometric embedding), is defined as a nonnegative, real extended valued, n Radon measure on Gk(M ), the Grassmannian manifold whose underlying set is the union of the n sets of k-dimensional subspaces of TxM as x varies on M . This point of view has a consequence: more freedom on the regularity of the metric (i.e., our theory holds even for the case of only C2 metrics, when the Nash embedding does not exists). In fact, there is a gap in the theory of isometric N 2 2+α embeddings in R , precisely in the case when the metric is C but not C with α > 0 it is not known whether an isometric immersion into Euclidean space exists. On the other hand, whenever the metric is Ck,α with k + α > 2 there exists isometric embeddings of class Ck+α. If the metric is k,α 1+ k+α C with k + α < 2 then there are isometric embeddings C 2 . The first theorem is proved with the aid of the "hard implicit function theorem", à la Nash-Moser. The second is proved using the technics of the first paper by John Nash about isometric embeddings, compare [Nas54]. This freedom constitutes, as shown in [Nar18] and [NOA18], a powerful tool to tackle problems in noncompact Riemannian manifolds of bounded geometry, or compact Riemannian manifolds with variable metric and also, possibly noncompact Riemannian manifolds of bounded geometry and variable metrics.

Our main goal is to reproduce Allard’s Theorem 7.3 in this new context along the lines of the proofs of Theorems 18.5 and 18.6 of [Sim83]. To do so, as in [HS74], our ambient manifold must satisfy some geometric conditions, namely positive injectivity radius and sectional curvature 1.3 MONOTONICITY INEQUALITY 3 bounded above, which from now on we refer as bounded geometry, see Definition 4.1.1. On the other hand, since we are interested in varifolds satisfying the conditions of the celebrated Theorem 8.1 [All72] (from now on called Allard Conditions (AC), see Definition 3.2.3) is necessary to study the properties of varifolds satisfying these conditions. Roughly speaking, we are interested in varifolds whose first variation has no boundary term, having generalized mean curvature belonging to Lp with respect to kδV k for some p > k, and whose density ratio is close to that of an Euclidean disk in a small ball, and density bounded below far from 0, kV k-a.e. The main contributions of this paper are:

(i) to define rectifiable sets, exploiting the local structure of the ambient manifold, and so, define rectifiable varifolds as in Chapter 4 of [Sim83].

(ii) to give an intrinsic Lp-monotonicity formula valid for manifolds with bounded geometry and N not only in R , without using Nash’s isometric embedding Theorem. See Theorem1. (iii) to prove, intrinsically, Poincaré and Sobolev type inequalities for C1(S) non-negative functions k defined over a Hg -rectifiable set S ⊂ M. Compare Theorems2 and3. (iv) to prove that the best constant C > 0 in the extrinsic Sobolev inequality (1.4) depends only on the dimension k of V , which is a remarkable fact, since one a priori expects that C could depend also on the bounds of the geometry of the ambient manifold n, k, injM , b. For the meaning of the former constants see Definition 4.1.1.

The rest of this introduction will describe in more detail the contributions of this paper. 1.2 Monotonicity Inequality Given a general varifold V , the classic way to deduce a "monotonicity formula" is to compute the first variation of V at a perturbations, by smooth functions, of (Euclidean) radial vector fields, u then seems natural to consider vector fields of the form γ( s )(u∇u)(x), where γ(t) is a smooth real valued function which vanishes for large values of the variable t, and u(x) := dist(M,g)(x, ξ) for some ξ ∈ M. The classic calculation uses the fact that the divergence of a radial vector field over a k dimensional plane is exactly k, which is not true in general Riemannian manifolds, so it is here where the bounded geometry plays a central role, since the Rauch comparison Theorem gives bounds on such quantity. The price to pay in making our monotonicity formula using this comparison geometry argument is that there is no equality anymore, instead we have the following inequality.

Theorem 1. Let (M n, g) be a complete Riemannian manifold with g of class at least C2 having n bounded geometry and Levi-Civita connection ∇g, such that r0 cotb(r0) > 0, and let V ∈ Vk(M ) 1 satisfying (AC), then for any 0 < s < r0, and h ∈ C (M) non-negative. There exists a constant c = c(s, b) ∈]0, 1[ such that, if we set u(x) = rξ(x) = dist(M,g)(x, ξ) we have for all 0 < s < r0

2 ! ∇S⊥ u d 1 Z d Z h(y)dkV k(y) ≥ h(y) g dV (y, T ) ds sk ds k Bg(ξ,s) Gk(Bg(ξ,s)) rξ ! 1 Z + k+1 h∇h(y), (u∇u)(y)ig dV (y, T ) s Gk(Bg(ξ,s)) (1.1) (c(s) − 1) k Z + k h(y)dkV k(y) s s Bg(ξ,s) 1 Z + k+1 hHg, h(y)(u∇u)(y)ig dkV k(y). s Bg(ξ,s)

Here Hg is as in Proposition 3.2.1. 4 INTRODUCTION 1.4

1.3 Rectifiable varifolds Here we define the main object of our investigation, namely rectifiable varifolds. We follow here the approach of Chapter 4 of [Sim83] without treating the rectifiability of general varifolds under conditions of the first variation and the density of their weights. Before to give the following definition it is worth to recall here the classical Rademacher Theorem.

n n Theorem 1.3.1 (Rademacher). If f is Lipschitz on R , then f is diﬀerentiable L -almost everywhere; that is, the gradient ∇f(x) exists and

f(y) − f(x) − ∇f(x) · (y − x) lim = 0, y→x |y − x|

n n for L -a . e .x ∈ R . Definition 1.3.1 (cf. [AK00], Definition 5.3, pg. 536). Let (M n, g) be a complete Riemannian k manifold. We say that a Borel set S ⊂ M is countable Hg -rectifiable if there exists countable many k n Lipschitz functions fj : R → M such that   k [ k Hg S \ fj(R ) = 0. j∈N

S k −1 For any x ∈ fj( ) such that fj is differentiable at y = f (x), we define the approximate j∈N R j k k n tangent space Tan (S, x) as dfy(R ) ≤ TyM . k k Remark 1.3.1. Observe that the preceding definition is well posed, since, if x ∈ fj(R )∩fi(R ) and −1 k k yl ∈ fl (x) for l ∈ {i, j}, then dfyi (R ) = dfyj (R ). The reader could consult [AK00], Definition 5.5, pg. 536.

k Definition 1.3.2 (c.f. [Sim83] Chapter 4, pg. 77). Let S be a countably Hg -rectifiable, subset of n 1 k (M , g) and let θ ∈ Lloc(S, Hg ). Corresponding to such a pair (S, θ) we define the rectifiable k- ˜ ˜ ˜ k varifold v(S, θ) to be, the equivalence class of all pairs (S, θ), where S is countably Hg -rectifiable k ˜ k ˜ with Hg ((S∆Se)) = 0 and where θ = θ, Hg − a.e. on S ∩ S. So, we can naturally induce a (general) varifold from a rectifiable one as follows.

k n Definition 1.3.3. Given an Hg -rectifiable varifold v(S, θ) on M there is a corresponding (general) n k-varifold V ∈ Vk(M ) (also denoted by v(S, θ)), defined by

k n V (A) := v(S, θ)(A) = Hg xθ(π({(x, TxM): x ∈ S∗} ∩ A)),A ⊂ Gk(M ), where S∗ is the set of x ∈ S such that S has an approximate tangent space TxS with respect to θ at k x. Evidently v(S, θ), so defined, has weight measure kv(S, θ)k = Hg xθ. 1.4 Poincaré and Sobolev inequalities In the special case in which, V = v(S, θ) is a rectifiable varifold such that the mean vector field belongs to a certain Lebesgue space Lp with p > k, the Theorem1 implies that, for h ∈ C1(S), h ≥ 0, S ! ∗ 1 Z Z ∇ h h(ξ) ≤ e(Λ+c k)ρ ρk hdkV k + g dkV k , ω k−1 k Bg(ξ,ρ) Bg(ξ,ρ) rξ for all ξ ∈ spt kV k and for all 0 < ρ < r0. From this, together with an approximation argument and Fubini’s Theorem we deduce the following Poincaré inequality. 1.5 STRUCTURE OF THIS THESIS 5

Theorem 2. Let (M n, g) be a complete Riemannian manifold with bounded geometry, let V := 1 v(S, θ) a rectiﬁable varifold satisfying (AC). Suppose: h ∈ C (M), h ≥ 0, Bg(ξ, 2ρ) ⊂ Bg(ξ, r0) for ξ ∈ S ﬁxed, |Hg|g ≤ Λ for some Λ > 0, θ > 1 kV k-a.e. in Bg(ξ, r0) and for some 0 < α < 1

k (Λ+c∗k)ρ kV k ({x ∈ Bg (ξ, ρ): h(x) > 0}) ≤ ωk (1 − α) ρ and, e ≤ 1 + α. (1.2)

Suppose also that, for some constant Γ > 0

k kV k (Bg(ξ, 2ρ)) ≤ Γρ . (1.3)

1 Then there are constants β := β (k, α, r0, b) ∈]0, 2 [ and C := C (k, α, r0, b) > 0 such that Z Z hdkV k ≤ Cρ ∇Sh dkV k. g g Bg(ξ,2ρ) Bg(ξ,ρ)

Here Hg is as in Proposition 3.2.1. Finally, the Sobolev inequality follows from Theorem1 again in the special case in which V = v(S, θ) is a rectiﬁable varifold such that the mean vector ﬁeld belongs to a certain Lebesgue space and a standard covering argument (c.f. [Sim83] Theorem 3.3 pg. 11).

Theorem 3. Let (M n, g) be a complete Riemannian manifold with bounded geometry, let V = 1 v(S, θ) a k-rectiﬁable varifold satisfying (AC). Suppose h ∈ C0 (M) non negative, and θ ≥ 1 kV k- a.e. in S. Then there exists C := C(k) > 0 such that

k−1 Z k Z k S ∗ h k−1 dkV k ≤ C ∇ h + h |H | − c k dkV k. (1.4) g g g g S S

Here Hg is as in Proposition 3.2.1. 1.5 Structure of this thesis In the Section2 we make a review of the basic concepts of the classical geometric measure theory and Riemannian geometry which in the sequel will be used, the experienced reader may omit this section. In general in this section we give precise references for the classical results, and encourage the unaware reader to look them. In Section3 the definition of general varifold of [All72] is extended to a complete Riemannian manifold as well as the first variation of them, extending the geometric theory of varifolds to this setting. In Section4 Theorem1 is proved, by testing the first variation of a general varifold with a suitable radial deformation field. Here the bounded geometry assumptions of the ambient manifold, play an essential role. In Section4 are also proved several results concerning the monotonicity of the density ratio of the wight measure, under Lp-type assumptions on the weak mean curvature field Hg. Finally, in Section5 Theorem2 and Theorem3 are proved, as a consequence of Theorem1, standard covering argument, and several technical results from classical analysis. 6 INTRODUCTION 1.5 Chapter 2

Preliminaries

The aim of this chapter is to introduce the basic notations and classical results fo the Geometric Measure Theory and Riemannian Comparison Geometry, in an intrinsic setting, in order to cover all the basics, to tackle in Chapter4, the intrinsic theory of general and rectiﬁable varifolds. Since our goal is to obtain estimates on functions deﬁned on a varifold which ambient space is a Riemnnian manifold, exploiting the geometry of the ambient space, and the monotonicity behaviour of the density of varifolds, the following results will be in the context of Riemannian manifold, or more generally in separabel metric spaces. 2.1 Basic Geometric Measure Theory 2.1.1 Radon Measures Let us begin with the basics on Radon measures and their relation with continuous functions, those link between the topology of the set of continuous functions and the space of Radon measure is known as the Riesz representation theorem, and is the cornerstone of the geometric measure theory. The proof of the Theorem in this section, can be founded in [Mag12] Chapter 4 (from where was taken) or in [Sim83] Ch 1.

Deﬁnition 2.1.1. An outer measure µ is a Radon measure in X, if it is a locally ﬁnite (i.e. µ(K) < ∞ for all K ⊂ X compact), Borel regular measure on X.

If µ is a Radon measure, in particular is Borel regular, is well known that

µ(E) = inf{µ(A): E ⊂ A, A open set} = sup{µ(K): K ⊂ E, K compact set}, for every E ∈ B(X). Thus by Borel regularity a Radon measure µ is characterized on M(µ) by its behaviour on compact (or on open) sets. It is clear that, if µ is a Borel regular measure on X, and E ∈ M(µ) is such that µ(E) < ∞, then µxE is a Radon measure on X. n Deﬁnition 2.1.2. An outer measure µ on R is concentrated on E ⊂ X if µ(X \ E) = 0. The intersection of the closed sets E such that µ is concentrated on E is denoted spt µ and called the support of µ. Furthermore, if X is metric, locally compact separable,

X \ spt(µ) = {x ∈ X : µ(B(x, r)) = 0 for some r > 0}.

Deﬁnition 2.1.3. Let X be metric, locally compact, and separable, and H a real Hilbert space, 0 consider L : Cc (X; H) → R a linear functional, we deﬁne the total variation of L as the set function |L| : P(X) → [0, ∞], such that, for any A ⊂ X open set

0 |L|(A) := sup{hL, ϕi : ϕ ∈ Cc (A; H), |ϕ| ≤ 1},

7 8 PRELIMINARIES 2.1 and for any arbitrary E ⊂ X

|L|(E) := inf{|L|(A): E ⊂ A, A open set}.

Theorem 2.1.1 (Riez Representation Theorem). Let X be a metric, locally compact and separable 0 space, H a real Hilbert space, if L : Cc (X; H) → R is a bounded linear functional (hence continuous), then its total variation |L| is a positive Radon measure on X and there exists a |L|-mesurable function ν : X → H, with |ν| = 1, |L|-a.e. on X and Z 0 hL, ϕi = hϕ, νid|L|, for all ϕ ∈ Cc (X,H), X that is, L = g|L|. Moreover, for every open set A ⊂ X we have Z 0 |L|(A) = sup hϕ, gid|L| : ϕ ∈ Cc (A, H), |ϕ| ≤ 1 , X and for each arbitrary subset E ⊂ X the following formula holds

|L|(E) := inf{|L|(A): E ⊂ A and A is open}.

2.1.2 Convergence of Radon measures Once establish the relation between radon measures and continuous functions, we begin to exploit the topology of this space to investigate compactness properties of the Radon measures, the proof in this section can be found in [Mag12] Chapter 4 and 5.

Let X be a metric, locally compact and separable space, and H a real Hilbert space, let L be a 0 monotone linear functional on Cc (X,H), i.e., L(ϕ1) ≤ L(ϕ2) whenever ϕ1 ≤ ϕ2, then L is bounded 0 on Cc (X,H), and by Theorem 2.1.1 it holds Z hL, ϕi = hϕ, νi d|L|, X where ν : X → H is |L|-mesurable, with |ν| = 1 |L|-a.e. on X, Note also that if two Radon measures n µ1, µ2 on R coincides as linear functionals, that is Z Z 0 n ϕdµ1 = ϕdµ2, ∀ϕ ∈ Cc (R ), n X R then we have µ1 = µ2, so Radon measures can be unambiguously identiﬁed with monotone linear 0 functionals on Cc (X,H).

Now, let Bb(X) denote the family of bounded Borel sets of X and B(E) the family of bounded 0 n sets contained in E ⊂ X. If L is a bounded linear functional on Cc (X; R ) (we are considering the n n special case when H = R ), then L induces a R -valued set function

ν : Bb(X) → R Z E 7→ ν(E) := νd|L|, E that enjoys the σ-additivity property, i.e. ! [ X ν En = ν(En), n∈N n∈N on every disjoint sequence {En}n∈N ⊂ B(K) for some K compact set in X. 2.1 BASIC GEOMETRIC MEASURE THEORY 9

0 n m n Thus bounded linear functionals on Cc (X; R ) naturally induces a R -valued set function on R that are σ-additive on bounded Borel sets, then is natural to deﬁne a notion of convergence on it.

