The Gaussian Isoperimetric Problem and the Self-Shrinkers of Mean Curvature Flow

THE GAUSSIAN ISOPERIMETRIC PROBLEM AND THE SELF-SHRINKERS OF MEAN CURVATURE FLOW

by Matthew McGonagle

A dissertation submitted to Johns Hopkins University in conformity with the requirements for the degree of Doctor of Philosophy

Baltimore, Maryland July 2014

We record work done by the author joint with John Ross [27] on stable smooth solutions to the gaussian isoperimetric problem, and we also record work done by the author [26] on two-dimensional self-shrinkers of any co-dimension.

The guassian isoperimetric problem is related to minimizing a gaussian weighted surface area while preserving an enclosed gaussian weighted volume. The work joint with the author and John

Ross [27] studies smooth, stable local minimizers to the gaussian isoperimetric problem. We show that for complete critical solutions of polynomial volume growth, the only stable solutions are the hyper-planes.

As in McGonagle-Ross[27], we also consider the incomplete case of stable solutions to the gaussian isoperimetric problem contained in a ball centered at the origin with their boundary contained in the boundary of the ball. We show that for general dimension, large enough ball, and appropriate euclidean area conditions on the hyper-surface, we get integral decay estimates for the curvature.

Furthermore, for the two-dimensional critical solutions in three-dimensional euclidean balls, using a de Giorgi-Moser-Nash type iteration [3,4, 19] we get point-wise decay estimates for similar conditions. These estimates are decay estimates in the sense that when everything else is held constant, we get zero curvature as the radius of the ball goes to infinity.

We also record work done by the author [26] on the application of gaussian harmonic one-forms to two dimensional self-shrinkers of the mean curvature flow in any co-dimension. Gaussian harmonic

2 one-forms are closed one-forms minimizing a weighted Lµ norm in their cohomology class. We are able to show a type of rigidity result that forces the genus of the hyper-surface to be zero if the curvature is smaller than a certain size. This bound for the curvature is larger than that given in the rigidity result of Cao-Li [7] which classifies the self-shrinker as either a sphere or cylinder ifthe curvature is small enough. We also show that if the self-shrinker satisfies an appropriate curvature condition, then the genus gives a lower bound on the index of the self-shrinking Jacobi operator.

READERS: Professor William P. Minicozzi II (Advisor) and Professor Joel Spruck

ii Acknowledgments

I would like to thank my advisor Professor William Minicozzi for his years of mentorship, assistance, and guidance. This work would not be possible without him. His hand has helped mold me into the researcher that I am today and the one I want to be in the future.

I would also like to thank Professor Joel Spruck and Professor Chikako Mese for their support, guidance, and conversations. They have also had great influence over my academic growth during my stay at Johns Hopkins. I would also like to thank Professor Jacob Bernstein for stimulating conversation during departmental social events.

Special thanks goes to my love, companion, and collaborator Ling Xiao. She has carried me through hard times and made me into a better person. With her support, I have found the mental stamina to tolerate the periods of frustration that seem to be a common occurrence during the course of my research.

I would like to thank my family. My adventure into the unknown world of graduate school would be impossible without their love and support.

I would like to thank my friend and collaborator John Ross. Without our partnership, much of the material contained in this work would not be possible.

I owe a lot of gratitude to Sabrina Raymond. Without her help, I would be too lost to get my foot out of the door.

I would also like to thank my companions and colleagues: Tim Tran, Vitaly Lorman, Longzhi

Lin, Sinan Ariturk, Hongtan Sun, Xuehua Chen, Lu Wang, Chris Kauffman, Poyao Chang, Jordan

Paschke, and many others. During our friendship, we were in the thick of it, and I wish fortune to those who still are. I would like to thank Professor Jian Kong for welcoming me to attend holiday events at Chinese restaurants.

I would also like to thank friends and acquaintances from other parts of the university that I have met through GRO happy hours and other activities: Chris, Fumbeya, Rob, Amy, Christina, Ian, and others. Thank you for the brief breaks from the world of pure mathematics and the exposure

iii to other things.

iv Contents

Abstract ii

Acknowledgments iii

List of Figures vi

1 Introduction 1

1.1 Notation...... 4

2 Background 6

2.1 The f-Area Functional and f-Minimal Surfaces...... 6

2.2 Isoperimetric Problems...... 8

2.3 Mean Curvature Flow and Self-Shrinkers...... 10

2.4 The Gaussian Isoperimetric Problem...... 12

2.5 Harmonic One Forms...... 15

3 Gaussian Isoperimetric Problem 18

3.1 Stability of Planar Hyper-surfaces...... 18

3.2 Complete Non-planar Hyper-surfaces...... 19

3.3 Stable Critical Hyper-surfaces in a Ball...... 24

3.3.1 Integral Curvature Estimates...... 26

3.3.2 Pointwise Curvature Estimates for Three-dimensional Case...... 28

4 Gaussian Harmonic One Forms 41

Appendix 49

Curriculum Vitae 59

v List of Figures

2.1 A Steiner Symmetrization...... 8

2.2 The V functional...... 9

2.3 Set up of the Jordan curves...... 16

3.1 Set up of the iteration...... 34

3.2 Set up of θiR ...... 39

vi 1

Introduction

Many problems in geometric analysis are of a variational nature.These problems often involve finding sub-manifolds or hyper-surfaces inside some ambient space that are local or global extrema of some functional associated with their geometry. Here, we will be investigating particular variational problems related to considering the geometry in euclidean space Rn+1 along with a guassian weight 2 − |x| e 4 .

 i 2 n n+1 The usual metric for euclidean space is gE = (dx ) , and for every hyper-surface Σ ⊂ R , i we have a natural area measure A(Σ) inherited from Rn+1. Likewise for every open region Ω ⊂ Rn+1 we have a natural volume measure V (Ω). We will be considering the gaussian weighted area measure 2 2 − |x| − |x| dAµ = e 4 dA and gaussian weighted volume measure dVµ = e 4 dV .

The traditional isoperimetric problem considers the process of fixing a volume V0 and finding regions Ω with V (Ω) = V0 such that Ω minimizes the area of its boundary. Classic results of Schwarz [34] and Steiner [35] show that spheres are the unique minimizers of euclidean space.

One can also consider a local version of an isoperimetric problem. In this case, instead of finding a minimizer Ω0 and comparing it to all regions Ω with V (Ω) = V0, we consider finding a minimizer

Ω0 such that it is a minimizer for all regions Ω close to Ω0 and with V (Ω) = V0. The boundary of such an Ω0 is considered a stable solution to the isoperimetric problem. For the euclidean case, Barboas-do Carmo [2] show that the only compact stable solutions are again the spheres.

Here we record work joint between the author and John Ross [27] on the local solutions to the gaussian isoperimetric problem. We consider local minimizers Σ0 of Aµ that enclose a fixed weighted volume Vµ,0 among hyper-surfaces Σ close to Σ0 and enclosing the same weighted volume. The first derivative test for this variation tells us that Σ0 = ∂Ω0 must satisfy the mean curvature relation

1 1 H = 2 ⟨x, N⟩ + C where C is a constant. Borell [5] and Sudakov & Cirel’son[36] show that hyper- planes are global minimizers. We use work done joint with John Ross [27] to show that hyper-planes are in fact the only smooth, two-sided, and complete local minimizers of polynomial volume growth.