Deﬁnition 2.1.4. Let X as above, {µn}n∈N and µ Radon measures with values in X. We say that ? µn converge to µ, and we write µn * µ, if Z Z 0 n ϕdµ = lim ϕdµn, for all ϕ ∈ Cc (X; R ). X n→∞ X Notice that this notion of covergence coincides with the waek* convergence for the space 0 m Cc (X, R ) 0; and by the comments above we know that the space of Radon measures is con- 0 m tained in Cc (X, R ) 0 this is the natural notion of convergence for Radon measures. The following two Theorems, are important results concerning to weak* convergence of Radon measures and the compactness of this space, and will be used along the text repeatedly, the full proof of that can be found in [Sim83]

Proposition 2.1.1. Let X a metric, locally compact and separable space. If {µn}n∈N and µ are Radon measures on X, then the following are equivalents ? 1. µn * µ. 2. If K is a compact set and A is an open set, then

µ(K) ≥ lim sup µn(K), n→∞

µ(A) ≤ lim inf µn(A). n→∞

3. If E is a bounded Borel set with µ(∂E) = 0, then

µ(E) = lim µn(E). n→∞

? Moreover if µn * µ the for every x ∈ spt µ there exists {xn}n∈N ⊂ X with

lim xn = x, xn ∈ Sptµn, for all n ∈ . n→∞ N

Theorem 2.1.2 (Compactness Criteria for Radon Measures). Let X a metric, locally compact and separable space. If {µn}n∈N is a sequence of Radon measures on X such that, for all compact set K in X sup µn(K) < ∞, n∈N then there exists a Radon measure µ on X and a sequence n(k) → ∞, as k → ∞ such that

? µn(k) * µ.

2.1.3 Covering Theorems In this section we deal with some covering theorems, which will allow to deduce global properties of a Radon measure from local ones with respect to some subset. Since the main goal is to deal with "measure theoretic" objects deﬁned on a Riemannian manifold, we need a machinery which deals with the fact that we are not in an Euclidean space anymore to infer key properties of suitable Radon measures in the large. This theory was developed in more general context by H. Federer (c.f. [Fed69] Chapter 2). From now, we will denote (X, d) a locally compact and separable metric space, and a µ a Radon measure on X. we also denote Bd(x, r) := {y ∈ X : d(x, y) < r} the open ball centered at x ∈ X of radius r > 0 on X with respect to de distance d. 10 PRELIMINARIES 2.1

Theorem 2.1.3. Let B = {Bd(xα, rα)}α∈Λ any collection of closed balls in X, with

sup{diam(Bd(xα, rα))} < ∞, α∈Λ

0 0 then, there exists a Λ ⊂ Λ such that the subcollection B := {Bd(xβ, rβ)}β∈Λ0 ⊂ B satisﬁes:

0 [ [ Bd(xβ, rβ) ∩ Bd(xη, rη) = ∅ if β, η ∈ Λ with β 6= η and, Bd(xα, rα) ⊂ Bd(xβ, 5rβ). α∈Λ β∈Λ0

In fact we can arrange the following stronger property: if Bd(xα, rα) ∈ B then there exists Bd(xβ, rβ) ∈ 0 B , such that Bd(xα, rα) ∩ Bd(xβ, rβ) = ∅, and Bd(xα, rα) ⊂ Bd(xβ, 5rβ). For a proof of this result the interested reader may see [Sim83] Theorem 1.3.3 pg. 11. Nevertheless this result is very useful in presence of some "regularity hypothesis" of a Radon measure , and for most applications it is good enough to obtain global properties from local ones. However, deal with problems where no such regularity hypothesis is at hand, this forces the use of another type of covering theorems, which impose no special hypothesis on the measure, but require boundedness of the set of centers of a collection of closed balls and a geometric condition on the metric d, namely.

Deﬁnition 2.1.5. A set A in a metric space X is said to be directionally (δ, C) limited if for each a ∈ A there exists at most C distinct points ξ1, . . . , ξC in Bd(a, δ)∩A such that for i 6= j and x ∈ X, the following holds: if

d(a, x) + d(x, ξi) = d(a, ξi) and d(a, ξj) = d(a, x), then 1 d(x, ξ ) ≥ d(a, ξ ). j 4 j n The idea behind this deﬁnition occurs when we consider the special case in which X = A = R , n with the usual metric induced from the canonical norm k·k. To see that R is directionally (δ, C(n)) n limited, for all δ > 0 and C(n) a dimensional constant, assume a = 0, and take ξi, ξj ∈ R ∩Bd(0, δ) distinct such that kξik ≥ kξjk, then setting

ξ kξ k x = i j , kξik we have that

d(0, ξj) = kξjk

ξjkξik = kξik = kxk, and

ξikξjk d(0, x) + d(x, ξi) = kxk + − ξi kξik

ξikξjk = kξjk + − ξi kξik 2.1 BASIC GEOMETRIC MEASURE THEORY 11

kξ kkξ k + kξ kξ k − ξ kξ kk = j i i j i i kξik kξ kkξ k + kξ k (kkξ k − kξ kk) = j i i j i kξik kξik = (kξjk + kkξjk − kξikk) kξik

= kξjk − kξjk + kξik

= d(0, ξi), where we use that kξik ≥ kξj. Thus, d(x, ξj) 1 ξikξjk ξi ξj = − ξj = − . d(0, ξj) kξik kξjk kξik kξjk On the other hand, since n n S := {ξ ∈ R : d(0, ξ) = 1}, C(n) n is compact, there exists C(n) finite integer, and {Uk}k=1 ⊂ R a collection of open sets such that n SC(n) 1 S ⊂ k=1 Uk and diam(Uk) ≥ 4 , for all k = 1,...,C(n). Then, there exists i, j ∈ {1,...,C(n)} ξi n ξj n ξi such that ∈ Ui ∩ and ∈ Uj ∩ , therefore, if we assume that neither ∈ Uj nor kξik S kξj k S kξik ξj ∈ Ui, we have that kxj k d(x, ξ ) 1 j ≥ , d(0, ξj) 4 then, there exists at most C(n) different points fulfilling the conditions. Finally, since δ is arbitrary, and without loss of generality the argument can be repeated for all n n a ∈ R , follows that R is directionally (δ, C(n)) limited.Thus, the word "directionaly" make sense, since there exists at most finite directions fulfiling the conditions.

Another interesting special case (whose study led to formulate the general concept) arises when X = (M, g) is a complete Riemannian manifold with g of class greater or equal than 2 and A = Bg(ξ,˜ r0) is the geodesic closed ball centered at ξ˜ ∈ M and r0 > 0 is such that

expξ˜ : {v ∈ TξM : kvk ≤ r0} ⊂ TξM → expξ˜({v ∈ TξM : |v| ≤ r0}), is a diﬀeomorphism (by the completeness of (M, g), such r0 always exists and is positive). Since ˜ expξ˜ is a diﬀeomorphism, and Bg(ξ, r0) is a compact set, it is an bi-Lipschitz map with constant λ > 0, i.e. for all vi, vj ∈ Tξ˜M 1 δ(v , v ) ≤ g(v , v ) ≤ λδ(v , v ), λ i j i j i j n ∼ where δ(·, ·) stands for the canonical metric on R = Tξ˜M. Hence, setting, viδ(vj, vj) ξi = expξ˜(vi), ξj = expξ˜(vj), and x = expξ˜ , δ(vi, vi) where r0 ≥ kvik ≥ kvjk, we have that,

dg(x, ξj) 2 dδ(x, vj) ξi ξj ≤ λ = − , dg(ξ,˜ ξj) dδ(0, vj) kξik kξjk 12 PRELIMINARIES 2.1

n where dδ and dg are the distance induced respectively by δ and g. Then, reasoning as in the R case, we have that there exists C(n, λ) positive integer and 0 < ε(ξ˜) < r0 such that, for ξi, ξj ∈ Bg(ξ,˜ ε(ξ˜)) and x as above, where i, j ∈ {1,...,C(n)} and if

dg(ξ,˜ x) + dg(x, ξi) = dg(ξ,˜ ξi) and dg(ξ,˜ ξj) = d(ξ,˜ x), then 1 d (x, ξ ) ≥ d (ξ,˜ ξ ). g j 4 g j

Finally, by compactness of Bg(ξ,˜ r0), ε(ξ˜) can be chosen uniform, satisfying that, for all ξ ∈ Bg(ξ,˜ r0), the argument above apply in the smaller ball Bg(ξ, ε) (this is possible by the compactness of the unit tangent vectors at the points of B (ξ,˜ r )), thus, taking C := sup C(n, λ), we have that g 0 λ∈R Bg(ξ,˜ r0) is directionally (ε, C) limited. The notion of directionally limited sets, permits to generalize a classical covering Theorem, known as the Besicovitch coverig theorem, to more general settings, as a closed balls of a Riemannian manifold with regular enough metric (c.f. [Mag12] Theorem 5.1 for the Euclidean case).

Theorem 2.1.4 (Federer-Besicovitch covering theorem). Let B := {Bd(xα, rα)}α∈Λ a family of closed balls such that {xα}α∈Λ = A is a directionally (δ, C) limited subset of X, and 0 < rα < δ for all α ∈ Λ, then there are 2C + 1 disjointed subfamilies G1,..., G2C+1 of B such that

2C+1 [ [ A ⊂ B.

i=1 B∈Gi

Corollary 2.1.4.1 (Vitali’s symmetric property). Let µ be a Radon measure on X, and B := {Bd(xα, rα)}α∈Λ a collection of closed non-degenerate balls such that A = {xα}α∈Λ is µ -measurable, and directionally (δ, C) limited, and, for every xα ∈ A,

inf diam(Bd(xα, rα)) = 0, α∈Λ then there exists a countable disjoint subfamily G := {Bd(xβ, rβ)}β∈Λ0⊂Λ of B such that   [ µ A \ Bd(xβ, rβ) = 0. β∈Λ0

Deﬁnition 2.1.6. Let (X, d) be locally compact and separable metric space and aµ a Radon measure on X satisfying the hypothesis of the Corollary 2.1.4.1, we say that X satisfying has the Vitali’s symmetric property with respect to µ.

Remark 2.1.1. Since we have seen that given a Riemannian manifold (M n, g) with C2 (or more regular) metric the geodesic balls Bg(ξ, r) with r < inj(M n,g)(ξ) are directionally (ε, C) limited ˜ ˜ for suitable ε and C(n), for each ξ ∈ Bg(ξ, inj(M n,g)(ξ)), the Theorem 2.1.4 is valid, and then ˜ ˜ (Bg(ξ, inj(M n,g)(ξ))) satisfy the Vitali’s symmetric property for a regular enough Radon measure µ on M. In fact every compact subset of a Riemannian manifold with metric of class greater than or equal than 2, c.f. [Fed69] 2.9.

2.1.4 Densities and Differentiation of measures Definition 2.1.7. Let X be a metric space, for any outer measure µ on X, any subset A ⊂ X, and ∗n n any point x ∈ X, we define the n -dimensional upper and lower densities Θ (µ, A, x), Θ∗ (µ, A, x) 2.1 BASIC GEOMETRIC MEASURE THEORY 13 by

∗n µ (A ∩ B(x, ρ)) Θ (µ, A, x) = lim sup n ρ↓0 ωnρ

n µ (A ∩ B(x, ρ)) Θ∗ (µ, A, x) = lim sup n ρ↓0 ωnρ

If both limits exists and coincides, the common value is called the n-dimensional density of µ at n ∗n n x ∈ A, and is denoted by Θ (µ, A, x) In case A = X we simply write Θ (µ, x) and Θ∗ (µ, x) to ∗n ∗n n n denote these quantities, so that Θ (µ, A, x) = Θ (µxA, x), Θ∗ (µ, A, x) = Θ∗ (µxA, x). Deﬁnition 2.1.8 (Upper and lower µ-densities). Let µ and ν be Radon measures on X, metric space, the upper µ density and the lower µ density of ν are functions

+ Dµ ν : spt µ → [0, ∞], − Dµ ν : spt µ → [0, ∞], deﬁned respectively as

+ ν(B(x, r)) Dµ ν(x) := lim sup , x ∈ spt µ, r↓0 µ(B(x, r))

− ν(B(x, r)) Dµ ν(x) := lim inf , x ∈ spt µ. r↓0 µ(B(x, r))

If the two limits exists and are ﬁnite, then we denote by Dµν(x) their common value and call it the µ-density on ν at x. Thus we have deﬁned a function

− + Dµν : {x ∈ sptµ : Dµ ν = Dµ ν} → [0, ∞].

The following result is a classic result concerning with diﬀerentiation of Radon measures, and for the proof can be found in [Fed69] in full generality, or in [Sim83]. Once we have discuss how geodesic balls in a Riemannian manifold of class greater than or equal than 2 satisﬁes the Vitali’s symmetric property in the full generality, the Simons approach is what we follow.

Theorem 2.1.5 (Lebesgue-Besicovitch-Federer diﬀerentiation Theorem). Let µ and ν are Radon measures on X, a metric space satisfying theVitali’s symetric property for ν. Then Dµν is deﬁned 1 µ-a.e. on X, Dµν ∈ Lloc(X, µ), and, in fact Dµν is a Borel measure on X. Furthermore

s ν = (Dµν)µ + νµ, on M(µ), (2.1)

s where the Radon measure νµ is concentrated on the Borel set

n + Y = R \{x ∈ sptµ : Dµ ν(x) < ∞} n + = (R \ sptµ) ∪ {x ∈ sptµ : Dµ ν(x) = ∞},

s in particular; νµ ⊥ µ. 2.1.5 Grassmanian manifold To end this "beginners guide" in geometric measure theory, we look up to the Grassmanian manifold (the fact that this is in fact a diﬀerentiable manifold, is a well know exercise on topology of manifolds, and can be founded in [War83]) and study their metric properties in the particular case n in which we have k dimensional subspaces of a tangent space TxM of an n dimensional manifold M. 14 PRELIMINARIES 2.2

Deﬁnition 2.1.9. Let (M n, g) a n-dimensional Riemmanian manifold, the space of all k-dimensional (k < n) subspaces of TxM, for x ∈ M ﬁxed will be denoted as Gr(k, TxM) endowed with the usual topology.

n Since for x ﬁxed TxM ' R the topology is the same. Suppose S ∈ Gr(k, TxM) We will use pS to denote orthogonal projection of TxM onto S. That is, pS ∈ Hom (TxM,TxM) is deﬁned as

k X pS(u) = hu, τiig τi, i=1 where u ∈ TxM for x ﬁxed, and {τ1, . . . , τk} is an orthonormal basis for S ∈ Gr(k, TξM).

ijn Then, the associated matrix e i,j=1 to pS is deﬁned as

ij e = hei, τji , where, as before {τ1, . . . , τk} is an onrthonormal basis for S, and {e1, . . . , en} is an orthonormal basis for TxM. Notice that pS satisﬁes

∗ pS ◦ pS = pS, (pS) = pS and Im (pS) = pS.

ijn Therefore e i,j=1 satisﬁes

ij2 ij ijn e = e and trace e i,j=1 = n.

On the other hand, we can induce an inner product on Hom (TxM,TxM) as

∗ L1 • L2 = trace (L1 ◦ L2) , and this inner product naturally induces a norm, denoted by | · |. Notice that, if we consider the operator norm on Hom(TxM,TxM) deﬁned as

kLk = sup{|L(v)|g : v ∈ TxM and |v| ≤ 1}, we have kLk ≤ |L| ≤ n1/2kLk. 2.2 Riemannian Geometry In this section the basics on Riemannian geometry to prove the Lemma 3.6 of [HS74] are covered. Such result is a comparison theorem for the Hessian of the distance function (naturally defined in an n-dimensional Riemannian manifold). Because this result is a comparison of the partial divergence of a radial vector field, we define,

Definition 2.2.1. Let (M n, g) be an n-dimensional Riemannian manifold, with Levi-Civita connection ∇, let T ∈ Gr(k, TξM) and {e1, . . . , ek} an orthonormal basis for T . The partial divergence of a vector field X over T is defined as

n X divT X(x) = h∇ei X(x), eiig . i=1

Readily follows that, in the case in which T = TξM for some ξ ∈ M, this partial divergence coincide with the classical deﬁnition of the divergence of a vector ﬁeld.

Lemma 2.2.1. Let (M n, g) be a complete n-dimensional Riemannian manifold, with Levi-Civita connection ∇, let b ∈ R such that Secg ≤ b, let 0 < r0 < inj(M, g) such that br0 < π and let ξ ∈ M. 2.2 RIEMANNIAN GEOMETRY 15

Then, setting u(x) := rξ(x) := dist(M,g)(x, ξ),

divT (u∇u)(x) ≥ ku(x) cotb(u(x)), for all x ∈ Bg(ξ, r0) := x ∈ M : dist(M,g)(x, ξ) ≤ r0 , and for all T ∈ Gr(k, TξM). Remark 2.2.1. First to all, recall that, given a Riemannian manifold (M, g), it is possible to this deﬁne a distance function induced by the metric g as follows:

dist(M,g)(p, q) = inf {`g(γ): γ :[a, b] → M smooth curve with γ(a) = p and γ(b) = q} , where `g(γ) denotes the length of the curve γ, which in a coordinate patch U ⊂ M is deﬁned as

Z b Z b q Z b q `g(γ) = |γ˙ (t)|gdt = hγ˙ (t), γ˙ (t)igdt = gij(γ(t))γ ˙i(t)γ ˙j(t)dt. a a a

In general such distance is not well deﬁned, but, if we consider such distance function on Bg(ξ, r0) is well deﬁned since the exponential map expξ : {v ∈ TξM : |v|g < r0} → M is a difeomorphism.

On the other hand, the result involves the quantity cotb(s) which is deﬁned as:

csb(s) cotb(s) = , snb(s) where snb(s) is the unique solution to  x00(s) + bx(s) = 0  x(0) = 0 (2.2) x0(0) = 1,

0 and csb(s) = snb(s). By the classical theory of second order ODE, to obtain explicit solutions to (2.2), by analyze the characteristic polynomial associated to it, namely

λ2 + b = 0.

Then, we have three cases Case 1: Assume b = 0, then

x00(s) = 0, then, the solution is x(s) = s, and s cotb(s) = 1.

Case 2: Assume b > 0, then √ λ = ± bi, 16 PRELIMINARIES 2.2 then the solution to (2.2) is √ sin bt x(s) = √ , b so, √ √ cos bs s cotb(s) = s b · √ . sin bs

And an asymptotic analysis when s → 0 gives

√ bs2 4 ! 1 − 2! + O(s ) s cotb(s) ∼ bs √ √ b bs3 5 bs − 3! + O(s ) 2 1 − bs + O(s4) ∼ 2! bs2 4 1 − 3! + O(s ) bs2 bs2 ∼ 1 − + O(s4) 1 − + O(s4) 2! 3! bs2 bs2 ∼ 1 + − + O(s4) 3! 2!

b b ∼ 1 + − s2 + O(S4) 6 2 s2 ∼ 1 − + O(s4) 3 ∼ 1 − O(s2)

Case 3: Assume b < 0, then

λ = ±p|b|, then the solution to (2.2) is

√ √ p e |b|s − e− |b|s sinh |b|s x(s) = = , 2p|b| p|b| so, cosh p|b|s p s cotb(s) = |b|s · . sinh p|b|s 2.2 RIEMANNIAN GEOMETRY 17

As in the previous case, an asymptotic analysis when s → 0 gives

√ bs2 4 ! 1 + 2! + O(s ) s cotb(s) ∼ bs √ √ b bs3 5 bs + 3! + O(s ) 2 1 + bs + O(s4) ∼ 2! bs2 4 1 + 3! + O(s ) bs2 bs2 ∼ 1 + + O(s4) 1 − + O(s4) 2! 3! bs2 bs2 ∼ 1 − + + O(s4) 3! 2! b b ∼ 1 + − s2 + O(S4) 2 6 s2 ∼ 1 + + O(s4) 3 ∼ 1 + O(s2).

The asymptotic analysis above will play a key role later in the Section4.

In order to prove the Lemma 2.2.1, we will use the well known Rauch Theorem and a Hessian comparison theorem derived from it.

2.2.1 Lie Derivatives and the Metric This section is inspired in the Chapter 2 of [Pet06], so the reader who is not familiarized with this, is encouraged to seek for the proofs there. Let X be a vector field and F t the flow associated to X on a smoothmanifold M (which locally always exists). Hence, F t(p) is well defined for t small and so, the curve t 7→ F t(p) is the integral curve for X that passes through p at t = 0. The Lie derivative of a tensor T in the direction of X is defined as the first-order term in a suitable expansion of T when moved by the flow of X. However, the precise formula depends on what kind of tensor is T .

Definition 2.2.2. Let (M n, g) an n-dimensional Riemannian manifold with Levi-Civita connection ∇, and let T a (0,k)-tensor. Define the Lie derivative of T along the vector field X as:

F t∗ T − T (∇X T )(Y1,...,Yk) = lim , t→0 t

t∗ t t t where F T (Y1,...,Yk) = T (DF (Y1),...,DF (Yk)) is the pull-back of T by F . The following proposition shows that the Lie derivative is in fact a derivation

Proposition 2.2.1. Let (M n, g) an n-dimensional Riemannian manifold with Levi-Civita connection ∇.

(i) If X is a vector ﬁeld and T a (0, k) -tensor on M, then

k X (∇X T )(Y1,...,Yk) = ∇X (T (Y1,...,Yk)) − T (Y1,..., ∇X Yi,...,Yk) . i=1

(ii) If T1 and T2 be (0, ki) -tensors, then

∇X (T1 · T2) = (∇X T1) · T2 + T1 · (∇X T2) 18 PRELIMINARIES 2.2

(iii) If T is a (0, k)-tensor and f : M → R a function, then

k X ∇fX T (Y1,...,Yk) = f∇X T (Y1,...,Yk) + (∇Yi f) T (Y1,...,X,...,Yk) . i=1

The Lie derivative of a tensor gives us the notion of the Hessian of a function f : M → R on a Riemannian manifold as a (0,2) -tensor:

Definition 2.2.3. Let (M n, g) an n-dimensional Riemannian manifold, with Levi-Civita connection ∇. The Hessian of a function f : M → R is defined as the (0, 2)-tensor Hess f such that: 1 Hess f(X,Y ) = (∇ g)(X,Y ), 2 ∇f where ∇f := grad(f) is the vector field satisfying hv, ∇fig = df(v) for all v ∈ TM.

The special case in which f ≡ rξ will be studied in the next section trough the machinery of the Jacobi ﬁelds. On the other hand we can analyse right now the Hessian of the square of the distance function in terms of the Hessian of the distance, this calculations will enter at the scene in the proof of the Lemma 2.2.1. Since the Lie derivative of a tensor is in fact a derivation, we have, in view of the Proposition 2.2.1, that, given X and Y vector ﬁelds on M

1 2 1 Hess rξ (X,Y ) = ∇∇ 1 r2 g (X,Y ) 2 2 2 ξ 1 = ∇ g (X,Y ) 2 rξ∇rξ 1 = r ∇ g(X,Y ) + ∇ r g(∇r ,Y ) + ∇ r g(X, ∇r ) 2 ξ ∇rξ X ξ ξ Y ξ ξ 1 = r Hess r (X,Y ) + (∇ r g(∇r ,Y ) + ∇ r g(X, ∇r )) . ξ ξ 2 X ξ ξ Y ξ ξ

Therefore, if {E1,...,Ek} ⊂ TM is an orthonormal frame, letting Ei(ξ) := ei ∈ TξM for all i = 1, . . . n, then {e1, . . . , ek} is an orthonormal basis of some T ∈ Gr(k, TξM) and 1 Hess r2(e , e ) = r Hess r (e , e ) + ∇ r h∇r , e i = r Hess r (e , e ) + h∇r , e i2 . (2.3) 2 ξ i i ξ ξ i i ei ξ ξ i g ξ ξ i i ξ i g 2.2.2 Jacobi Fields and Rauch Comparison Theorem Definition 2.2.4. Let γ : I → M be geodesic. A vector field X along γ is called a Jacobi field if

∇ dγ ∇ dγ X + R(X, γ˙ )γ ˙ = 0. dt dt As an abbreviation, we shall sometimes write

X˙ = ∇ dγ X, X¨ = ∇ dγ ∇ dγ X, dt dt dt then the above expression becomes X¨ + R(X, γ˙ )γ ˙ = 0, where R stands for the curvature tensor.

The following lemma shows the existence and uniqueness of Jacobi ﬁelds with given initial data. For this, as in the case of geodesics, the Jacobi equation will be interpreted as a system of (n = dim M) linear second order ODEs.

Lemma 2.2.2. Let γ :[a, b] → M be a geodesic. For any v, w ∈ Tγ(a)M, there exists a unique 2.2 RIEMANNIAN GEOMETRY 19

Jacobi field X along γ with X(p) = v, X˙ (p) = w. Since the existence and uniques of Jacobi fields are ensure for given initial values of it along a geodesic, a natural question is how to construct a Jacobi field along a given geodesic. the next theorem shows how to construct Jacobi fields.

Theorem 2.2.3. Let γ : [0,T ] → M be geodesic, deﬁned up to a time T > 0. Let c(t, s) be a variation of γ(t), i.e. γ(·, ·) : [0,T ]×] − ε, ε[→ M, for which all curves γ(·, s) =: γs(·) are geodesics, too. Then,

∂ X(t) := c(t, s) ∂s s=0 is a Jacobi ﬁeld along γ(t) = γ0(t). Conversely, every Jacobi ﬁeld along γ(t) may be obtained in this way, i.e. by a variation of γ(t) through geodesics.

Let γ(t, s) = γs(t) be a family of geodesics parametrized by s, i.e. ∂γ ∇ ∂γ (t, s) = 0 for all s ∂t ∂t Then also ∂γ ∇ ∂∂ ∇ ∂ (t, s) = 0, ∂s ∂t ∂t ∂c and this implies that X(t) = ∂s (t, s) s=0 satisﬁes the Jacobi equation. Consequently, the Jacobi equation is the linearization of the equation for geodesic curves. Thus, If a particular proper variation of a geodesic through geodesics is given, then also the 2nd derivative of the length and energy functionals with respect to the family parameter vanish.

Corollary 2.2.3.1. Let γ : [0,T ] → M be a geodesic, deﬁned up a given time T > 0 and p = γ(0), i.e. γ(t) = expp tγ˙ (0).

For w ∈ TpM, the Jacobi ﬁeld X along γ with X(0) = 0, X˙ (0) = w then is given by:

X(t) = D exp [tw], p tγ˙ (0) where

(D exp ) : T (T M) → T M p tγ˙ (0) tγ˙ (0) p expp(tγ˙ (0))

tv 7→ (D expp)tγ˙ (0)[tv].

Now, let (M n, g) be an n-dimensional Riemannian manifold with curvature K satisfying, for ± b ∈ R b− ≤ K ≤ b+ and assume b− ≤ 0, b+ ≥ 0. We must estimate a Jacobi field in M by the Jacobi field in Hn(−b−) (The space form with negative curvature) from above, with initial values of same lengths, and from below by the corresponding one in Sn(b+). Thus the distance between geodesics and also the derivative of the exponential map of M can be controlled by the geometry of the model spaces Hn(−b−) and Sn(b+). Instead of consider Jacobi fields with vanishing tangential component we shall assume b− ≤ 0 and b+ ≥ 0. 20 PRELIMINARIES 2.2

As in Remark 2.2.1, we put for b ∈ R  √ cos( bt) if b > 0  cn (t) := 1 if b = 0 b √ cosh( −bt) if b < 0, and √  1  √ sin( bt) if b > 0  b snb(t) := t if b = 0  √  √1 sinh( −bt) if b < 0  −b As seen in the remark 2.2.1, these functions are solutions of a second order ODE, which coincides with the Jacobi equation for constant sectional curvature b, namely

f¨(t) + bf(t) = 0 with initial values f(0) = 1, f˙(0) = 0,respectively f(0) = 0, f˙(0) = 1. Henceforth γ(t) will always be a geodesic on M parametrized by arc length, i.e. satisfying

|γ˙ |g ≡ 1.

Let J(t) be a Jacobi ﬁeld along γ(t).

Theorem 2.2.4 (Rauch Comparison Theorem I ). Suppose K ≤ b, and γ be parametrized by arc length. Assume either b ≥ 0 or that the tangential component of J is identically zero along γ. ˙ Let fµ := |J(0)|gcµ + |J|g(0)sµ solve f¨+ µf = 0 ˙ · with f(0) = |J(0)|g, and f(0) = |J|g(0),. If fµ(t) > 0 for 0 < t < τ then

hJ, J˙igfµ ≥ hJ, Jigf˙µ on [0, τ],

|J (t1)|g |J (t2)|g 1 ≤ ≤ , if 0 < t1 ≤ t2 < τ, (2.4) fµ (t1) fµ (t2) · |J(0)|gcnb(t) + |J|g(0)snb(t) ≤ |J(t)|g for 0 ≤ t ≤ τ.

The proof of this result can be found in [Jos17] Theorem 4.5.1, from where this version was taken, as well as the next result, which is the case in which the sectional curvature is bounded from bellow (c.f. [Jos17] Theorem 4.5.2)

Theorem 2.2.5 (Rauch Comparison Theorem II). Assume b− ≤ K ≤ b+ and either b− ≤ 0 or that the tangential component of J is identically zero along γ; also, as always, γ is paramatrized by arc length. Moreover, let J(0) and J˙(0) be linearly dependent. Assume

sn 1 − + > 0 on ]0, τ[. 2 (b +b ) Then, for 0 ≤ t ≤ τ · |J(t)|g ≤ |J(0)|gcnb− (t) + |J|g(0)snb+ (t). (2.5) 2.2.3 Geometric Applications of Jacobi Fields Estimates Finally, it is time to put all together, in order to prove the Lemma 2.2.1, before that, we start by proving an estimate on the Hessian of the distance function rξ as a consequence of the Rauch 2.2 RIEMANNIAN GEOMETRY 21 comparison theorems, and for clarifying purposes we will prove it, following the proof of the Theorem 4.6.1 of [Jos17], where the reader will ﬁnd this comming theorem.

Theorem 2.2.6. Let the exponential map exp p : TpM → M be a diﬀeomorphism on {v ∈ TpM : |v|g ≤ ρ}. Let the curvature of M in the ball

Bg(ξ, ρ) := {q ∈ M : dg(ξ, q) ≤ ρ} satisfying b− ≤ K ≤ b+, with b− ≤ 0, b+ ≥ 0, and suppose π ρ < √ in case b+ > 0. 2 b+

1 2 Let rξ(x) := dg(x, ξ). Then 2 rξ is smooth on Bg(ξ, ρ) and satisﬁes

1 ∇ r2 (x) = − exp−1(ξ) (2.6) 2 ξ x and therefore 1 2 ∇ rξ (x) = rξ(x). 2 g (2.7) 2 1 2 r (x) cot + (r (x))|v| ≤ Hess r (v, v) ≤ r(x) cot − (r(x))|v| , ξ b ξ g 2 ξ (−b ) g for all x ∈ Bg(ξ, ρ), v ∈ TxM and cotb± be deﬁned as in the remark 2.2.1

1 2 1 2 Proof. Since ∇ 2 rξ (x) is, by deﬁnition, orthogonal to the level surfaces of 2 rξ , and those are the spheres Sg(ξ, R) := {q ∈ M : dg(ξ, q) = R} = expξ {v ∈ TξM : kvk = R} , where R < ρ, we have 1 ∇ r2 (x) = − exp−1 ξ, 2 ξ x

1 2 and, in particular, ∇ 2 rξ has length dg(x, ξ), proving the ﬁrst assertion (2.6).