The fact that hyper-planes are stable is recorded in Theorem1 of Section3.

Theorem 1. Hyper-planes are stable critical hyper-surfaces to the gaussian isoperimetric problem.

In fact, we show that bounds on the index of the associated Jacobi operator − L for a critical Σ0 allow us to make statements about Σ0 splitting off a linear space. Here the index is defined to bethe maximal dimension of sub-spaces of weighted volume preserving variations for which − L is negative definite (for example,0 Σ is stable if and only if the index is 0), and a critical Σ0 is defined to be satisfying the first derivative test of the gaussian isoperimetric problem. This result is recorded in the following theorem.

Theorem 2. Let Σ ⊂ Rn+1 be a non-planar critical hyper-surface for the gaussian isoperimetric problem with Index = I for all compact weighted volume preserving variations. Also, we require

0 ≤ I ≤ n. Then

i Σ = Σ0 × R , (1.0.1) where i ≥ n + 1 − I. In particular, there are no non-planar stable solutions.

From Theorem1 and Theorem2, we immediately have the following corollary.

Corollary 1. The only smooth stable solutions of the gaussian isoperimetric problem of polynomial growth are the hyper-planes. There are no solutions of index one.

n+1 We also consider the problem of local solutions Σ ⊂ BR(0) ⊂ R such that ∂Σ ⊂ ∂BR(0). For n  2 any dimension Σ , we obtain estimates for Σ |A| dAµ for certain natural lower bounds on R and upper bounds on the euclidean area of Σ; see Proposition2. For the three-dimensional case Σ 2 ⊂ R3, these integral estimates may be turned into point-wise estimates for |A|. McGonagle-Ross [27] uses a Choi-Schoen type argument [8] to establish these point-wise estimates. However, we will use an alternative method based on de Giorgi-Moser-Nash iteration [3,4, 19]. The point-wise estimates are recorded in the following theorem. We use OC ≡ 1 + |C|, OR ≡ 1 + R, and AE represents the euclidean area of Σ.

3 Theorem 3. Let Σ ⊂ BR(0) ⊂ R be a smooth stable critical hyper-surface of the gaussian isoperi- 1 metric problem with H = 2 ⟨x, N⟩ + C and ∂Σ ⊂ ∂BR(0). Consider any 0 < θ < 1. We have

2 that there exists a constant D (independent of Σ, θ, and R) such that if θ(1 − θ)R > DOC and 2 2 (1−θ)R −DOC n 2 2 AE(Σ) < De OC (1 − θ) R e 16 , then

1 2 1 1 2 2 2 D0OC −4 4 − 32 (1−θ )R sup |A| ≤ DOC e (1 − θ) AE(1 + AE)ORe . (1.0.2) BθR(0)

One of the main technical considerations for creating these point-wise estimates is that we must be careful to get estimates that have exponential decay in R for any sub-ball BθR(0). Furthermore, we want that it is sufficient that the euclidean area be bounded by exponential growth in R.

Finally we consider the work in McGonagle [26] on two-dimensional self-shrinkers of any co- dimension, M 2 ⊂ Rn. A self-shrinker of mean curvature flow is a sub-manifold that moves by dilations under the mean curvature flow, and so it must satisfy the mean curvature condition H⃗ =

1 N 2 ⃗x , where ⃗x is the position vector. We use an analog of harmonic one-forms that we call gaussian harmonic one-forms. Gaussian harmonic one-forms are constructed from minimizing the functional

 2 M |ω| dAµ over closed one-forms ω in a particular cohomology class. We show that these forms satisfy certain Euler-Lagrange equations that allow us to use them as test functions much like Palmer

[31], Ros [33], and Urbano [37] use harmonic one-forms for minimal surfaces.

Our first main result in this direction is a rigidity-type theorem for the genusof M depending on a particular norm of the second fundamental form A. It is expected that bounds on |A| give bounds on the topology of a self-shrinker Σ. An argument pointed out to the author by Professor

2 1 William Minicozzi shows that if |A| ≤ C|x| for C < 8 , then Σ has finite topology [29]. For any two-dimensional self-shrinker Σ, we have that |H|2 ≤ 2|A|2. For a self-shrinker, we then get that

|xN |2 ≤ 8C|x|2. So, we have that |xT |2 ≥ (1 − 8C)|x|2 > 0 for |x| sufficiently large. So, from Morse

Theory, we have that the topology of Σ ∩ BR(0) is constant for large enough R. Our first main result is the following.

Theorem 4. If M 2 ⊂ RN is an orientable self-shrinker of polynomial volume growth with genus g ≥ 1, then 1 sup Aα(v, i)Aα(i, v) ≥ . (1.0.3) x∈M,v∈TxM,|v|=1 2

This theorem should be compared with a rigidity theorem of Cao-Li [7]. Cao-Li show that if

1 |A| ≤ 2 (here we have renormalized their result to match our definition of self-shrinker) then M must be a sphere or cylinder. Our result shows that for M 2 ⊂ R3, we have that if the maximum of 1 the absolute value of the principal curvatures |κi| ≤ 2 , then the genus must be zero. Our second large result in this direction is a result on how the genus of M gives us lower bounds

3 on the index of the appropriate Jacobi operator L if M satisfies a certain curvature condition.

Theorem 5. Let M 2 ⊂ Rn be an orientable self-shrinker of polynomial volume growth. If  2 2 sup inf |κα1 − κα2| ≤ δ < 1, then the index of L acting on scalar functions on M has a p∈M {ηα} orth. α lower bound given by g Index (L) ≥ . (1.0.4) M n

1.1 Notation

Here we give a consolidated list of notation that will be used throughout our arguments.

Generally we use M m to be a sub-manifold of N n of any co-dimension. We denote hyper-surfaces of N n by Σ.

The covariant derivative on M or Σ is denoted by ∇. The covariant derivative on N is denoted by ∇. So, ∇v = (∇v)T . The covariant derivative on the normal bundle of M is denoted by ∇N , so that ∇N v = (∇v)N .

We will often need to make use of the second covariant derivative. We will use the notation ∇2 2 for the second covariant derivative for the ∇ connection on the sub-manifold M, ∇ for the second covariant derivative connection on the ambient manifold N, and ∇N,2 for the second covariant derivative of the normal connection ∇N . For example, ∇N,2 V = ∇N (∇N V ) − ∇N V. X,Y X Y ∇X Y m n N For the general co-dimensional case of M ⊂ N , we will be using Aij = −∇i ∂j. Let ηα be a α N frame for the normal bundle. We will often need to use the notation Aij = −⟨∇i ∂j, ηα⟩. m m+1 For the co-dimension one case M ⊂ N , we make use of the standard convention Aij =

−⟨∇¯ iN, ∂j⟩.

For the case of N n = Rn, we will be needing to make use of certain second order elliptic operators. Let x be the position vector in Euclidean space. For scalar functions u ∈ C2(M m), we define

1 Lu = △u − ∇ T u (1.1.1) 2 x 1 L u = Lu + |A|2u + u. (1.1.2) 2

T For any C1 Euclidean vector field V defined along M m, we define ∇V = ∇V  . We are interested in the euclidean case where we denote the ambient euclidean connection by ∇E. If V is

4 C2, we define:

M 2 1 L V = ∇ V − ∇ T V, (1.1.3) ii 2 x E E,2 1 E L V = ∇ V − ∇ T V, (1.1.4) ii 2 x 1 LE V = LE V + |A|2V + V, (1.1.5) 2 (1.1.6)

As explained in Section 2.3, all variations are considered to be normal.

While computing estimates for stable solutions to the gaussian isoperimetric problem in BR(0), we will often be making use of arbitrary constants. We will often need to increase the size of constants so that we may use results from other lemmas or propositions. To accomplish this, we write every statement in such a way that they remain valid if we increase the size of the arbitrary constants.

While investigating gaussian harmonic forms, we will often need to make use of euclidean vectors and their euclidean components. We will use indices such as a, b, ... to denote euclidean vectors or

n components. For example, the vectors {∂a} are an orthonormal basis in R , and for any tangential a vector v we may write v = v ∂a.

5 2

Background

2.1 The f-Area Functional and f-Minimal Surfaces

Consider a manifold N n and a sub-manifold M m ⊂ N n. In geometric analysis, natural quantities to consider are the m-dimensional areas of compact subsets of M. Very important questions are related to how these m-dimensional areas vary as we bend or change M on compact sets. Using a parametric approach, we model these variations by considering smooth functions F (x, t): M m × R → N n such m that F (x, 0) = x, and for each compact time interval [t0, t1], the set {x ∈ M |F (x, t) ̸= x and t0 ≤ t ≤ t1} is contained in a compact set. Also, for clarity of notation, we use Ut ≡ F (U, t), and for

m ∂F ∂F coordinates xi on M we use Fi ≡ and Ft ≡ . ∂xi ∂t We will also need to consider both weighted areas and weighted volumes. For now, we will only consider weighted areas, and we will hold our discussion of weighted volumes until we review the

Gaussian Isoperimetric Problem.

Definition 1. For a smooth function f : M m → R, we define the weighted f-area of a compact subset U ⊂ M by  f Af (U) = e dA. (2.1.1) U One should note that by a conformal change of metric, it is possible to consider the weighted f-area to be the regular area functional under a different metric. However, when studying the

Gaussian Isoperimetric Problem, we will have need to consider the area functional in the form given by (2.1.1).

We wish to consider manifolds M that are minimizers or local minimizers for the f-area functional. Consider a compact variation F (x, t) and some compact domain U ⊂ M such that on some

6 compact time interval I containing 0, we have that F (x, t) = x on (M \ U) × I. Then, we have

2  d  d  that dt Af (Ut) = 0 and dt2 Af (Ut) ≥ 0. Therefore, to consider the local minimizers of Af t=0 t=0 we must consider the first and second variations of Af . Since the f-area functional is invariant under diffeomorphic re-parameterization of M, it is often

∂ convenient to consider variations F (x, t) such that ∂t F (x, t) is normal to the tangent space of TF (·, t). Moving from a general compact variation to a compact normal variation is accomplished by solving the appropriate ODE that arises from re-parameterizing by diffeomorphisms.

By considering M as a domain of parameterization for every time slice F (·, t), we have that

 f◦F Af (Ut) = e dAt. (2.1.2) U

Here dAt is the euclidean area element of each time slice. A standard computation [9] gives that

d  ∂F  dt dAtt=0 = ⟨H, ∂t ⟩t=0 . So we clearly have

  ∂  f   Af (Ut) = e ∇f + H,Ft dV. (2.1.3) ∂t t=0 U

We see that a necessary condition for M to be a local minimizer for Af is that ∇f + H ≡ 0.

Definition 2. A sub-manifold M ⊂ N is called f-minimal if ∇f + H ≡ 0.

The special case of f ≡ 0 gives the regular area functional, and it is standard to call these sub-manifolds minimal.

Now, a standard computation (see proof of Lemma2 given in the appendix) gives the second variation of the f-area for an f-minimal sub-manifold.

Lemma 1. Let M m ⊂ N n be an f-minimal sub-manifold with compact normal variation F (x, t).

Let U be a compact domain of M and Ut = F (U, t). We have that

2   ∂  f  N,2   2 Af (Ut) = − e ∇ii Ft + ⟨Aij,Ft⟩Aij,Ft + Ric(Ft,Ft) dA ∂t t=0 U  (2.1.4) f − e Hessf (Ft,Ft) dA. U

When the co-dimension is one and M is two sided, we use scalar functions to represent variations

7 F (x, t). We use u ≡ ⟨Ft,N⟩ and then have that (2.1.4) becomes

2   ∂  f  2  2 Af (Ut) = − e u △u + |A| u + Ric(N,N)u + Hessf (N,N)u dA. (2.1.5) ∂t t=0 U

For the case of unweighted euclidean area, we use f ≡ 0, and get

2   ∂  2 2 2 2 A(Ut) = − u△u + |A| u + Ric(N,N)u dV. (2.1.6) ∂t t=0 U

Therefore, a minimal hyper-surface is stable if and only if the operator −(△ + |A|2 + Ric(N,N)) is positive.

The property of a hyper-surface Σ being minimal and stable is strong. It puts restrictions on Σ in many ways including topology and singularities. For example, the only stable complete minimal hyper-surfaces in R3 are the planes [12, 18, 32, 33].

2.2 Isoperimetric Problems

 For a region Ω ⊂ N, the volume of Ω is defined by the n-dimensional volume in N, dVN . If ∂Ω is Ω C1, then the area of Ω is defined by the area element on ∂Ω.

Isoperimetric problems are concerned with minimizing the area of the boundary of a region in space over all regions with a fixed volume V . More specifically, two questions of importance in isoperimetric problems are the following:

1. Given a fixed volume V , what is the infimum of boundary area over all regions of volume V

in a manifold M?

2. For a fixed volume V , is the infimum realized by an actual minimizer? In other words, does

there exist a region Ω whose boundary ∂Ω has area realizing the infimum?

Σ Σ′ Ω′

Ω

Figure 2.1: A Steiner Symmetrization

For the Euclidean case of N = Rn+1, a classic result of Schwarz [34] and Steiner [35] show that balls are the only minimizers. The proof uses a rearrangement technique called Steiner Sym-

8 metrization to show that any minimizer must be spherically symmetric. A Steiner Symmetrization is a vertical rearrangement over a hyperplane that preserves volume, reduces area, and makes the domain symmetric about the hyperplane. See Figure 2.1 above.

A local minimizer for an isoperimetric problem is a region Ω of ﬁxed volume V that has minimal boundary area amongst all regions that are close to Ω. Another important question for isoperimetric problems is the following:

3. Do there exist local minimizers that aren’t global minimizers?

An important tool for studying solutions to these three questions uses functions to parameterize compact variations that preserve the enclosed volume. Consider Σ = ∂Ω(soΣistwo-sided)thatis

2 at least C . For a given variation F (x, t), we again use a scalar function u ≡Ft,N.