1 2 On the other hand, since Hess 2 rξ is symmetric, it can hence be diagonalized, without loss 1 2 of generality, it thus suﬃces to prove (2.7) for each eigen direction v of Hess 2 rξ . Let η(s) be the curve in M with η(0) = x, η0(0) = v

−1 γ(t, s) := expη(s) t expη(s) ξ in particular γ(0, s) = η(s), η(1, s) ≡ ξ. Then, 1 2 ∂ ∇ rξ (γ(s)) = − γ(t, s) , 2 ∂t l=0 22 PRELIMINARIES 2.2 hence 1 2 ∂ ∂ ∇v ∇ rξ (x) = − ∇ γ(t, s) 2 ∂s ∂t t=0,s=0

∂ ∂ = − ∇ γ(t, s) , ∂t ∂s t=0,s=0 therefore, ∂ J(t) := γ(t, s) , ∂s |s=0 is a Jacobi ﬁeld along the geodesic, parametrized by arc length, from x to ξ with J(0) =γ ˙ (0) = v, J(1) = 0 ∈ TpM, thus 1 ∇ ∇ r2 (x) = −J˙(0) v 2 ξ i.e. 1 2 1 2 ˙ Hess rξ (v, v) = ∇v ∇ rξ , v = −hJ(0),J(0)ig (2.8) 2 2 g 1 2 1 2 ˙ since v is an eigen direction of Hess 2 rξ , ∇v 2 rξ and v, i.e. J(0) and J(0) are linearly dependent. Finally (2.4) and (2.5) imply for t = 1(J(1) = 0) (notice that we are assuming that the geodesic which J is been deﬁed along to, is parametrized by arc length, this is not always true, see the remark below)

· · |v|gcnb+ (r(x)) + |J|g(0)snb+ (r(x)) ≤ 0 ≤ |v|gcnb− (r(x)) + |J|g(0)sn(−b−)(r(x)), thus,

· |v|g cotb+ (r(x)) + |J|g(0) ≤ 0, and · (2.9) |v|g cot(−b−)(r(x)) + |J|g(0) ≥ 0, since, D E D E J˙(0),J(0) J˙(0),J(0) d q · g g |J|g(0) = hJ(t),J(t)ig = = , dt t=0 |J(0)|g rξ(x)|v|g then, from (2.8) and (2.9), 2 1 2 r (x) cot + (r (x))|v| ≤ Hess r (v, v) ≤ r(x) cot − (r(x))|v| , ξ b ξ g 2 ξ (−b ) g which completes the proof.

Remark 2.2.2. If we do not assume |η˙|g ≡ 1, (where η is the geodesic from x to ξ) in all the preceding estimates, t has to be replaced by t|η˙|g as argument of snb± , cnb± , fb± etc. Namely, let

t η˜(t) = η |η˙|g

˜ t be the reparametrization of η by arc length. Then J(t) = J |η˙| is the Jacobi ﬁeld along η˜ with

J˙(0) J˜(0) = J(0), J˜(0) = ; |η˙|g 2.2 RIEMANNIAN GEOMETRY 23 since J satisﬁes the Jacobi equation, J˜ satisﬁes

J˜¨ + R(J,˜ η˜˙)η˜˙ = 0.

Thus, estimates for J˜ yield corresponding estimates for J.

Finally, as promised, we will prove the Lemma 2.2.1, as a corollary of the theorem above, and the calculation in (2.3).

Proof of Lemma 2.2.1: First to all, notice that, since r0 < inj(M,g)(ξ), we know that the exponential map: expξ : {v : |v| ≤ r0} ⊂ TξM → Bg(ξ, r0) ⊂ M, is a diﬀeomorphism, and since by hypothesis K := Secg ≤ b for some b ∈ R on Bg(ξ, r0) we are in the hypothesis of the Theorem 2.2.6.

Now, by (2.3) we know that, given T ∈ Gr(k, TξM) and {e1, . . . , ek} an orthonormal basis for it,

k X divT (u∇u) = h∇ei (u∇u) , eiig i=1 k X = h∇ei u · ∇u, eiig + hu∇ei (∇u) , eiig i=1 n k X X 2 = u h∇ei (∇u) , eiig + h∇u, eiig k=1 i=1 k X 2 = u divT ∇u + h∇u, eiig i=1

k k X 1 X = u ∇ g(e , e ) + h∇u, e i2 2 ∇u i i i g i=1 i=1 k X 2 = u Hess u(ei, ei) + h∇u, eiig i=1 k X 1 = Hess r2(e , e ), 2 ξ i i i=1 and, by the Theorem 2.2.6 (using the only left side of (2.7), and setting b+ = b), we have that

1 Hess r2(e , e ) (x) ≥ r (x) cot (r (x)), 2 ξ i i ξ b ξ

Therefore divT (u∇u) ≥ ku(x) cotb(u(x)), for all x ∈ Bg(ξ, r0), and for all T ∈ Gr(k, TξM). Remark 2.2.3. Notice that we only use the estimate from above of the Theorem 2.2.6, which will be enough for our purposes, but if we instead use the estimate from below. To do so, we must assume − + − + that in Lemma 2.2.1, that b ≤ Segg ≤ b , where b ≤ 0 and b ≥ 0 and sn 1 − + > 0 on η, the 2 (b +b ) geodesic along J is deﬁned in the proof of Theorem 2.2.6, and we will get, that for all x ∈ Bb(ξ, r0) 24 PRELIMINARIES 2.2

and T ∈ Gr(k, TξM)

ku(x) cot(b+)(u(x)) ≤ divT (u∇u) ≤ ku(x) cot(−b−)(u(x)). (2.10) Chapter 3

Intrinsic Riemannian Theory of General Varifolds

3.1 General Varifolds Now we introduce the notations and concepts relative to varifolds that we need to make a Riemannian intrinsic theory of varifolds. In this respect we closely follow [All72], [Sim83], [Lel12], [NOA18]. In what follows V will always denote a varifold and dvg the Riemannian measure of (M n, g).

n Deﬁnition 3.1.1. For any k, n ∈ N, n ≥ 2, 1 ≤ k ≤ n − 1, let M a n-dimensional manifold. We say that V is a k-dimensional varifold in M, if V is a nonnegative, real extended valued, Radon n measure on Gk(M ) the Grassmannian manifold whose underlying set is the union of the sets n Gr(k, TxM), where Gr(k, TxM) denotes the set of k-dimensional subspaces of TxM , as x varies on n n M (compare with section 2.6 of [All72]). For every k ∈ {1, ..., n − 1}, we deﬁne Vk(M ) to be the n 0 n space of all k-dimensional varifolds on M endowed with the weak topology induced by Cc (Gk(M )), n say the space of continuous compactly supported functions on Gk(M ) endowed with the compact open topology.

n n Deﬁnition 3.1.2. Let V ∈ Vk(M ), g is a Riemannian metric on M , we say that the nonnegative n Radon measure on M , ||V || is the weight of V , if ||V || = π#(V ), here π indicates the natural ﬁber n n n n bundle projection π : Gk(M ) → M , π :(x, S) 7→ x, for every (x, S) ∈ Gk(M ), x ∈ M , S ∈ Gr(k, TxM), ||V ||(A) := V (π−1(A)).

n π n Remark 3.1.1. Recall that π is a proper map because the ﬁbers of the ﬁber bundle Gk(M ) → M are compact.

As the reader has noticed, an abstract varifold can be a quite strange object, because it is hard n to work with Borel sets on Gk(M ) in an operative way (here operative way has to be understate in a sense to be specified later in this section); but in the sequel, we also define the weight of V which is a Radon measure on Σ obtained from V by ignoring the fiber variable. The next theorem illustrates how to "simplify" a varifold in an operative way. This result is a direct application of a well known disintegration Theorem, which can be found in [AFP00] Theorem 2.28. However the following is an adaptation of it into the context of our intrinsic varifolds.

Theorem 3.1.1 (Disintegration Theorem for Varifolds). Let (M n, g) a n-dimensional Riemannian n n n manifold, let V ∈ Vk(M ), and π : Gk(M ) → M be the canonical projection onto M . Then there exists a family of Radon measures {πx}x∈M n such that, the map x 7→ πx is ||V ||-measurable and the

25 26 INTRINSIC RIEMANNIAN THEORY OF GENERAL VARIFOLDS 3.2 following relations are satisﬁed

V (Bg(x, r) × B) n πx(B) := lim , for all B ∈ B(Gr(k, TxM )), (3.1) r↓0 ||V ||(B(x, r)) n n n πx (Gr(k, TxM ) \{S : S ⊂ T an(M , x)}) = 0, and πx(Gr(k, TxM )) = 1, (3.2) 1 n n f(x, ·) ∈ L (Gr(k, TxM ), πx), for ||V || − a.e. x ∈ M , (3.3) Z 1 n x 7→ f(x, S)dπx(S) ∈ L (M , ||V ||), is ||V || − measurable, (3.4) n Gr(k,TxM ) Z Z Z ! f(x, S)dV (x, T ) = f(x, S)dπx(S) d||V ||(x), (3.5) n n n Gk(M ) M Gr(k,TxM )

1 n 0 for any f ∈ L (Gk(M ),V ). Moreover, if πx is any other ||V ||-measurable map satisfying (3.4) and 0 n (3.5) for every bounded Borel function with compact support and such that x 7→ πx(Gr(k, TxM )) ∈ 1 n 0 n L (M , ||V ||), then πx = πx for ||V ||-a.e. x ∈ M . n Corollary 3.1.1.1. Let V ∈ Vk(M ), with the same notation of the Theorem 3.1.1, the equality

V = ||V || ⊗ πx, holds. 3.2 The First Variation of a Varifold According to [All72], we associate to each varifold V a vector valued distribution, which depends on the metric g of the ambient space, called the first variation of V . If this first variation is (as Radon measure measure) absolute continuous with respect to weight measure kV k in analogy with the smooth case we define the generalized mean curvature vector Hg. Definition 3.2.1. Let (M n, g) a n-dimensional Riemannian manifold with Levi-Civita connection 1 ∇, Xc (M) the set of differentiable vector fields on M and V ∈ Vk(M) a k-dimensional varifold 1 (k ≤ n). We define the first variation of V along the vector field X ∈ Xc (M) as Z δV (X) := h∇X(x) ◦ pT , pT ig dV (x, T ), Gk(M) where the inner product in the integrand is the one defined in Hom (TxM,TxM), and ∇X : X(M) → X(M) such that ∇X(Y ) := ∇Y X.

Let {τ1, . . . , τk} an orthonormal basis of T ∈ Gr(k, TxM) for x given, and {τ1, . . . , τk, τk+1, . . . τn} 1 the completion to an orthonormal basis for TxM, then for given X ∈ Xc (M), Z ∗ δV (X) = trace ((∇X(x) ◦ pT ) ◦ pT ) dV (x, T ) Gk(M) k Z X = hτi, ∇τi X(x)ig dV (x, T ), Gk(M) i=1

1 so, we have that, for all X ∈ Xc (M), the deﬁnition (3.2.1) is equivalent to say, Z δV (X) = divT X(x)dV (x, T ), (3.6) Gk(M)

where, for given x, and {τ1, . . . , τk} an orthonormal basis of a ﬁxed T ∈ Gr(k, TxM),

k X divT X(x) = hτi, ∇τi X(x)ig . i=1 3.2 THE FIRST VARIATION OF A VARIFOLD 27

Then (3.6) give us formula with more geometric meaning than the merely definition, on the other hand, by analogy with the smooth case, we desire to related in some way the first variation of a varifold with a "mean curvature" vector, to do so, let us first analyze the total variation of the first variation.

Deﬁnition 3.2.2. Let V ∈ Vk(M) with (M, g) a n-dimensional Riemannian manifold, we say that V has locally bounded ﬁrst variation in U ⊂ M open set, if for all W ⊂⊂ U open set, there exists a constant C := C(W ) such that

|δV (X)| ≤ CkXkL∞(W,V ),

0 for all X ∈ Xc (W ). 0 Remark 3.2.1. Notice that the definition of bounded first variation is valid for each X ∈ Xc (M) 1 0 while the first variation, δV , is defined on Xc (M). However, defining an extension δVf : Xc (M) → R of δV as δVf (X) := lim δV (Xε), ε↓0 1 n 0 1 1 where (X ,...,X ) = X ∈ Xc (M), and Xε ∈ Xc (M) is a "C approximation" of X defined as follows: Let {(Φi,Ui)}i∈Λ an atlas of M, since we are interested in compactly vector fields we can choose N ` Λ = {1,...,N} such that {Ui}i=1 is an open covering for spt X, and consider {ψ}i=1 a partition N of unity subordinate to {Ui}i=1, then

`  N  X X i ε ∂ Xε(p) :=  ψj X ? ϕ (p) , Ui ∂x i=1 j=1 i

ε n where ϕ is an standard approximation of the identity in R , and ? denotes the usual convolution. Clearly Xε is independent of the choice of charts and deﬁned on spt X. Furthermore by standard theory of convolutions, we know that,

i ε i ψj X ? ϕ → ψj X Ui Uj uniformly in compacts, when ε → 0. Hence

kX − XεkL∞ → 0, when ε → 0,

1 and Xε ∈ Xc (M). 1 2 1 1 1 On the other hand, given Xε ,Xε ∈ Xc (M) as above such that Xε → X and Xε → X, assume that V has bounded ﬁrst variation on M, then

1 2 1 1 δV (Xε − Xε ) ≤ CkXε − Xε kL∞ → 0 when ε → 0, which implies that 0 δVf : Xc (M) → R 1 is well defined and δVf ≡ δV on Xc (M). By abuse os notation we will identify δVf with δV . n n Proposition 3.2.1. Let V ∈ Vk(M ) with locally bounded first variation in (M , g). Then the total variation kδV k is a Radon measure. Furthermore, there exist a kV k-measurable function Hg : M → TM, and Z ⊂ M with kV k(Z) = 0, such that Z Z δV (X) = − hX,Hgig dkV k + hX, νig dkδV ksing, M n M n 28 INTRINSIC RIEMANNIAN THEORY OF GENERAL VARIFOLDS 3.2 where ν is a kδV k-measurable function with |ν(x)|g = 1, and kδV ksing = kδV kxZ. We call Hg the generalized mean curvature vector of V . Proof. Since by hypothesis V has locally bounded first variation, for all open set W ⊂⊂ M

0 kδV k(W ) = sup{ |δV (X)| : X ∈ Xc (W ), |X| ≤ 1} ≤ C < ∞.