Instead of dealing directly with the volume of the enclosed region Ωt ≡ F (Ω,t), we deal with the signed diﬀerence in volume. See Figure 2.2 below. If ω is the volume form of N, then the signed volume change is given by V (t)= F ∗ω. (2.2.1)

[0,t]×Σ

Note, that since {(x, s)|{F (x, s) = x and s ∈ [0,t]} is compact, we have that (2.2.1) is well-deﬁned.

V = −

Figure 2.2: The V functional

Barbosa, do Carmo, and Eschenburg [1] show that

∂ V (t) = udA. (2.2.2) ∂t t=0 Σ

∈ 1 Therefore, variations u that preserve volume are precisely represented by those u C0 (Σ) such that Σ udA=0. Hence, for a hyper-surface Σ to be a local minimum among hyper-surfaces enclosing the same ∈ 1 volume, it is necessary that Σ HudA =0forallu C0 (Σ) satisfying Σ udA= 0. This represents a ﬁrst derivative test for the area functional under the constraint of using hyper-surfaces enclosing a ﬁxed volume, and it implies that the mean curvature H is constant on Σ.

9 Barbosa, do Carmo, and Eschenburg [1] show that for a critical hyper-surface Σ (a hyper-surface of constant mean curvature), one has that the second variation of the area functional is also given by (2.1.6). In earlier work on the Euclidean case, Barbosa and do Carmo [2] use the generators of homotheties u = ⟨x, N⟩ and constant functions to show that spheres are the only compact, stable local minima of the isoperimetric problem in Euclidean space. Barbosa, do Carmo, and Eschenburg

[1] later generalize this to show that if the ambient manifold is simply connected and of constant curvature, then the only compact, stable local minima are geodesic spheres.

2.3 Mean Curvature Flow and Self-Shrinkers

A smooth family of sub-manifolds Mt ⊂ N parameterized by F (x, t): M0 × R → N is said to be moving by the mean curvature flow if

 N ∂F⃗ (x, t) = −H.⃗ (2.3.1) ∂t

Note that geometrically we can only specify the normal part of the variation; we will have the same family of sub-manifolds if we change each Mt by a diffeomorphism of itself. That is, geometrically, we can not specify the tangential part of the variation. This will receive more explanation when we introduce self-shrinkers of the mean curvature flow.

The definition of the mean curvature flow that we have introduced is a parametric model.It is assumed throughout the existence of the flow that the topology is unchanging. Non-parametric models of geometric flow allow for an analysis of the mean curvature flow where the topologyis changing. These models include level-set mean curvature flow [14] and Brakke flow6 [ ]. It should be noted that these approaches are not equivalent.[22] The price we pay for using a parametric flow is that as Mt approaches a change in topology, the solution will become non-smooth as its curvature blows up.

For a compact hyper-surface, the non-existence of the flow for all time is guaranteed bythe maximum principle and the fact that a sphere of radius R will collapse by homothety to its center

R2 in time T = 2n . Therefore, for a random initial sub-manifold, it is expected that a singularity will occur during the mean curvature flow.

To understand the mean curvature flow for non-parametric models, we need to understand the behavior of the parametric model as Mt approaches the singularity. For convenience, we consider the time of the flow to be defined− [ T, 0) with a singularity at t = 0. An important type of sub-manifold

10 for understanding this behavior are the self-shrinkers of mean curvature flow.

Definition 3. A sub-manifold M ⊂ N is a self-shrinker of the mean curvature flow if

⃗xN H⃗ = . (2.3.2) 2

A self-shrinker M flows under mean curvature flow by dilations to the origin; so, ifwechoose √ M−1 = M, then Mt = −tM. Note, that the derivatives of this parameterization of the flow are not orthogonal to the tangent space of Mt, and so it is not true that ∂F/∂t is the mean curvature vector H⃗ . However, it is true that this evolution of a self-shrinker satisfies the mean curvature flow

(2.3.1).

A singularity of the mean curvature flow is defined to beof Type I if

C sup |A|2 ≤ . (2.3.3) Mt −t

In the case of a flow of compact hyper-surfaces Σt, Huisken [20] shows that if a singularity is of Type I, then the flow is asymptotic to a self-shrinker as it approaches the singular time.

Ilmanen [22] and White [38] show that in general, Mt weakly converges as varifolds to a self- shrinker in the sense of Brakke mean curvature flow. Around the singularity (x0, 0), one performs

λ parabolic blow-up Mt = λ(Mλ−2t−x0). Ilmanen[21] and White[38] both show that for every sequence

λij λi → 0, there is a subsequence λij such that Mt converges weakly as varifold flows to an integral

λij Brakke flow. Note that since Mt is a parabolic blow-up, we have that any limit is invariant under parabolic dilations. So the limit must represent a self-shrinking Brakke flow [22].

Recent progress on the uniqueness of the tangent flows arising from these parabolic blow-ups have been made by Colding-Minicozzi [11]. They show that if the first singular time has a blow up that is a multiplicity one cylinder, then the limits of the many different sequences of parabolic blow-ups must be unique. That is, all other converging parabolic blow-ups must converge to the same multiplicity one cylinder.

Let

2 −m/2 −|x−x0| /4(t−t0) ρ(x, t) = [4π(t − t0)] e (2.3.4) be the backwards m-dimensional heat kernel. Huisken [20] shows that we have a monotonicity formula    N 2 d  (⃗x − ⃗x0)  ρ(x, t) dA = − ρ(x, t) H⃗ −  dA. (2.3.5) dt  2(t − t0)  Mt Mt

11 This monotonicity formula is an important tool used by Huisken [20], Ilmanen [22], and White [38] to establish the above results on the convergence of mean curvature flows to self-shrinkers.

The m-dimensional backwards heat kernel is also related to self-shrinkers by the fact that self-

n 2 −|x|2/2m  2 shrinkers are minimal submanifolds of R for the conformal gaussian metric ds = e i dxi [10]. Thus, we see that self-shrinkers are related to the geometry of conformal changing the euclidean metric by a gaussian function.

Variations for self-shrinkers are often represented by functions defined using the euclidean metric 2 − |x| u ≡ ⟨F,NE⟩E. Using the appropriate f = e 4 in (2.1.4), one obtains that

2   ∂  −|x|2/4 2 Af (Ut) = − e u L u dA, (2.3.6) ∂t t=0 U

1 2 1 where L u = △u − 2 ∇xT u + |A| u + 2 u. Colding-Minicozzi [10] analyze the operator L and its eigenfunctions to show that the only generic self-shrinkers are spheres and cylinders Sn−k × Rk.

2.4 The Gaussian Isoperimetric Problem

n+1 2 −|x|2/2(n+1)  2 Consider Euclidean space R with a gaussian conformal metric ds = e i dxi . We n+1 −|x|2/4 have a weighted volume form on R given by ωµ = e ωE where ωE is the standard euclidean

2 volume form for Rn+1. We let µ be the weighted volume measure defined by dµ ≡ e−|x| /4 dLn+1. −|x|2/4 n n We also let Au be the weighted area measure defined by dAµ ≡ e dH where H is the n-

n+1 dimensional Hausdorff measure on R . Note that since both µ and Aµ are defined using the same weight, they can’t both be simultaneously identified as being the area and volume measures fora single conformal change of the euclidean metric.

n+1 Now, consider a region Ω ⊂ R , and let Ωϵ be the ϵ-neighborhood of Ω. Fix a weighted volume D. The Gaussian Isoperimetric Problem considers finding regions Ω of fixed weighted volume

µ(Ω) = D such that the ϵ-neighborhoods Ωϵ all have minimal weighted volume. That is, for any

′ ′ ′ other region Ω with µ(Ω ) = D, one has µ(Ωϵ) ≤ µ(Ωϵ) for small enough ϵ. Borell [5] and Sudakov & Cirel′son [36] show that the absolute minimizers of the guassian isoperimetric problem are half-spaces. That is, for 0 < D ≤ µ(Rn+1) and any given half-space S with

µ(S) = D, we have µ(Sϵ) ≤ µ(Ωϵ) for any Ω with µ(Ω) = D. Just like the Euclidean case, we may consider a local form of the gaussian isoperimetric problem.

We consider regions Ω with smooth boundary Σ = ∂Ω, and Ω minimizing the weighted volume of its ϵ-neighborhoods when compared with regions Ω′ close to Ω. First, from the formula for the first

12 2 variation of the weighted volume (2.4.2), we have that µ(Ωϵ) = µ(Ω) + Aµ(Σ)ϵ + OΣ(ϵ ). Therefore, we see that if we compare two regions Ω and Ω′ (Σ′ = ∂Ω′) of the same weighted volume, then in order for Ω to have ϵ-neighborhoods of smaller weighted volume, it is necessary that

′ Aµ(Σ) ≤ Aµ(Σ ). Hence, if Ω is a local minimizer of the gaussian isoperimetric problem, then it is necessary that Aµ(Σ) ≤ Aµ(Ft(Σ)) for all compact, weighted volume preserving variations F (t, x).

Thus, we are lead to studying the problem of finding local minimizers Σ for Aµ(Σ) and compact variations preserving weighted volume.

For determining those functions representing variations preserving weighted volume, we consider a weighted volume functional similar to (2.2.1).

Definition 4. For a given compact variation F (x, t), the gaussian weighted volume functional Vµ is defined by  ∗ Vµ(t) = F ωµ. (2.4.1)

[0,t]×Σ

In the case that Σ = ∂Ω, we have that for small |t|, Vu(t) measures the change in the weighted volume from µ(Ω) = µ(F0(Ω)) to µ(Ft(Ω)). Much like the Euclidean case [2], one has

  ∂  Vµ(t) = u dAµ. (2.4.2) ∂t t=0 Σ

From (2.1.3) we see that the first variation of Aµ is given by

    ∂  ⟨x, N⟩ Aµ(t) = H − u dAµ. (2.4.3) ∂t t=0 2 Σ

 1 Therefore, we seek hyper-surfaces such that Σ(H − ⟨x, N⟩/2)u dAµ = 0 for all u ∈ C0 (Σ) such that  Σ u dAµ = 0. This occurs exactly when H − ⟨x, N⟩/2 is a constant function on Σ. So, we are lead to the following definition.

Definition 5. A hyper-surface Σ is a critical hyper-surface to the Gaussian Isoperimetric

Problem if and only if ⟨x, N⟩ H = + C, (2.4.4) 2 where C is a constant.

For the second variation of Aµ at a critical hyper-surface of the Gaussian Isoperimetric Problem,

2 we will make use of the operators Lu = △u − (1/2)∇xT u and L u = Lu + |A| u + (1/2)u acting

2 on functions u ∈ C0 (Σ). As discussed before, these operators are also of importance in the case of

13 the self-shrinkers of the mean curvature flow. The operator L gives the second variation of Aµ for a self-shrinker [10]. In fact, we also have that if Σ is a critical point to the Gaussian Isoperimetric

Problem, then the operator L gives the second variation of Aµ for variations preserving the weighted volume. We represent compact normal variations F (x, t) of Σ by functions u ≡ ⟨Ft,N⟩. Much like the euclidean case [2], we may use (2.4.2) to show that u represents a weighted volume preserving  compact normal variation if and only if Σ u dAµ = 0.

Lemma 2. If Σ is a critical hyper-surface to the Gaussian Isoperimetric Problem and let F (x, t) be a compact volume preserving normal variation. Let suppt Ft ⊂ U ⊂ Σ, Ut = F (U, t), and u = ⟨Ft,N⟩.  Note that Σ u dAµ = 0. We have that

2   ∂  2 Aµ(Ut) = − u L u dAµ. (2.4.5) ∂t t=0 U

Therefore, we are lead to consider the stability of the operator L for the space of functions

2  u ∈ C0 (Σ) such that Σ u dAµ = 0.

Definition 6. A critical hyper-surface Σ of the Gaussian Isoperimetric Problem is said to be stable if  − u L u dAµ ≥ 0 (2.4.6) Σ

2  for all u ∈ C0 (Σ) such that Σ u dAµ = 0.

Furthermore for critical hyper-surfaces Σ, we will also be concerned with the consequences of bounds on the maximal dimension of sub-spaces of weighted volume preserving variations u where

− L is negative definite.

Definition 7. For a critical hyper-surface Σ of the Gaussian Isoperimetric Problem and a compact, connected domain Ω ⊂ Σ, the IndexU L is defined to be the maximal dimension of subspaces V

2   contained in the space {u ∈ C0 (U)| Σ u dAµ = 0} such that Q(u) ≡ − Σ u L u dAµ is negative definite.

For any increasing exhaustion Ωi of Σ by compact, connected domains, we define

IndexΣ L = sup IndexUi L . (2.4.7) Ui

Remark 1. Note, that the sequence IndexUi L is increasing since Ui ⊂ Ui+1.

n n We will also be concerned with curvature decay estimates for Σ ⊂ BR(0) ⊂ R with ∂Σ ⊂

14 ∂BR(0). Curvature estimates have many uses. For example, they allow one to estimate the size of domain for which a hyper-surface is a graph over a particular tangent plane, and they also measure in a certain sense how far a hyper-surface is from planar [9]. By a decay estimate, we mean that if

1 we fix the size of the constant C in the curvature condition H = 2 ⟨x, N⟩ and fix a reasonable area growth condition, then as R → ∞ our estimates give us that the curvature goes to |A| = 0.

In Proposition2, we show that for large enough θR (here, 0 < θ < 1) and for smooth, stable crit-

 2 ical points satisfying certain area growth bounds, that we get decay estimates for |A| dAµ. BR(0)∩Σ  The starting point for the derivation of these estimates is the stability inequality − Σ u L u dAµ ≤ 0  for Σ u dAµ = 0. Then using a De Giorgi-Moser-Nash iteration type argument, we show in Theorem 2 3 3 that for the two dimensional case of Σ ⊂ BR(0) ⊂ R , we get point-wise decay estimates for |A| on sub-balls BθR(0).