0 Then by the Riesz Representation Theorem for Xc (M) (where we are identifying δVf with δV ) there 0 exists ν : M → TM kδV k-measurable function with |ν(x)|g = 1 kδV k-a.e. and for all X ∈ Xc (M) Z δV (X) = hX, νig dkδV k, M and kδV k is a Radon measure. This last assertion prove the first part of the theorem as in the Euclidean case. To prove the second part of the theorem we need to make use of the fact that the Vitali symmetry property holds for every closed metric balls of (M n, g). This guarantees that we can apply the Lebesgue Differentiation Theorem to space Bg(ξ, injM n,g(ξ)) because kV k is a regular enough Radon measure (see Remark 2.1.1). Thus, by the Lebesgue Differentiation Theorem we get

kδV k (Bg(x, ρ)) DkV kkδV k = lim ρ↓0 kV k (Bg(x, ρ)) exists kV k-a.e and (writing −Hg(x) := DkV kkδV k(x)ν(x)) we are lead to the following formula Z Z δV (X) = − hX(x),Hg(x)ig dkV k(x) + hX(x), ν(x)ig dkδV ksing, M where kδV ksing = kδV kx x : DkV kkδV k = +∞ . n Remark 3.2.2. Let (M , g) be an n-dimensional Riemannian manifold, let V ∈ Vk(M) such that for some C > 0, p > k, 0 < ρ < injξ(M, g) for ﬁxed ξ ∈ M

q−1 ! q Z q δV (X) ≤ C |X| q−1 dkV k . (3.7) Bg(ξ,ρ)

Then, by Hölder inequality we have that

∞ |δV (X)| ≤ C1kXkL (kV k,spt X∩Bg(ξ,ρ)),

q−1 where C1(ξ, ρ) := C (kV k (spt X ∩ Bg(ξ, ρ))) q . Then V has locally bounded ﬁrst variation, furthermore, kδV k is a Radon measure on M. Roughly speaking, if we assume that spt X ⊂ A, for A a given kV k-negligible set, and take the supremum over all such X’s with kXkL∞(kV k) = 1 then by (3.7) we get that q−1 kδV k(A) ≤ C (kV k(A)) q = 0, which implies that kδV k kV k. In fact, technically to prove the preceding equation we need to be more carefull, i.e., we need to take a sequence of open sets A ⊂ Uj ⊆ Bg(ξ, ρ), ∀j such that Uj → A, then using (3.7) we have that

q−1 ! q Z q q−1 1 δV (X) ≤ C |X| dkV k , ∀X ∈ Xc (Uj). Uj

Then taking limits, it easy to check that

q−1 kδV k(A) = lim kδV k(Uj) ≤ lim C (kV k(Uj)) q = 0. j→∞ j→∞ 3.2 THE FIRST VARIATION OF A VARIFOLD 29

1 n Hence, the Radon-Nikodym Theorem gives the existence of a L (M , kV k) function Hg such that Z Z δV (X) = divSX(y)dV (y, T ) = − hHg(y),X(y)ig dkV k(y). (3.8) Gk(M) M

q On the other hand, (3.7) means that δV (X) is a bounded linear functional on Lloc(Γ(TM), kV k) q of the Lloc vector ﬁelds on M with respect to kV k. So by well known theorems of measure theory (compare Theorem 6.16 of [Rud87]) stating that the topological dual of Lq is Lp, i.e., (Lq)∗ = Lp we have that there exists a function f ∈ Lp such that δV (X) = R hf, Xi and the best constant Bg(ξ,ρ) g C > 0 in (3.7) is given by ||f||Lp . Combining (3.7) and (3.8) we get that f = Hg and so that p 1 Hg ∈ Lloc(Γ(TM), kV k). Finally, notice that, for all X ∈ Xc (M) (again using the Hölder inequality)

δV (X) ≤ kHgkLp(kV k,M)kXk q . L q−1 (kV k,M)

At this stage we have proved that (3.7) implies that V has locally bounded first variation, no p singular part and the generalized mean curvature vector field Hg, satisfies an L condition. Definition 3.2.3 (Allard’s Conditions). Given V a k-dimensional varifold, we say that V satisfies an Allard’s type condition for the generalized mean curvature, if V satisfies (3.7). From now on, denoted as (AC).

This kind of conditions will play a key role in the section (4.1) when we study the monotonicity behaviour of the density ratio. 30 INTRINSIC RIEMANNIAN THEORY OF GENERAL VARIFOLDS 3.2 Chapter 4

Monotonicity and Consequences

The aim of this section is to obtain information about V from their ﬁrst variation δV (X), for 1 an appropriate choice of X ∈ X0(U). This kind of result is known as "Monotonicity Formula", and is the step zero in any interior regularity theory.

4.1 Weighted Monotonicity Formulae for Abstrac Varifolds

Let (M, g) a Riemannian manifold with Levi-Civita connection ∇ and V ∈ Vk(M) satisfying (AC), ﬁx ξ ∈ M and assume injξ(M, g) > 0. Choose r0 > 0 such that r0 < injξ(M, g), and, for 1 ﬁxed ε > 0 let γε ∈ Cc (] − ∞, 1[), such that ( 1 if y ≤ ε, 0 γε(y) := and γ (y) < 0 if ε < y < 1, 0 if y > 1, ε

Then we can consider the radial perturbation

u(x) X˜ (x) = γ (u∇u) (x), for 0 < |s| < r , s,ε ε s 0

T T where u(x) = rξ(x) = dist(M,g)(x, ξ). Now, let y ∈ M given, and T ∈ Gr(k, TyM). Let {e1 , . . . , ek } an orthonormal basis for S, then

k ˜ X D T u E divT Xs,ε = ei , ∇eT γ (u∇u) i s g i=1 k k 0 u u X T 2 u X D T E = γε ei , ∇u + γ ei , ∇eT (u∇u) s s g s i g i=1 i=1 2 u 0 u u > = γ divT (u∇u) + γ ∇ u , s s s g where ∇T u is the orthogonal projection of ∇u onto S. Let ∇T ⊥ u denote the orthogonal projection of ∇u onto T ⊥, then, T 2 T ⊥ 2 2 |∇ u|g + |∇ u|g = |∇u|g = 1, since u is a distance function. Therefore

⊥ 2 ˜ u 0 u u 0 u u S divT Xs,ε = γ divT (u∇u) + γ − γ ∇ u . s s s s s g Although this choice is enough to get many useful information, let us consider a general case, which will be used in the sequel. Let h ∈ C1(U) a non-negative function, and consider

u(y) X (y) := h(y)X˜ (y) = h(y) γ (u∇u) (y), for 0 < |s| < r . (4.1) s,ε s ε s 0

31 32 MONOTONICITY AND CONSEQUENCES 4.1

S T Then, if {e1 , . . . , ek } is as above,

k X D T ˜ E divT Xs,ε(y) = ei , ∇eT h(y)Xs(y) i g i=1 k k X D T ˜ E X D T ˜ E (4.2) = h(y) ei , ∇eT Xs(y) + ∇eT h(y) ei , Xs(y) i g i g i=1 i=1 D T E = h(y) divT X˜s,ε + ∇ h(y), X˜s(y) . g By the deﬁnition of the ﬁrst variation, we know that

Z δV (Xs,ε) : = divT Xs,ε(y)dV (y, T ) Gk(U) Z Z D T E = h(y) divT X˜s(y)dV (y, T ) + ∇ h(y), X˜s(y) dV (y, T ). g Gk(U) Gk(U) Then, replacing (4.1) in (4.2) and the information above, we have

Z u(y) δV (Xs,ε) = h(y)γε divT (u∇u)(y)dV (y, T ) Gk(U) s Z h(y)u(y) 0 u(y) + γε dV (y, T ) G (U) s s k (4.3) Z 2 h(y)u(y) 0 u(y) T ⊥ − γε ∇ u dV (y, T ) g Gk(U) s s Z u(y) T + γε ∇ h(y), (u∇u)(y) g dV (y, T ) Gk(U) s If we compare (4.3) with the Euclidean case (see Cf. [Sim83] (4.14) and (4.24)) we note the lack of a dimensional term, then, in order to have an intrinsic result, we need to compare it in some way with the Euclidean case. To do this we use the Rauch’s comparison theorem, applied as in Lemma 3.6 of [HS74], which states:

Lemma 4.1.1. Let (M, g) be a complete Riemannian manifold, with Levi-Civita connection ∇, let b ∈ R such that Secg ≤ b, assume br0 < π. Then

divT (u∇u)(x) ≥ ku(x) cotb(u(x)), for all x ∈ Bg(ξ, r0). Remark 4.1.1. Notice that under the conditions of Lemma 4.1.1 we can compare the divergence of the radial ﬁeld u∇u in the ambient manifold M n with the same quantity calculated in a space-form, so, is natural to guess that this is the framework to get some "monotonicity" behaviour (inspired in the results from the euclidean case). Furthermore, in view of Remark 2.2.1 above, we expect to recovery in some sense the euclidean case, this is why from now we will be in the setting of the Lemma 4.1.1, i.e.

Deﬁnition 4.1.1 ( Bounded Geometry). We say that a complete Riemannian manifold (M n, g) has bounded geometry (or satisfy (BG)), if Secg < b for some constant b ∈ R and for every ξ ∈ M there is r0 such that 0 < r0 < injξ(M, g) and r0b < π. 4.1 WEIGHTED MONOTONICITY FORMULAE FOR ABSTRAC VARIFOLDS 33

In view of the Remarks 2.2.1 and 4.1.1 we can deﬁne c(s) := c(s, b) as: ( √ √ s b cot bs b > 0 c(s) = 0 b ≤ 0, where b ∈ R. Then, by the Lemma 4.1.1,

divT ((u∇u)(x)) ≥ kc(u(x), b), for all x ∈ Bg(ξ, r0), furthermore we can assume r0 cotb(r0) > 0, since we are interested in small geodesic balls around ξ. Then, substituting in (4.3) we have

Z u(y) δV (Xs,ε) ≥ kch(y)γε dV (y, T ) Gk(M) s Z u(y) T + γε ∇ h(y), (u∇u)(y) g dV (y, T ) Gk(M) s Z h(y)u(y) 0 u(y) + γε dV (y, T ) Gk(M) s s Z 2 h(y)u(y) 0 u(y) T ⊥ − γε ∇ u dV (y, T ). g Gk(M) s s

On the other hand, sincec(s) is decrrasing, c(u(x)) ≥ c(s) ≥ c(r0) whenever 0 ≤ u(x) ≤ s ≤ r0 = r0(b). Hence,

Z Z 2 u hu 0 u hu 0 u T ⊥ − hkγε + γε dV ≥ − γε ∇ u dV g Gk(M) s s s Gk(M) s s Z u T + γε ∇ h, (u∇u) g dV Gk(M) s Z u + (c(s) − 1) hkγε dV Gk(M) s

− δV (Xs,ε),

now, dividing by sk+1,

Z Z 2 h(y) ∂ u(y) S⊥ − h(y)I(s)dV (y, T ) ≥ γε ∇ u dV (y, T ) k g Gk(M) Gk(M) s ∂s s Z ! 1 u(y) T + k+1 γε ∇ h(y), (u∇u)(y) g dV (y, T ) s Gk(M) s (c(s) − 1) k Z u(y) + k h(y)γε dV (y, T ) s s Gk(M) s δV (X ) − s,ε , sk+1 34 MONOTONICITY AND CONSEQUENCES 4.1

where k u(y) u(y) u(y) I(s) : = γ + γ0 sk+1 ε s sk+2 ε s d 1 u(y) 1 d u(y) = − γ − γ0 ds sk ε s sk ds ε s d 1 u(y) = − γ . ds sk ε s

Diﬀerentiating under the sign of integral we have

! Z Z 2 d 1 u h ∂ u T ⊥ hγε dV ≥ γε ∇ u dV k k g ds s Gk(M) s Gk(M) s ∂s s Z ! 1 u T + k+1 γε ∇ h, (u∇u) g dV s Gk(M) s (c(s) − 1) k Z u + k hγε dV s s Gk(M) s δV (X ) − s,ε . sk+1 Now, by Theorem 3.1.1,

Z u(y) Z u(y) h(y)γε dV (y, T ) = h(y)γε dkV k(y) Gk(M) s M s and Z u(y) Z u(y) h(y)kγε dV (y, T ) = h(y)kγε dkV k(y). Gk(M) s M s Thus,

Z Z 2 d 1 u h ∂ u T ⊥ hγε dkV k ≥ γε ∇ u dV k k g ds s M s Gk(M) s ∂s s ! 1 Z u + γ ∇T h, (u∇u) dV sk+1 ε s g Gk(M) (4.4) (c(s) − 1) k Z u + k hγε dkV k s s Gk(M) s δV (X ) − s,ε . sk+1 Remark 4.1.2. Notice that, by the Remark 2.2.1 the behavior of c(s) := c(s, b), we have that

(c(s) − 1) = −O(s) = O(s), as s → 0+ s furthermore, this is decreasing, and

(c(s) − 1) (c(r0) − 1) ∗ ≥ := c1 s r0 Theorem 4.1.2 (Fundamental Weighted Monotonicity Inequality). Let (M n, g) be a complete Riemannian Manifold with Levi-Civita connection ∇, satisfying (BG), and r0 cotb(r0) > 0. Let n V ∈ Vk(M ) satisfying (AC), then for any 0 < s < r0. There exists a constant c = c(s, b) ∈]0, 1[ 4.1 WEIGHTED MONOTONICITY FORMULAE FOR ABSTRAC VARIFOLDS 35

such that, if we set u(x) = rξ(x) = dist(M,g)(x, ξ) we have for all 0 < s < r0

Z ! Z ! d 1 1 T k hdkV k ≥ k+1 ∇ h, (u∇u) g dV ds s Bg(ξ,s) s Gk(Bg(ξ,s)) (c(s) − 1) k Z + hdkV k s sk Bg(ξ,s) (4.5) 1 Z + k+1 hH, h(u∇u)ig dkV k s Bg(ξ,s) d + J(s), ds where 2 T ⊥ Z ∇ u g J(s) := h(y) k dV (y, T ). Gk(Bg(ξ,s)) rξ

1 Proof. Let {εj}j∈N a sequence, such that εj ↑ 1 when j → ∞ and {γεj } ⊂ C0 (] − ∞, 1[) a sequence of molliﬁers, such that γεj → χ]−∞,1[, pointwise from below. Considering Xs,εj as in (4.1) by the previous reasoning (see (4.4)) we have for every j ∈ N d 1 Z u Z h ∂ u ⊥ 2 T hγεj dkV k ≥ γεj ∇ u dV k k g ds s M s Gk(M) s ∂s s ! 1 Z u + k+1 γεj h∇h, (u∇u)ig dV s Gk(M) s (c(s) − 1) k Z u + k hγεj dkV k s s Gk(M) s δV (X ) − s,ε . sk+1

Letting j → ∞, in virtue of Lebesgue dominated convergence theorem, we have for all 0 < s < r0, Z u(y) Z lim h(y)γεj d||V ||(y) = h(y)χBg (ξ,s)(y)d||V ||(x) j→∞ M s M Z (4.6) = h(y)d||V ||(y). Bg (ξ,s) On the other hand, since γ0 u(x) < 0, for all j ∈ and for all x ∈ M such that ε ≤ u(x)/s ≤ 1, εj s N j and 0 < s < r . Then 0 ∂ u(x) 0 u(x) −u(x) γε = γ > 0. ∂s j s εj s s2 Therefore, since h ≥ 0

k k ε ∂ u(y) ⊥ 2 ε ∂ u(y) ⊥ 2 j T j T h(y) γεj ∇ u(y) ≤ h(y) γεj ∇ u(y) sk ∂s s g u(y)k ∂s s g 1 ∂ u(y) ⊥ 2 T ≤ h(y) γεj ∇ u(y) , sk ∂s s g for all n ∈ N. Integrating over Gk(M),

k k Z ε ∂ u ⊥ 2 Z ε ∂ u ⊥ 2 j T j T h γεj ∇ u dV ≤ h γεj ∇ u(x) dV k g k g Gk(M) s ∂s s Gk(M) u ∂s s Z 1 ∂ u ⊥ 2 T ≤ h γεj ∇ u dV, k g Gk(M) s ∂s s 36 MONOTONICITY AND CONSEQUENCES 4.1 and making j → ∞, we have in distributional sense Z 1 ∂ u(y) ⊥ 2 d lim h(y) γ ∇S u dV (y, T ) = J(s). (4.7) k εj j→∞ M s ∂s s g ds Finally, since V satisﬁes (AC), we have (see Remark 3.2.2)

Z u(y) δV (Xs,εj ) = − h(y)γεj hHg(y), (u∇u)(y)ig dkV k(y), M s then, taking Xs(y) = h(y)χBg(ξ,s)(y)(u∇u)(y), Z u(y) u(y) |δV (Xs,εj − Xs)| ≤ |h(y)| γεj − χ]−∞,1[ |Hg|gdkV k(y). M s s

1 By the Lebesgue Dominated Donvergence Theorem, we have γεj → χ]−1,1[ in L sense, then, letting 1 ≤ p0 < +∞ be the conjugate exponent to 1 < p ≤ +∞, by Hölder inequality we have that

γε − χ]−1,1[ p0 → 0, j L (kV k,Bg(ξ,s))

0 for all s ∈]0, r0[ and 1 ≤ p ≤ +∞. Therefore, Z u kH k|h| γ − χ dkV k ≤ khk ∞ kH k p kγ − χ k 0 . g εj ]−1,1[ L (kV k) g L εj ]−1,1[ Lp M s Hence, 1 Z lim δV (Xs,ε ) = hH, h(y)(u∇u)(y)i dkV k(y), (4.8) j→∞ j k+1 g s Bg (ξ,s) and then the result follows from (4.6), (4.7), and (4.8)

Corollary 4.1.2.1 (Weighted Monotonicity Formula). Under the hypothesis of the theorem above, if 0 < σ < ρ < r0. Then

Z Z Z ρ Z ! 1 1 1 T k hdkV k − k hdkV k ≥ − k |∇ h|dV ρ Bg(ξ,ρ) σ Bg(ξ,σ) σ s Bg(ξ,s) Z ρ Z ! 1 ∗ − k h (|Hg|g − kc1) dkV k ds σ s Bg(ξ,s) + (J(ρ) − J(σ)) , where J(s) is deﬁned as in Theorem 4.1.2.