2.5 Harmonic One Forms

A harmonic one-form ω is a closed one-form (dω = 0) on a manifold M such that ω satisfies the Euler-

 2 Lagrange equations for minimizing the norm M |ω| dV . In particular is the case of constructing harmonic one-forms Ω on a two-dimensional manifold M 2.

Let τi be an orthonormal frame of one-forms on M. We have that d = τi ∧ ∇i [24]. Therefore, we see that the condition of ω being closed is equivalent to ∇ω being symmetric. If ω0 minimizes

 2  M |ω| dA in its cohomology class, then we see that it necessary that M ⟨ω, df⟩ dA = 0 for all ∞ f ∈ C0 (Σ). Therefore, we have that δω = 0. However, for one-forms we have that δ = − Div, and therefore we have that it is necessary that Tr ∇ω = 0. So, it is standard to make the following definition.

Definition 8. For a manifold M, a one-form ω on M is called a harmonic one-form if

1. ω is closed (∇ω is symmetric).

2. ω is co-closed (Tr ∇ω = 0).

For an orientable two-dimensional manifold M 2, the genus is defined to be the maximal number of embedded Jordan curves that don’t separate M into two or more open domains. The genus g of M gives lower bounds on the dimension of the subspace of harmonic one-forms [15, 23]. Using any non-trivial Jordan curve γ that doesn’t separate M, we may construct two linearly independent harmonic-one forms. First consider a neighborhood N of γ such that γ separates N into two

∞ components N1 and N2. We define a function f ∈ C (M \ γ) such that f ≡ 0 on M \ N1, and

15 γ ′ N1 ∩ N N2

N1 γ2

Figure 2.3: Set up of the Jordan curves

′ ′ such that there is a neighborhood N of γ such that f ≡ 1 on N1 ∩ N . See Figure 2.3 above. We

∞ now define the one-form ω1 ≡ df, and we note that ω1 is C0 (M). Also, note that since γ does not  separate M, there exists another Jordan curve γ2 such that ω1 ̸= 0. γ2

Using the Hodge star operator ∗, we can then construct the one-form ω2 = ∗ω1. Note that   for a Jordan curve γ3 close to γ, we have that ω2 ̸= 0 while ω1 = 0. Therefore, ω1 and γ3 γ3  2 ω2 are in different cohomology classes. So, by minimizing the norm M |ω| dV in their respective cohomology classes, we obtain two linearly independent harmonic one-forms. Furthermore, the above construction works for g disjoint non-separating Jordan curves, and we see that if M has genus g then we may construct at least 2g linearly independent L2 harmonic one-forms.

Harmonic one-forms have important applications, including their use to create lower estimates for the index of geometric operators. Palmer [31] uses harmonic one-forms to show that for compact constant mean curvature surfaces Σ ⊂ R3, one has that the operator associated to the second 2   2  variation of the energy of the gauss map N :Σ → S E(Σ) = Σ |dN| dA has index bounded below by ⌊1 + g/2⌋. Ros [33] uses harmonic one-forms to show for two-dimensional minimal surfaces

Σ (regardless of orientability) in quotients of R3, one has a lower bound for the index of the Jacobi operator depending on the genus of either Σ or the orientable double cover of Σ. This and other tools show that there are no stable two-sided minimal surfaces in R3. Urbano [37] uses harmonic one-forms to study the index of minimal surfaces immersed in quotients of S3 and S2 × R.

For studying two-dimensional self-shrinkers M 2 ⊂ Rn, instead of using harmonic one-forms, we  2 will make use of closed forms that minimize the weighted norm M |ω| dAµ in their cohomology class. Here we record the author’s work from [26] on such forms. Closed one-forms satisfying the respective Euler-Lagrange equation will be called gaussian-harmonic one-forms. The derivation of the Euler-Lagrange equation is similar to the euclidean case above, and we make the following definition.

16 Definition 9. For a manifold M ⊂ Rn, a one-form on M is called a gaussian-harmonic one- form if

1. ω is closed (∇ω is symmetric),

1 T 2. ω satisfies Tr ∇ω = 2 ω(x ).

Just like the euclidean case, we consider the effect of the associated Jacobi operator Lonthe euclidean coordinate functions of gaussian harmonic one-forms. This will allow us to link a curvature bound to the genus of the self-shrinker, and we will also link the genus of a two-dimensional self- shrinker to lower bounds on the index of L for specific curvature bounds.

17 3

Gaussian Isoperimetric Problem

Here we consider the work joint between the author and John Ross [27] on the gaussian isoperimetric problem. We consider smooth orientable hyper-surfaces Σ ⊂ Rn+1 such that Σ satisfies polynomial volume growth.

3.1 Stability of Planar Hyper-surfaces

Theorem 1. Hyper-planes are stable critical hyper-surfaces to the gaussian isoperimetric problem.

′ Proof. First, note that by rotation we may consider the plane {xn+1 = D}. If we let x = (x , xn+1), 2 2 ′ 2 − |x| − D − |x | then on the plane {xn+1 = D}, we have that e 4 = e 4 e 4 . Therefore, the stability of

{xn+1 = D} is entirely equivalent to the stability of {xn+1 = 0}. Following Kapouleas-Kleene-Møller [25], we compare the second order operator L to the harmonic oscillator. We have that

2  2   2  |x| |x| 1 − |x| L u = e 8 △ − + (n + 2) e 8 u . (3.1.1) 16 4

The operator |x|2 1 H = △ − + (n + 2), (3.1.2) 16 4 may be viewed as a shifted version of the harmonic oscillator operator H ≡ △−|x|2. Using a change

18 of variables x = 2y, we have that

1 |y|2 1 H = △ − + (n + 2), (3.1.3) 4 y 4 4 1 1 = H + (n + 2). (3.1.4) 4 y 4

The eigenvalues of Hy are well-known to be n + 2k for integers k ≥ 0, and their eigenfunctions 2 − |y| are formed by the product of Hermite polynomials with e 2 . Therefore, the eigenvalues of H are

n + 2k − n − 2 −λ = , (3.1.5) 4 k − 1 = . (3.1.6) 2

2 − |y| The lowest eigenfunction is e 2 , and so from (3.1.1) we see that the lowest eigenfunction 2 2 |x| − |y| of L is u = e 8 e 2 = 1. Therefore, the constant functions are the lowest eigenfunctions with eigenvalue −λ = −1. We see from (3.1.6) that the rest of the eigenvalues are non-negative. Note  that a variation v is weighted volume preserving ( Σ v dAµ = 0) if and only if it is orthogonal to the constant functions. Therefore, we see that the hyper-planes are stable.

3.2 Complete Non-planar Hyper-surfaces

Let V ⊂ C∞(Σ) denote the subspace of functions defined by V ≡ {⟨v, N⟩|v ∈ Rn+1 is a constant vector}. Colding-Minicozzi [10] consider functions u ∈ V for the self-shrinkers of the mean curvature

1 flow. They show that L u = 2 u. We also have that u ∈ V are eigenfunctions of L for critical hyper- surfaces of the gaussian isoperimetric problem.

Lemma 3. Let Σ be a critical hyper-surface to the gaussian isoperimetric problem, and u ∈ V .

Then, we have that 1 L u = u. (3.2.1) 2

Proof. Let v ∈ Rn+1 be a constant vector such that u = ⟨v, N⟩. Consider a point fixed point p ∈ Σ T,k  kl  and geodesic normal coordinates at p. First, note that at p, we have ∇iv = ∇i g ⟨v, ∂l⟩ =

kl T,k −g Ailu. We have that ∇ju = Ajkv . Taking another derivative, we have at p that

2 T,k T,k ∇iju = ∇iAjkv + Ajk∇iv , (3.2.2)

T,k k = ∇iAjkv − AjkAi u. (3.2.3)

19 Therefore, Codazzi’s equation gives us that

2 △u = ∇vT H − |A| u. (3.2.4)

1 1 1 T,j Since H = 2 ⟨x, N⟩ + C, we have that ∇iH = 2 ∇i⟨x, N⟩ = 2 Aijx . So, we have that ∇vT H = 1 T,i T,j 1 1 2 Aijv x = 2 ∇xT ⟨v, N⟩ = 2 ∇xT u. Therefore, we have

1 2 △u = ∇ T u − |A| u. (3.2.5) 2 x

1 Hence, we have that L u = 2 u.

Next, we record some formulas that will be useful later.

∞ ∞ Lemma 4. For any hyper-surface Σ, ϕ ∈ C0 (Σ), and f ∈ C (Σ), we have that

   2 2 2 ϕf L(ϕf) dAµ = ϕ f L f dAµ − |∇ϕ| f dAµ. (3.2.6) Σ Σ Σ

n n+1 1 If Σ ⊂ R is a critical hyper-surface for the gaussian isoperimetric problem (H = 2 ⟨x, N⟩ + C), ∞ n+1 then for any ϕ ∈ C0 (Σ) and any constant vector v ∈ R , we have that

  2 2 T ϕ |A| ⟨v, N⟩ dAµ = 2 ϕA(∇ϕ, v ) dAµ, (3.2.7) Σ Σ and

L|x|2 = 2n − |x|2 − 2C⟨x, N⟩. (3.2.8)

Remark 2. Equation (3.2.7) measures the effect of using cut-off functions to make the following for-

1  1  mal argument rigorous. Consider a function u with L u = 2 u. So, we have Σ L u dAµ = 2 Σ u dAµ.    1 2  However, formally applying integration by parts we get Σ L u dAµ = Σ Lu + 2 u + |A| u dAµ =   1 2   2 Σ 2 u + |A| u dAµ. So, formally, we have Σ |A| u dAµ = 0. The right hand side of (3.2.7) is the term we pick up from rigorously using a cut-off function.

Proof. Let us first show (3.2.6). We first work with the operator L. We get

   2 2  ϕfL(ϕf) dAµ = f ϕLϕ + 2ϕf⟨∇ϕ, ∇f⟩ + ϕ fLf dAµ, (3.2.9) Σ Σ   1  = f 2ϕLϕ + ⟨∇ϕ2, ∇f 2⟩ + ϕ2fLf dA . (3.2.10) 2 µ Σ

20 Then, using integration by parts we have

  1  = f 2ϕLϕ − f 2Lϕ2 + ϕ2fLf dA , (3.2.11) 2 µ Σ   2 2 2  = −f |∇ϕ| + ϕ fLf dAµ. (3.2.12) Σ

2 1 Then we use L = L + |A| + 2 to get (3.2.6). Now, we prove (3.2.7) for the case that Σ is a critical hyper-surface for the gaussian isoperimetric

1 problem. Therefore, we have that L⟨v, N⟩ = 2 ⟨v, N⟩. So, using integration by parts we have

1   ϕ2⟨v, N⟩ dA = ϕ2 L⟨v, N⟩ dA , (3.2.13) 2 µ µ Σ Σ   1  = |A|2 + ϕ2⟨v, N⟩ dA + ϕ2L⟨v, N⟩ dA , (3.2.14) 2 µ µ Σ Σ   1  = |A|2 + ϕ2⟨v, N⟩ dA − 2gijϕ∇ ϕ∇ ⟨v, N⟩ dA . (3.2.15) 2 µ i j µ Σ Σ

Therefore, we get

  2 2 ij ϕ |A| ⟨v, N⟩ = 2 g ϕ∇iϕ∇j⟨v, N⟩ dAµ, (3.2.16) Σ Σ  ij T = 2 g ϕ∇iϕA(v , ∂j) dAµ. (3.2.17) Σ

So we get (3.2.7).

Finally, to show (3.2.8), we note that △|x|2 = 2n − 2⟨x, N⟩H. Since Σ is a critical hyper-

1 surface of the gaussian isoperimetric problem, we have that H = 2 ⟨x, N⟩ + C. Therefore, we 2 N 2 2 T 2 have that △|x| = 2n − ⟨x, N⟩C − |x | . Also, note that ∇xT |x| = 2|x | . Hence, we get L|x|2 = 2n − ⟨x, N⟩C − |x|2.

Now, consider the subspace W ⊂ C∞(Σ) defined by

n+1 W = {d + ⟨v, N⟩|d ∈ R and v ∈ R is a constant vector}. (3.2.18)

∞ ∞ For any fixed ϕ ∈ C0 (Σ), we may consider the subspace ϕW ⊂ C0 (Σ) given by the product of functions in W with the function ϕ.

21 Also, consider the quadratic form defined by

 Q(u) = − u L u dAµ. (3.2.19) Σ

∞ We show that there exists a choice of fixed ϕ ∈ C0 (Σ) such that Q is negative definite on ϕW .

Proposition 1. Let Σ be a critical hyper-surface to the gaussian isoperimetric problem. Then,

∞ there exists ϕ ∈ C0 (Σ) such that the quadratic form Q is negative definite on ϕW where W is the sub-space of C∞(Σ) defined by (3.2.18). Furthermore, we also have Dim ϕW = Dim W .

Proof. Consider any d + ⟨v, N⟩ ∈ W with |d|2 + |v|2 = 1. For now, let us compute Q(uϕ) for any

∞ ϕ ∈ C0 (Σ). Using (3.2.6) we have that

  2 2 Q(uϕ) = − ϕ u L u dAµ + |∇ϕ| dAµ, (3.2.20) Σ Σ  1  1   = − ϕ2ud + |A|2 dA − ϕ2u⟨v, N⟩ dA + |∇ϕ|2 dA . (3.2.21) 2 µ 2 µ µ Σ Σ Σ

Then, using that u = d + ⟨v, N⟩ and grouping terms we obtain

 1  1  Q(uϕ) = − ϕ2d (d + ⟨v, N⟩) + |A|2 dA − ϕ2 (d + ⟨v, N⟩) ⟨v, N⟩ dA 2 µ 2 µ Σ Σ  2 2 + |∇ϕ| u dAµ, (3.2.22) Σ  1 1   = − ϕ2 d2 + d⟨v, N⟩ + ⟨v, N⟩2 dA − ϕ2d2|A|2 dA 2 2 µ µ Σ Σ   2 2 2 − ϕ |A| d⟨v, N⟩ dAµ + |∇ϕ| dAµ, (3.2.23) Σ Σ 1    = − ϕ2u2 dA − ϕ2|A|2d2 dA − ϕ2|A|2d⟨v, N⟩ dA 2 µ µ µ Σ Σ Σ  2 2 + |∇ϕ| u dAµ. (3.2.24) Σ