Proof. Integrating over [σ, ρ] ⊂ [0, r0], from Theorem 4.1.2 we have,

Z ρ Z !! Z ρ Z !! d 1 1 T k hdkV k ds ≥ k+1 ∇ h, (u∇u) g dV ds σ ds s Bg(ξ,s) σ s Gk(Bg(ξ,s)) ! Z ρ (c(s) − 1) k Z + k hdkV k ds σ s s Bg(ξ,s) ! Z ρ 1 Z + k+1 hHg, h(u∇u)ig dkV k ds σ s Bg(ξ,s) Z ρ d + J(s)ds. σ ds Applying the Fundamental Theorem of Calculus to the left-hand term, we have 4.1 WEIGHTED MONOTONICITY FORMULAE FOR ABSTRAC VARIFOLDS 37

!! Z ρ d 1 Z 1 Z 1 Z k hdkV k = k hdkV k − k hdkV k. (4.9) σ ds s Bg(ξ,s) ρ Bg(ξ,ρ) σ Bg(ξ,σ) To estimate the right-hand term, we precede in four steps. For the last term, again from the Fundamental Theorem of Calculus we have,

Z ρ d J(s)ds = J(ρ) − J(ρ). (4.10) σ ds For the ﬁrst term, we have

Z ρ Z !! Z ρ Z ! 1 T 1 T k+1 ∇ h, (u∇u) g dV ds ≥ − k |∇ h|gdV ds (4.11) σ s Gk(Bg (ξ,s)) σ s Gk(Bg (ξ,s))

Now, estimating the third and fourth term, (see Remark 4.1.2) we have ! ! Z ρ 1 Z Z ρ c(s) − 1 Z k+1 hHg, h(u∇u)ig dkV k ds + k+1 hdkV k ds σ s Bg (ξ,s) σ s Bg (ξ,s) Z ρ Z ! Z ρ ∗ Z ! 1 kc1 ≥ − k |Hg|ghdkV k ds + k hdkV k ds (4.12) σ s Bg (ξ,s) σ s Bg (ξ,s) Z ρ Z ! 1 ∗ ≥ − k h(|Hg(y)|g − kc1)dkV k ds. σ s Bg (ξ,s)

Hence, the result follows from (4.9), (4.10), (4.11), and (4.12)

Remark 4.1.3. Let h : spt kV k ⊂ M → R such that, h ≡ 1, and J(s) as in Theorem 4.1.2 then, under the hypothesis of Theorem 4.1.2 as a particular case we have that there exists c ∈]0, 1[ such that, fort all 0 < s < r0

Z d kV k (Bg(ξ, s)) d (c(s) − 1) k k ≥ J(s) + k dkV k ds s ds s s B (ξ,s) g (4.13) 1 Z + k+1 hHg, h(u∇u)ig dkV k. s Bg(ξ,s)

Now, assume that V has locally bounded ﬁrst variation in Bg(ξ, 2ρ0) for some ρ0 < r0, and let Λ ≥ 0 such that kδV k(Bg(ξ, ρ)) ≤ ΛkV k(Bg(ξ, ρ)), for 0 < ρ < ρ0, notice that the hypothesis above are much stronger than (AC), and those implies that for all X ∈ 0 Xc (M) with spt X ⊂ Bg(ξ, sρ0), Z Z δV (X) = divSX(y)dV (y, T ) = − hHg(y),X(y)ig dkV k(y). Gk(M) M

Then is not so hard to see that (4.13) remains valid and |Hg(y)|g ≤ Λ for kV k-a.e. y ∈ Bg(ξ, ρ0). ∗ Then, setting c1 as in the Remark 4.1.2, 38 MONOTONICITY AND CONSEQUENCES 4.2

∗ Z Z d kV k (Bg(ξ, s)) d kc1 1 |u∇u|g k ≥ J(s) + k dkV k − k |Hg|gdkV k ds s ds s Bg(ξ,s) s Bg(ξ,s) s Z d 1 ∗ ≥ J(s) − k (|Hg(y)|g − kc1) dkV k(y) ds s Bg(ξ,s) d kV k (B (ξ, s)) ≥ J(s) − (Λ − kc∗) g . ds 1 sk Hence,

d kV k (B (ξ, s)) kV k (B (ξ, s)) d g + (Λ − kc∗) g ≥ J(s) > 0. ds sk 1 sk ds (Λ−kc∗)s Finally, multiplying by e 1 we have (Λ−kc∗)s d kV k (Bg(ξ, s)) ∗ (Λ−kc∗)s kV k (Bg(ξ, s)) e 1 + (Λ − kc ) e 1 ≥ 0, ds sk 1 sk then d (Λ−kc∗)s kV k (Bg(ξ, s)) e 1 ≥ 0. (4.14) ds sk Therfore, the function

f :]0, ρ0[ → [0, +∞[

(Λ−kc∗)s kV k (Bg(ξ, s)) s 7→ e 1 sk in non-decreasing. In the sequel we exploit the Lp condition over the generalized mean curvature H to get similar results, concerning to the monotonicity of this ratio. 4.2 Lp Monotonicity Formula In this section we prove the monotonicity behaviour of the density ratio, as in Remark 4.1.3, but instead of the hypothesis of such we assume Lp boundedness for H, i.e. under the hypothesis (AC).

First to all, from now on we assume (M, g) to be an n-dimensional complete Riemannian manifold satisfying (BG), and let ∗ ∗ c(b, r0) − 1 c = c (b, r0) = , r0 where c(b, s) is as in Remark ??.

n Notice that, if V ∈ Vk(M ) satisfy (AC), and considering the special case on the Remark 4.1.3 we have in distributional sense that, Z d kV k (Bg(ξ, s)) d 1 ∗ k ≥ J(s) + k+1 kc s + hHg, u∇uig dkV k, (4.15) ds s ds s Bg(ξ,s) where J(s) is as in Theorem 4.1.2. The equation above is the Intrinsic Riemannian version of the fundamental monotonicity identity of [Sim83], and from now on we refer to this as fundamental monotonicity inequality , also notice that, since

2 T ⊥ Z ∇ u g J(s) := k dV and kV k(Bg(ξ, s)), Gk(Bg(ξ,s)) rξ 4.2 LP MONOTONICITY FORMULA 39 are increasing in s, the inequality (4.15) holds also in the classical sense. Furthermore, from the analysis of this inequality, naturally follows a monotone behaviour of the density ratio.

Theorem 4.2.1. Given ξ ∈ M, let 0 < α ≤ 1 and Λ be a positive constant. Let V ∈ Vk(M) satisfying (AC) and for all 0 < s < r0

1 Z s α−1 |Hg(x)|gdkV k(x) ≤ Λ kV k (Bg(ξ, s)) . α Bg(ξ,s) r0 Then the function kV k (B (ξ, s)) f(s) := eλ(s) g , sk is a non-decreasing function, where ! s α−1 λ(s) := Λ − c∗k s, r0 and in fact, for 0 < σ < ρ ≤ r0, kV k (B (ξ, ρ)) kV k (B (ξ, σ)) eλ(ρ) g − eλ(σ) g ≥ J(ρ) − J(σ). (4.16) ρk σk Proof. Notice that it is enough to prove (4.16) to guarantee that f is non-decreasing. To prove (4.16), we multiply the fundamental monotonicity inequality (4.15) by eλ(s) ≥ 1, we have

λ(s) Z λ(s) d kV k (Bg(ξ, s)) λ(s) d e ∗ e k ≥ e J(s) + k+1 kc s + hHg, u∇uig dkV k ds s ds s Bg(ξ,s) λ(s) Z d e ∗ ≥ J(s) − k+1 (|Hg|g|u∇u|g − kc s) dkV k ds s Bg(ξ,s) λ(s) Z d e |u∇u|g ∗ ≥ J(s) − k |Hg|g − kc dkV k ds s Bg(ξ,s) s λ(s) Z d e ∗ ≥ J(s) − k (|Hg|g − kc ) dkV k. ds s Bg(ξ,s) On the other hand, by hypothesis

Z α−1 ! ∗ s ∗ − (|H|g − kc ) dkV k ≥ − αΛ − kc kV k(Bg(ξ, s)), Bg(ξ,s) r0 therefore α−1 ! λ(s) d kV k (Bg(ξ, s)) s ∗ λ(s) kV k(Bg(ξ, s)) d e k + αΛ − kc e k ≥ J(s). ds s r0 s ds

Finally, since d s α−1 λ(s) = αΛ − kc∗, ds r0 hence d kV k(B (ξ, s)) eλ(s) g ≥ J(s) ≥ 0, ds sk and (4.16) follows readily by integrating over the interval [σ, ρ]. 40 MONOTONICITY AND CONSEQUENCES 4.2

Theorem 4.2.2. Given ξ ∈ M, let p > k ≥ 1 and Γ be a possitive constant. Let V ∈ Vk(M) satisfying (AC) such that Z !1/p p |Hg|gdkV k ≤ Γ. Bg(ξ,r0) ∗ ∗ Then there exists a positive constant c2 = c2(p, k, b, r0) such that

1 1 (kV k(B (ξ, σ))) p (kV k(B (ξ, ρ))) p Γ + c∗ g − g ≤ 2 ρ1−k/p − σ1−k/p , (4.17) σk ρk p − k for every 0 < σ < ρ ≤ r0.

Proof. From the Hölder inequality in the fundamental monotonicity inequality, Theorem (4.15) we have for all 0 < s < r0 that

Z d kV k(Bg(ξ, s)) d 1 ∗ k ≥ J(s) + k+1 kc s + hHg, u∇uig dkV k ds s ds s Bg(ξ,s) Z d 1 ∗ ≥ J(s) − k (|Hg|g − kc ) dkV k ds s Bg(ξ,s) Z d 1 ∗ kV k(Bg(ξ, s)) ≥ J(s) − k |Hg|gdkV k + kc k ds s Bg(ξ,s) s 1 d 1 0 ≥ J(s) − (kV k(B (ξ, s))) p kH k p ds sk g g L (Bg(ξ,s)) kV k(B (ξ, s)) + kc∗ g , sk where p0 is the conjugate exponent to p. Now, by hypothesis we know that

p p kHgkL (kV k,Bg(ξ,s)) ≤ kHgkL (kV k,Bg(ξ,r0)) ≤ Γ, d and, by (4.7) we also have that ds J(s) ≥ 0 for all 0 < s < r0, then

1 0 d kV k(Bg(ξ, s)) (kV k(Bg(ξ, s))) p 1 ≥ − Γ − kc∗ (kV k(B (ξ, s)).) p ds sk sk g

∗ Now, by Remarks 4.1.2 and ??, c ≤ 0, then for all 0 < s < r0

∗ 1 ∗ 1 Γ − kc (kV k(Bg(ξ, s))) p ≤ Γ − kc (kV k(Bg(ξ, r0))) p .

Hence 1 d kV k(B (ξ, s)) (kV k(B (ξ, s))) p0 g ≥ − g (Γ + c∗) , ds sk sk 2 ∗ ∗ where c2 = c2(k, p, b, r0) is a positive constant such that

1 ∗ p ∗ −kc (kV k(Bg(ξ, r0))) ≤ c2, such constant always exists because kV k is a Radon measure.

On the other hand, since

1 − 1 d kV k(B (ξ, s)) p 1 kV k(B (ξ, s)) p0 d kV k(B (ξ, s)) g = g g , ds sk p sk ds sk 4.2 LP MONOTONICITY FORMULA 41 we have that

1 − 1 1 ! p p0 p0 ∗ d kV k(Bg(ξ, s)) 1 kV k(Bg(ξ, s)) (kV k(Bg(ξ, s))) ∗ Γ + c2 k ≥ k − k (Γ + c2) ≥ − k , ds s p s s ps p and the result follows integrating the inequality above over [σ, ρ] ⊂]0, r0[.

p k Corollary 4.2.2.1. If H ∈ Lloc(M,TM, Hg ) for some p > k, then the density

k kV k(Bg(x, ρ)) Θ (kV k, x) := lim k , ρ↓0 ωkρ

k does exists at every x ∈ Bg(ξ, r0). Furthermore, Θ (kV k, ·) is an upper-semi-continuous function in M n, i.e. k k Θ (kV k, x) ≥ lim sup Θ (kV k, x) ∀x ∈ Bg(ξ, r0). y→x Proof. Let kV k(B (ξ, s))1/p Γ + c∗ f(s) := g + 1 s1−k/p. sk p − k

By the Theorem 4.2.2, for 0 < σ < ρ ≤ r0 ! kV k(B (ξ, σ))1/p kV k(B (ξ, ρ))1/p f(ρ) − f(σ) = − g − g σk ρk Γ + c∗ + 1 ρ1−1/p − σ1−1/p ≥ 0. p − k

Then f is a non-decreasing function, therefore, the limit of f(s) when s ↓ 0+, does exists and is nonnegative. Hence the limit kV k(B (ξ, s)) lim g , s↓0 sk also exists, furthermore, is equal to kV k(B (ξ, s)) lim g . s↓0 sk

On the other hand, let ε > 0 ﬁxed, such that, if 0 < σ < ρ, Bg(x, ρ+ε) ⊂ Bg(ξ, r0) and dg(x, y) < ε, then, we also deduce from Theorem 4.2.2 and the monotonicity of kV k that

kV k(B (y, σ))1/p kV k(B (y, ρ))1/p Γ + c∗ g ≤ g + 1 ρ1−k/p σk ρk p − k kV k(B (x, ρ + ε))1/p Γ + c∗ ≤ g + 1 ρ1−k/p. ρk p − k

Letting σ ↓ 0, we thus have

1/p kV k(B (x, ρ + ε))1/p Γ + c∗ Θk(kV k, y) ≤ g + 1 ρ1−k/p. k/p 1/p ωk(ρ + ε) ωk (p − k) Now, let δ > 0 be given and choose ε ρ < δ so that

kV k(B (x, ρ + ε)) 1/p g < Θk(kV k, x) + δ. k/p ωk(ρ + ε) 42 MONOTONICITY AND CONSEQUENCES 4.2

The inequality above gives

1/p 1/p Γ + c∗ Θk(kV k, y) ≤ Θk(kV k, x) + 1 ρ1−k/p, 1/p ωk (p − k) provided dg(y, x) < ε and the desired upper-semi-continuity follows straightforward.

Proposition 4.2.1. Let V ∈ Vk(M), ξ ∈ M ﬁxed, and assume that

n kV k(Bg(ξ, σ)) ≤ βσ for 0 < σ < r0, then Z α α−k kβρ dg(x, ξ) dkV k(x) ≤ , Bg(ξ,ρ) α for any ρ ∈]0, r0[ and 0 < α < k. The proof of this is based in the following technical lemma.

Lemma 4.2.3. Let (X, µ) a measure space, γ > 0, f ∈ L1(µ), f ≥ 0, then Z ∞ Z γ−1 1 γ t µ(Et)dt = f (x)dµ(x), 0 γ E0 where Et := {x : f(x) > t}. More generally Z ∞ 1 Z tγ−1µ(E )dt = (f γ(x) − tγ)dµ(x), t γ 0 t0 Et0 for each t0 ≥ 0. This is a clasical Lemma from measure theory, and its proof can be found in [Mat95] pg. 15.

−1 Proof of Proposition 4.2.1. Let in the previous Lemma, f(x) = |x − ξ| , γ = k − α and t0 = 1/ρ, then Z Z ∞ α−k α−k (k−α)−1 |x − ξ|g − ρ dkV k(x) = (α − k) t kV k(Et)dkV k(x). (4.18) E1/ρ 1/ρ

Notice that 1 Et := {x : f(x) > t} = x : > t = {x : |x − ξ|g < 1/t} := Bg(ξ, 1/t). |x − ξ|g On the other hand, by hypothesis, we have

Z ∞ Z ∞ (k−α)−1 α (k−α)−1 t β βρ t kV k(Bg(ξ, 1/t))dµ(x) ≤ k dkV k(x) = , 1/ρ 1/ρ t α for any 0 < 1/t < r0. Putting together the information above in (4.18), we have Z Z Z α−k α−k α−k α−k |x − ξ|g − ρ dkV k = |x − ξ|g dkV k − ρ dkV k Bg (ξ,ρ) Bg (ξ,ρ) Bg (ξ,ρ) Z α−k α−k = |x − ξ|g dkV k − ρ kV k(Bg(ξ, ρ)) Bg (ξ,ρ) (k − α)βρα ≤ , α 4.2 LP MONOTONICITY FORMULA 43 then,

Z α α−k (k − α)βρ α−k |x − ξ|g dkV k ≤ + ρ kV k(Bg(ξ, ρ)) Bg (ξ,ρ) α (k − α)βρα ≤ + βρk α kβρα = , α provided ρ ∈]0, r0[. 44 MONOTONICITY AND CONSEQUENCES 4.2 Chapter 5

Poincaré and Sobolev Type Inequalities for Intrinsic Var- ifolds

In this final section, a Poincaré and Sobolev-type inequalities are proved for non-negative func- n k tions defined on S ⊂ M , Hg -rectifiable sets as a consequence of the Fundamental Weighted Mono- 1 k tonicity Inequality, Theorem 4.1.2, in the particular case in which V := v(S, θ), with θ ∈ Lloc(Hg xS) 1 and mean curvature vector Hg ∈ Lloc(kV k). Therefore along this section we are under the assumptions of Theorem 4.1.2, i.e.

n (BG): We say that a complete Riemannian manifold (M , g) satisfy (BG), if Secg < b for some constant b ∈ R and for every ξ ∈ M there is r0 such that 0 < r0 < injξ(M, g) and in case b > 0 we assume furthermore br0 < π.

(AC): We say that V ∈ Vk(M) satisﬁes the (AC) condition, whenever

p−1 ! p Z p p−1 1 δV (X) ≤ C(V ) |X| dkV k , ∀X ∈ Xc (Bg(ξ, r0)), Bg(ξ,r0)

for some p > k.

Furthermore, we need to explain some notation: in what follows ξ ∈ M is a ﬁxed point and r0 < injξ(M, g) such that r0 cotb(r0) > 0, according to Deﬁnition 4.1.1. 5.1 Poincaré type inequality The Poincaré type inequality which we proof is a consequence of the following lemma, which is a direct application of the formula (4.1.2).

n n Lemma 5.1.1. Assume (M , g) satisfying (BG) and V ∈ Vk(M ) satisfying (AC). Let Λ > 0 1 such that |Hg|g ≤ Λ, and h ∈ C (M), then for kV k-a.e. ξ ∈ spt kV k, and for all 0 < ρ < r0

S ! ∗ 1 Z Z ∇g h h(ξ) ≤ e(Λ+c k)ρ ρk hdkV k + g dkV k , (5.1) ω k−1 k Bg(ξ,ρ) Bg(ξ,ρ) rξ where c∗ is as in Remark 4.1.2.

Proof. By Hypothesis (BG) and (AC) we are in the conditions of Theorem 4.1.2, then, for ξ ∈ spt kV k and for all 0 < s < r0 we have

45 46 POINCARÉ AND SOBOLEV TYPE INEQUALITIES FOR INTRINSIC VARIFOLDS 5.1

! d 1 Z u Z h ∂ u 2 hγ dkV k ≥ γ ∇⊥u dkV k k ε k ε g ds s Bg(ξ,s) s Bg(ξ,s) s ∂s s g 1 Z u + γ ∇Sh, u∇ u dkV k k ε g g g s Bg(ξ,s) s kc∗ Z u + k hγε dkV k s Bg(ξ,s) s 1 Z u + k+1 γε hhHg, u∇guig dkV k, s Bg(ξ,s) s

1 where γε ∈ C (R) is, such that, given 0 < ε < 1, ( 1, if s < ε γε(s) := 0, if s > 1, and 0 γε(s) < 0 if ε < s < 1. Then

! d 1 Z u 1 Z u hγ dkV k ≥ γ H h + ∇Sh, u∇ u dkV k k ε k+1 ε g g g g ds s Bg(ξ,s) s s Bg(ξ,s) s c∗k Z u + k hγε dkV k s Bg(ξ,s) s 1 Z u ≥ − γ hH + ∇Sh udkV k k+1 ε g g g s Bg(ξ,s) s c∗k Z u + k hγε dkV k, s Bg(ξ,s) s therefore ! d 1 Z u 1 Z u γ hdkV k ≥ − γ u ∇Sh dkV k k ε k+1 ε g g ds s Bg(ξ,s) s s Bg(ξ,s) s Z 1 u ∗ − k γε h |Hg|g − c k dkV k. s Bg(ξ,s) s

Since, by hypothesis, |Hg|g ≤ Λ, we get, ! d 1 Z u 1 Z u γ hdkV k ≥ − γ u ∇Sh dkV k k ε k+1 ε g g ds s Bg(ξ,s) s s Bg(ξ,s) s Λ + c∗k Z u − k γε hdkV k. s Bg(ξ,s) s

1 Now, let {εj}j∈N a sequence of real numbers such that ε ↑ 1 as j → ∞, and {γεj }j∈N ⊂ C (R) such that ( 1, if s < εj γε (s) := j 0, if s > 1, 5.1 POINCARÉ TYPE INEQUALITY 47 and 0 γεj (s) < 0, if εj < s < 1, and εj → χ]−∞,1[ pointwise from below. Thus, reasoning as above we have (as in the proof of Theorem 4.1.2) that for all j ∈ N, and for all 0 < s < r0

! d 1 Z u 1 Z u γ hdkV k ≥ − γ u ∇Sh dkV k k εj k+1 εj g g ds s Bg(ξ,s) s s Bg(ξ,s) s Λ + c∗k Z u − k γεj hdkV k. s Bg(ξ,s) s

Multiply the above inequality by eλ˜(s) and letting j → ∞ we have in distributional sense in s,

! ∗ ˜ d 1 Z ˜ Λ + c k Z eλ(s) hdkV k + eλ(s) hdkV k ds sk sk Bg (ξ,s) Bg (ξ,s) (5.2) ˜ 1 Z ≥ −eλ(s) u ∇Sh dkV k, k+1 g g s Bg (ξ,s) where λ˜(s) := (Λ + c∗k) s, (5.3) therefore, since λ˜0(s) = Λ + c∗k, we have ! d eλ˜(s) Z eλ˜(s) Z hdkV k ≥ − u ∇Sh dkV k, k k+1 g g ds s Bg(ξ,s) s bg(ξ,s) hence, integrating over [σ, ρ] ⊂]0, r0[, ! eλ˜(ρ) Z eλ˜(σ) Z Z ρ d eλ˜(s) Z k hdkV k − k hdkV k = k hdkV k ρ Bg (ξ,ρ) σ Bg (ξ,σ) σ ds s Bg (ξ,s) Z ρ eλ˜(s) Z ≥ − u ∇Sh dkV k k+1 g g σ s Bg (ξ,s) ρ ! ˜ Z 1 Z ≥ −eλ(ρ) u ∇Sh dkV k ds, k+1 g g σ s Bg (ξ,s) then, rearranging terms,

ρ ! ! 1 Z ˜ 1 Z 1 Z 1 Z hdkV k ≤ eλ(ρ) hdkV k + u ∇Sh dkV k ds . (5.4) k+1 g g ωkσk Bg (ξ,σ) ωkρk Bg (ξ,ρ) ωk σ s Bg (ξ,s)

Now, letting τ = 1/s in the inequality above, we have that

Z ρ k+1 Z ! Z 1 Z 1 S ρ k−1 S u ∇g h dkV k ds = − τ u ∇g h dkV k, s g 1 1 g σ Bg(ξ,s) σ Bg(ξ, τ ) then, if we let for A ⊂ M Z ν(A) := u ∇Sh dkV k, g g A 48 POINCARÉ AND SOBOLEV TYPE INEQUALITIES FOR INTRINSIC VARIFOLDS 5.1 since u is an increasing function, ν deﬁnes a measure on (M, g), so by Lemma 4.2.3

Z ρ Z ! Z 1 1 S ρ k−1 1 u ∇g h dkV k ds = − τ ν Bg ξ, dτ sk+1 g 1 τ σ Bg(ξ,s) σ 1 Z σ 1 = τ k−1ν x ∈ M : > τ dτ 1 u(x) ρ 1 Z 1 1 1 = − − dν k n 1 1 1 o u(x)k σk ρk ρ < u < σ Z 1 1 1 = k − k − k dν Bg(ξ,ρ)\Bg(ξ,σ) u σ ρ 1 Z 1 ≤ k dν k Bg(ξ,ρ) u Z u ∇Sh 1 g g = k dkV k k Bg(ξ,ρ) u Z ∇Sh 1 g g = k−1 dkV k. k Bg(ξ,ρ) u Thus S ! 1 Z ˜ 1 Z 1 Z ∇g h hdkV k ≤ eλ(ρ) hdkV k + g dkV k . ω σk ω ρk kω k−1 k Bg (ξ,σ) k Bg (ξ,ρ) k Bg (ξ,ρ) rξ Finally, let σ ↓ 0, and the result follows, since, ξ ∈ spt kV k implies Θk(kV k, ξ) = 1 for kV k-a.e. ξ, therefore ξ is a Lebesgue point kV k-a.e.

Theorem 5.1.2 (Poincaré-Type Inequality for Intrinsic Varifolds). Let (M n, g) be a complete Rie- mannian manifold satisfying (BG), let V := v(S, θ) a rectiﬁable varifold satisfying (AC). Suppose: 1 h ∈ C (M), h ≥ 0, Bg(ξ, 2ρ) ⊂ Bg(ξ, r0) for ξ ∈ S ﬁxed, |Hg|g ≤ Λ for some Λ > 0, θ > 1 kV k-a.e. in Bg(ξ, r0) and for some 0 < α < 1

k kV k ({x ∈ Bg (ξ, ρ): h(x) > 0}) ≤ ωk (1 − α) ρ and, ∗ (5.5) e(Λ+kc )ρ ≤ 1 + α.

Suppose also that, for some constant Γ > 0

k kV k (Bg(ξ, 2ρ)) ≤ Γρ . (5.6) 1 Then there are constants β := β (k, α, r0, b) ∈]0, 2 [ and C := C (k, α, r0, b) > 0 such that Z Z hdkV k ≤ Cρ ∇Sh dkV k. g g Bg(ξ,2ρ) Bg(ξ,ρ)

Proof. Take β ∈]0, 1/2[ an arbitrary parameter, to be speciﬁed later and λ˜(s) as in (5.3). Applying the previous Lemma with η ∈ Bg(ξ, βρ) ∩ spt kV k, in place of ξ, thence, for all 0 < ρ < r0

Z Z ∇Sh ! λ˜((1−β)ρ) 1 1 g g h(η) ≤ e k k hdkV k + k−1 dkV k . ωk(1 − β) ρ Bg (η,(1−β)ρ) kωk Bg (η,(1−β)ρ) rη 5.1 POINCARÉ TYPE INEQUALITY 49

Since ρ/2 < (1 − β)ρ < ρ, we have Bg(η, (1 − β)ρ) ⊂ Bg(ξ, ρ) and

Z Z ∇Sh ! λ˜(ρ) 1 1 g g h(η) ≤ e k k hdkV k + k k−1 dkV k . (5.7) ωk(1 − β) ρ Bg (ξ,ρ) kω Bg (ξ,ρ) rη

1 Now, let t0 ≥ 0 ﬁxed, and γ ∈ C (R) a ﬁxed non-decreasing function with γ(t) = 0, if t ≤ 0 and γ(t) ≤ 1 if t > 0, and apply (5.7) with f(x) := γ (h(x) − t0) in place of h, then

0 S ! Z Z γ (h − t0) ∇ h λ˜(ρ) 1 1 g g f(η) ≤ e k k fdkV k + k k−1 dkV k . ωk(1 − β) ρ Bg (ξ,ρ) kω Bg (ξ,ρ) rη

By hypothesis (5.5) we know that:

˜ eλ(ρ) ≤ 1 + α and that

k kV k ({x ∈ Bg(ξ, ρ): h(x) > 0}) ≤ ωk(1 − α)ρ ,

hence,

Z Z γ0 (h − t ) ∇Sh ! 1 1 0 g g f(η) ≤ (1 + α) k k fdkV k + k k−1 dkV k ωk(1 − β) ρ Bg (ξ,ρ) kω Bg (ξ,ρ) rη Z γ0 (h − t ) ∇Sh ! kV k ({h > 0} ∩ Bg(ξ, ρ)) 1 0 g g ≤ (1 + α) k k + k k−1 dkV k ωk(1 − β) ρ kω Bg (ξ,ρ) rη

2 Z γ0 (h − t ) ∇Sh 1 − α 1 + α 0 g g ≤ k + k−1 dkV k. (1 − β) kωk Bg (ξ,ρ) rη Now, take β := β(k, α) ∈]0, 1/2[ small enough, such that

1 − α2 α2 ≤ 1 − , (1 − β)k 2 then

2 Z γ0 (h(x) − t ) ∇Sh(x) α (1 + α) 0 g g + (f(η) − 1) ≤ k−1 dkV k(x). (5.8) 2 kωk Bg(ξ,ρ) rη

Therefore, for any η ∈ Bg(ξ, βρ) ∩ spt kV k such that f(η) ≥ 1,

Z γ0 (h(x) − t ) ∇Sh(x) 2(1 + α) 0 g g 1 ≤ 2 k−1 dkV k(x). (5.9) α kωk Bg(ξ,ρ) rη

Let ε > 0 given, and choose γ such that, γ(t) ≡ 1 for t ≥ ε, then f ≡ 1 for η ∈ Bg(ξ, βρ) ∩ Et0+ε, where Eτ := {x ∈ spt kV k : h(x) > τ}, and so (5.8) remains valid. Then integrating in booth sides 50 POINCARÉ AND SOBOLEV TYPE INEQUALITIES FOR INTRINSIC VARIFOLDS 5.2

over Bg(ξ, βρ) ∩ Et0+ε, we get

Z Z γ0 (h(x) − t ) ∇Sh(x) ! 2(1 + α) 0 g g kV k (Bg(ξ, βρ) ∩ Et0+ε) ≤ 2 k−1 dkV k(x) dkV k(η) α kωk r Bg (ξ,βρ)∩Et0+ε Bg (ξ,ρ) η ! 2(1 + α) Z Z 1 = γ0 (h(x) − t ) ∇Sh(x) dkV k(η) dkV k(x) 2 0 g g k−1 α kωk r Bg (ξ,ρ) Bg (ξ,βρ)∩Et0+ε η ! 2(1 + α) Z Z 1 ≤ γ0 (h(x) − t ) ∇Sh(x) dkV k(η) dkV k(x). 2 0 g g k−1 α kωk Bg (ξ,ρ) Bg (ξ,ρ) rη

k Now, since by hypothesis kV k (Bg(ξ, 2ρ)) ≤ Γρ , we have, by Proposition 4.2.1 for all 0 < ρ < r0 (notice that ρ has to satisfy 0 < ρ < 2ρ < r0) Z 1 k−1 dkV k(η) ≤ kΓρ, Bg(ξ,ρ) rη then 2(1 + α)Γ Z kV k (B (ξ, βρ) ∩ E ) ≤ ρ γ0 (h(x) − t ) ∇Sh(x) dkV k(x). g t0+ε 2 0 g g α ωk Bg(ξ,ρ)

Integrating over [0, ∞[ in t0, ! Z ∞ 2(1 + α)Γ Z ∞ Z kV k (B (ξ, βρ) ∩ E ) dt ≤ ρ γ0 (h(x) − t ) ∇Sh(x) dkV k(x) dt g t0+ε 0 2 0 g g 0 0 α ωk 0 Bg (ξ,ρ) ! 2(1 + α)Γ Z ∞ Z ∂ = ρ − (γ(h(x) − t )) ∇Sh(x) dkV k(x) dt 2 0 g g 0 α ωk 0 Bg (ξ,ρ) ∂t0 2(1 + α)Γ Z d Z ∞ = ρ ∇Sh(x) − γ(h(x) − t )dt dkV k(x) 2 g g 0 0 α ωk Bg (ξ,ρ) ds 0 2(1 + α)Γ Z ≤ ρ ∇Sh(x) γ(h(x))dkV k(x) 2 g g α ωk Bg (ξ,ρ) 2(1 + α)Γ Z ≤ ρ ∇Sh(x) dkV k(x). 2 g g α ωk Bg (ξ,ρ)

To calculate the left term, let A˜τ := {x ∈ spt kV k :(h − ε)(x) > τ} = Aτ+ε, then by Lemma 4.2.3 Z ∞ Z ∞ ˜ kV k (Bg(ξ, βρ) ∩ Et0+ε) = kV kx(Bg(ξ, βρ)) Aτ dt0 0 0 Z = (h − ε) dkV kx(Bg(ξ, βρ)) A˜0 Z = (h − ε) dkV k. Bg(ξ,βρ)∩Aε Sumarizing, Z Z (h(x) − ε) dkV k(x) ≤ Cρ ∇Sh(x) dkV k(x), g g Bg(ξ,βρ)∩Aε Bg(ξ,ρ) where 2(1 + α)Γ C := C(k, α, Γ, r0) := 2 . α ωk Finally, letting ε ↓ 0, the result follows. 5.2 SOBOLEV TYPE INEQUALITY 51

5.2 Sobolev type Inequality Finally, in order to prove the desired Sobolev type inequality, we use the Theorem 4.1.2 to derive an inequality in which the tangential derivative of h appears, this joint with a calculus lemma and the Lemma 4.2.3 gives a "local " version of the desired inequality, which via a covering argument (c.f. Theorem 2.1.3) implies a "global" Sobolev inequality, where the constant derived depends only on the dimension of the ambient manifold and does not depends on the geometry, nevertheless this constant is not optimal.

n k Lemma 5.2.1. Assume (M , g) satisfying (BG) and V := v(S, θ), with S ⊂ M Hg -rectiﬁable, and 1 1 θ ∈ Lloc(kV k), such that (AC) is satisﬁed. Let h ∈ C (M) non-negative, then for all ξ ∈ spt kV k and for all 0 < ρ < r0 ! 1 Z 1 Z Z ρ 1 Z hdkV k ≤ hdkV k + ∇Sh + h |H | − c∗k dkV k ds (5.10) k k k g g g g ωkσ Bg (ξ,σ) ωkρ Bg (ξ,ρ) σ s Bg (ξ,s) where c∗ is as in Remark 4.1.2.

Proof. By Hypothesis (BG) and (AC) we are in the conditions of Theorem 4.1.2, then, for ξ ∈ spt kV k and for all 0 < s < r0

! d 1 Z u Z h ∂ u 2 hγ dkV k ≥ γ ∇⊥u dkV k k ε k ε g ds s Bg(ξ,s) s Bg(ξ,s) s ∂s s g 1 Z u + γ ∇Sh, u∇ u dkV k k ε g g g s Bg(ξ,s) s kc∗ Z u + k hγε dkV k s Bg(ξ,s) s 1 Z u + k+1 γε hHgh, u∇guig dkV k, s Bg(ξ,s) s

1 where γε ∈ C (R) is, such that, given 0 < ε < 1, ( 1, if s < ε γε(s) := 0, if s > 1, and 0 γε(s) < 0 if ε < s < 1. Then

! d 1 Z u 1 Z u hγ dkV k ≥ γ H h + ∇Sh, u∇ u dkV k k ε k+1 ε g g g g ds s Bg(ξ,s) s s Bg(ξ,s) s c∗k Z u + k hγε dkV k s Bg(ξ,s) s 1 Z u ≥ − γ H h + ∇Sh udkV k k+1 ε g g g s Bg(ξ,s) s c∗k Z u + k hγε dkV k, s Bg(ξ,s) s 52 POINCARÉ AND SOBOLEV TYPE INEQUALITIES FOR INTRINSIC VARIFOLDS 5.2

therefore ! d 1 Z u 1 Z u γ hdkV k ≥ − γ ∇Sh + h |H | − c∗k dkV k. k ε k ε g g g g ds s Bg(ξ,s) s s Bg(ξ,s) s

1 Now, let {εj}j∈N a sequence of real numbers such that ε ↑ 1 as j → ∞, and {γεj }j∈N ⊂ C (R) such that ( 1, if s < εj γε (s) := j 0, if s > 1, and 0 γεj (s) < 0 if εj < s < 1, and εj → χ]−∞,1[ pointwise from below. Then, reasoning as in the proof of Theorem (4.1.2), for all j ∈ N, and for all 0 < s < r0 ! d 1 Z u 1 Z u γ hdkV k ≥ − γ ∇Sh + h |H | − c∗k dkV k. k εj k εj g g g g ds s Bg (ξ,s) s s Bg (ξ,s) s

Letting j → ∞ we have in distributional sense in s, ! d 1 Z 1 Z hdkV k ≥ − ∇Sh + h |H | − c∗k dkV k. k k g g g g ds s Bg(ξ,s) s Bg(ξ,s)

Finally, integrating over [σ, ρ] ⊂]0, r0[ we have ! Z ρ d 1 Z 1 Z 1 Z k hdkV k = k hdkV k − k hdkV k σ ds s Bg (ξ,s) ρ Bg (ξ,ρ) σ Bg (ξ,σ) ! Z ρ 1 Z ≥ − ∇Sh + h |H | − c∗k dkV k ds. k g g g g σ s Bg (ξ,s)

Therefore,

1 Z 1 Z k hdkV k ≤ k hdkV k ωkσ Bg(ξ,σ) ωkρ Bg(ξ,ρ) ! Z ρ 1 Z + ∇Sh + h |H | − c∗k dkV k ds. k g g g g σ s Bg(ξ,s)

Lemma 5.2.2. Suppse f and g are bounded non-decreasing functions on ]0, ∞[, and

1 1 Z ρ 1 1 ≤ k f(σ) ≤ k f(ρ) + k g(s)ds, (5.11) σ ρ 0 s where 0 < σ < ρ < ∞. Then, there exists ρ ∈]0, ρ0[ such that 1 f(ρ) ≤ 5kρ g(ρ), 2 0 where 1 k ρ0 = 2 lim f(ρ) . ρ→∞ 5.2 SOBOLEV TYPE INEQUALITY 53

Proof. We argue by contradiction. Assume that for every ρ ∈]0, ρ0[ 1 f(5ρ) > 5kρ g(ρ), 2 0 then, by (5.11),

1 1 Z ρ0 1 1 ≤ sup k f(σ) ≤ k f(ρ0) + k g(s)ds 0<σ<ρ0 σ ρ0 0 s 1 2 Z ρ0 1 < k f(ρ0) + k k f(5s)ds ρ0 5 ρ0 0 s 1 2 Z 5ρ0 5k ds = k f(ρ0) + k k f(s) ρ0 5 ρ0 0 s 5 1 2 Z ρ0 1 Z 5ρ0 1 = k f(ρ0) + k f(s)ds + k f(s)ds ρ0 5ρ0 0 s ρ0 s 1 2 1 2 Z 5ρ0 1 ≤ f(ρ0) + sup f(s) + lim f(s) ds k k s→∞ k ρ0 5 0

1 1 1 1 1 1 ≤ sup f(σ) < sup f(σ) + 3 lim f(s) < 4 lim f(s). k k k s→∞ k s→∞ 0<σ<ρ0 σ 2 0<σ<ρ0 σ ρ0 ρ0 Finally, since ρk = 2k lim f(s), 0 s→∞ and dividing by 2, we have 1 1 1 ≤ sup f(σ) < k−1 , 2 2 0<σ<ρ0 2 which clearly is a contradiction since k ≥ 2.

Now, we are ready to prove the Sobolev-type inequality, which we state bellow. 54 POINCARÉ AND SOBOLEV TYPE INEQUALITIES FOR INTRINSIC VARIFOLDS 5.2

Theorem 5.2.3 (Sobolev-Type Inequality for Intrinsic Varifolds). Let (M n, g) be a complete Rie- mannian manifold satisfying (BG), let V = v(S, θ) a rectiﬁable varifold satisfying (AC). Suppose 1 h ∈ C0 (S) non negative, and θ ≥ 1 kV k-a.e. in S. Then there exists C := C(k) > 0 such that

k−1 Z k Z k S ∗ h k−1 dkV k ≤ C ∇ h + h |H | − c k dkV k. (5.13) g g g g S S

Proof. Notice that, for Lemma 5.2.1, for all 0 < σ < ρ < r0 1 Z 1 Z k hdkV k ≤ k hdkV k ωkσ Bg(ξ,σ) ωkρ Bg(ξ,ρ) ! Z ρ 1 Z + ∇Sh + h |H | − c∗k dkV k ds, k g g g g σ s Bg(ξ,s)

1 but, since by hypothesis h ∈ C0 (Bg(ξ, r0)) it remains valid for all 0 < σ < ρ < ∞, hence we can apply the Lemma 5.2.2 with the choices 1 Z f(ρ) := hdkV k ωk Bg(ξ,ρ) 1 Z g(ρ) := ∇Sh + h |H | − c∗k dkV k, g g g g ωk Bg(ξ,ρ) provided that ξ ∈ spt kV k and h(ξ) ≥ 1. Thus for kV k-a.e ξ ∈ {x ∈ spt kV k : h(x) ≥ 1} we have

1 1 Z k ρ < 2 hdkV k , ωk M and

1 Z 1 Z k Z hdkV k ≤ 5k hdkV k ∇Sh + h |H | − c∗k dkV k. (5.14) g g g g Bg(ξ,5ρ) ωk M Bg(ξ,ρ)

Now, since {x ∈ spt kV | : h(x) ≥ 1} ⊂ spt kV k ∩ spt h which is compact, in virtue of Vitali’s ﬁve Lemma (cf. [Sim83] Theorem 3.3 pg. 11) we can select a collection of balls

{Bg(ξi, ρi)}i⊂Λ ,

0 such that, for all i ∈ Λ, ξi ⊂ {x ∈ spt kV k : h(x) ≥ 1} and, there exists Λ ⊂ Λ such that for every 0 i 6= j, i, j ∈ Λ it holds Bg(ξi, ρi) ∩ Bg(ξj, ρj) = ∅ and [ [ {x ∈ spt kV k : h(x) ≥ 1} ⊂ Bg(ξij , 5ρij ) = Bg(ξi, ρi). j∈Λ0 i∈Λ 5.2 SOBOLEV TYPE INEQUALITY 55

Then, by (5.14)

Z X Z hdkV k ≤ hdkV k {h≥1}∩spt kV k B (ξ ,5ρ ) j∈Λ0 g ij ij 1 ! X 1 Z k Z ≤ 5k hdkV k ∇Sh + h |H| − c∗k dkV k g g g ωk M B (ξ ,ρ ) j∈Λ0 g ij ij 1 (5.15) 1 Z k X Z ≤ 5k hdkV k ∇Sh + h |H| − c∗k dkV k g g g ωk M B (ξ ,ρ ) j∈Λ0 g ij ij 1 1 Z k Z ≤ 5k hdkV k ∇Sh + h |H| − c∗k dkV k. g g g ωk M M

1 Let γε ∈ C (R) a non-decreasing function such that, for given ε > 0, ( 1, if 0 < ε < t γε(t) := 0, if t ≤ 0, and consider, for t0 ≥ 0 given, f(x) := γε (h(x) − t0) , then, applying (5.15) with f instead of h, and setting Sα := {x ∈ S : h(x) > α}, we get, Z

kV k (St0+ε) = hdkV k {x∈M:h(x)−t0>ε} Z = fdkV k {x∈S:f(x)≥1} 1 1 Z k Z ≤ 5k fdkV k γ0 (h − t ) ∇Sh + f |H | − c∗k dkV k ε 0 g g g g ωk S S k Z 5 1 0 S ∗ ≤ (kV k(S )) k γ (h − t ) ∇ h + f |H | − c k dkV k, 1 t0 ε 0 g g g g k S ωk

1 now, multiplying by (t0 + ε) k−1 we have,

k 1 Z 1 5 k k 0 S ∗ (t + ε) k−1 kV k (S ) ≤ (t + ε) k−1 kV k(S ) γ (h − t ) ∇ h + f |H| − c k dkV k 0 t0+ε 1 0 t0 ε 0 g g g k S ωk 1 k Z ! k Z 5 k 0 S ∗ = (t + ε) k−1 dkV k γ (h − t ) ∇ h + f |H| − c k dkV k 1 0 ε 0 g g g k S S ωk t0 1 k Z ! k Z Z 5 k ∂ 0 S ∗ ≤ (h + ε) k−1 dkV k − γ (h − t ) ∇ h dkV k + f |H| − c k dkV k 1 ε 0 g g g k S S ∂t0 S ωk t0 1 k Z k Z Z 5 k ∂ 0 S ∗ ≤ (h + ε) k−1 dkV k − γ (h − t ) ∇ h dkV k + f |H| − c k dkV k . 1 ε 0 g g g k S S ∂t0 S ωk

Integrating the above inequality on t0 in the interval ]0, ∞[, we have

∞ Z 1 k−1 (t0 + ε) kV k (St0+ε) dt0 ≤ 0 1 k Z k Z ∞ Z Z 5 k ∂ 0 S ∗ (h + ε) k−1 dkV k − γ (h − t ) ∇ h dkV k + f |H| − c k dkV k dt . 1 ε 0 g g g 0 k S 0 S ∂t0 S ωk 56 POINCARÉ AND SOBOLEV TYPE INEQUALITIES FOR INTRINSIC VARIFOLDS

First, notice that the left hand side, by the Lemma 4.2.3, is equal to,

∞ Z 1 k − 1 Z k k − 1 Z k k−1 k−1 k−1 (t0 + ε) kV k (St0+ε) dt0 = (h − ε) dkV k = (h − ε) dkV k. 0 k S0 k {x∈S:h(x)>ε} on the other hand, we can estimate the right hand side as follows, ﬁrst by Fubini’s Theorem

Z ∞ Z ∂ Z d Z ∞ − γ0 (h(x) − t ) ∇Sh dkV k dt = ∇Sh − γ (h(x) − t ) dt dkV k ε 0 g g 0 g g ε 0 0 0 S ∂t0 S dt 0 Z = ∇Sh (γ (h(x))) dkV k g g ε S Z S ≤ ∇g h(x) dkV k. S And, again by Fubini’s Theorem, Z ∞ Z Z Z ∞ ∗ ∗ f(x) |Hg|g − c k dkV k(x) dt = |Hg|g − c k γε(h(x) − t0)dt0 dkV k 0 S S 0 Z Z −∞ ! ∗ = |Hg|g − c k −γε(w)dw dkV k S h(x) Z ∗ ≤ h |Hg|g − c k dkV k. S Therefore

1 Z k Z k Z k − 1 k 5 k S ∗ (h − ε) k−1 dkV k ≤ (h + ε) k−1 dkV k ∇ h + h |H | − c k dkV k, 1 g g g g k Mε k M M ωk then, letting ε → 0 we obtain

k−1 Z k Z k S ∗ h k−1 dkV k ≤ C ∇ h + h |H | − c k dkV k, g g g g S S where k5k C := C(k) := 1 . k (k − 1)ωk Bibliography

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