Then, we use (3.2.7) and |v|2 ≤ 1 to get

         2 2   T   ϕ |A| d⟨v, N⟩ dAµ = 2  ϕA(∇ϕ, v )d dAµ , (3.2.25)     Σ  Σ    2 2 2 2 ≤ ϕ |A| d dAµ + |∇ϕ| dAµ. (3.2.26) Σ Σ

22 Therefore, we have that

1   Q(uϕ) ≤ − ϕ2u2 dA + |∇ϕ|2 u2 + 1 dA , (3.2.27) 2 µ µ Σ Σ 1   ≤ − ϕ2u2 dA + 3 |∇ϕ|2 dA . (3.2.28) 2 µ µ Σ Σ

Next, we determine a family of cut-off functions that will be useful. We consider linear (in radius) cut-off functions defined by

  1 r ≤ R   ϕR(r) = 1 − 1 (r − R) R ≤ r ≤ 2R (3.2.29)  R   0 r ≥ 2R

where r is the euclidean distance to the origin. Observe that |∇ΣϕR| ≤ 1/R. We wish to now determine an appropriate range of R to use for our cut-off function. Note that

2 2 we may not have that Dim W = n + 1. Let ui = di + ⟨vi,N⟩ be a basis for W with di + |vi| = 1. i  i 2 Consider the set S ⊂ W defined by S ≡ {e ui| i(e ) = 1}. Note that since Dim W < ∞, we have that S is compact. Furthermore, there exists R0 such that for all u ∈ S we have that u ̸≡ 0 on

i BR0 (0). To see so, consider the case that for some sequence Rj → ∞ there exists a sequence ej with  i 2 i i i i i (ej) = 1 and ejui ≡ 0 on BRj (0). We have that ej → e∞, and e∞ui ≡ 0 on Σ. Then e∞ = 0, and we have a contradiction.

So, it is clear that for R ≥ R0, we have that Dim(ϕRW ) = Dim W , and

 2 2 ϕRu dAµ ≥ mR > 0, (3.2.30) Σ for all u ∈ S. Furthermore, mR is non-decreasing in R.

2 Finally, since Aµ(Σ) < ∞, we have that W is a finite dimensional subspace of Lµ(Σ). Therefore,  2 we have that Σ |∇ϕR| dAµ → 0 uniformly for u ∈ S as R → ∞. So, from (3.2.28), we get that for some R > R0 that Q(ϕRu) ≤ −mR/4 < 0 for all u ∈ S.

Theorem 2. Let Σ ⊂ Rn+1 be a non-planar critical hyper-surface for the gaussian isoperimetric problem with Index = I for all compact weighted volume preserving variations. Also, we require

0 ≤ I ≤ n. Then

i Σ = Σ0 × R , (3.2.31)

23 where i ≥ n + 1 − I. In particular, there are no non-planar stable solutions.

Remark 3. The restriction we put on the index is only to highlight for which values of the index does the theorem say something meaningful.

Proof. Again consider the subspace W ⊂ C∞(Σ) defined in (3.2.18). First, we consider the di-

n+1 mension of W . Let d0 be any constant function such that d0 ∈ Span{⟨v, N⟩}v∈R . Since 1 n+1 Span{⟨v, N⟩}v∈R is an eigenspace of L, we have that L d0 = 2 d0. However, since d0 is a constant 1 2 2 function, we also have that L d0 = 2 d0 + |A| d0. Hence, we have that |A| d0 ≡ 0. Since Σ is

n+1 non-planar, we have d0 = 0. Therefore, Dim W = 1 + Dim Span{⟨v, N⟩}v∈R . ∞ By Proposition1 we have for some ϕ ∈ C0 (Σ) that Dim(ϕW ) = Dim W and Q is negative  definite on ϕW . Consider the functional u → Σ u dAµ. Considering the kernel of this func- ⊥ n+1 tional and by counting dimensions, we see that Dim(ϕW ∩ 1 ) ≥ Dim Span{⟨v, N⟩}v∈R . Hence, Dim Span{⟨v, N⟩} ≤ I.

Now, consider the linear map v ∈ Rn+1 → ⟨v, N⟩ ∈ C∞(Σ) and its kernel K. Let k = Dim K. Note that Dim Span{⟨v, N⟩} = n + 1 − k and so we have that n + 1 − k ≤ I. Finally, note that Σ splits off the linear space K, and

Σ = Σ0 × K (3.2.32)

where Σ0 is a (n − k)-dimensional slice of Σ.

3.3 Stable Critical Hyper-surfaces in a Ball

Now, we consider two-sided hyper-surfaces Σ with boundary ∂Σ; specifically we consider Σ ⊂ BR(0) with ∂Σ ⊂ ∂BR(0). That is, Σ is contained in a ball centered at the origin, and ∂Σ is contained in the boundary of the ball. We require that Σ be proper immersed, but we do not directly require any volume growth conditions.

n+1 Like in the complete case, the functions {⟨v, N⟩}v∈R will play an important role. Often when working with stable minimal hyper-surfaces, it is productive to use certain test functions with a stability inequality. For the minimal case, one is free to construct any compactly supported test function that one wishes to use. For the gaussian isoperimetric problem, we are limited by the  fact that the stability inequality 0 ≥ − Σ u L u dAµ only applies to those compact variations u

n+1 that preserve weighted volume. The functions {⟨v, N⟩}v∈R will serve the purpose of aiding us in recovering a form of the stability inequality that applies to more convenient compact variations. The price we pay for a more general stability inequality is that our new inequality is more cumbersome

24 to work with.

We will need to make use of an average of the normal N that depends on a choice of cut-off

∞  function ϕ with suppt ϕ ⊂⊂ BR(0). Let ϕ ∈ C0 (Σ) and Σ ϕ dAµ ̸= 0. We let

 ϕN dAµ Σ Nϕ =  . (3.3.1) ϕ dAµ Σ

We now show a modified form of the stability inequality that applies to more convenient variations.

Lemma 5. Let Σ ⊂ Rn+1 (possibly with boundary) be a stable critical hyper-surface of the gaussian ∞  isoperimetric problem. Let ϕ ∈ C0 (Σ) (suppt(ϕ) ∩ ∂Σ = ∅) such that Σ ϕ dAµ ̸= 0. Then, we have that    2 2 2 2 2 2 ϕ |N − Nϕ| dAµ + |Nϕ| ϕ |A| dAµ ≤ 4(n + 2) |∇ϕ| dAµ. (3.3.2) Σ Σ Σ

n+1  Proof. Note that for any v ∈ R we have that Σ ϕ⟨v, N − Nϕ⟩ dAµ = 0. Therefore, we have that

Q(ϕ⟨v, N − Nϕ⟩) ≥ 0 and we use (3.2.6) to get that

  2 2 2 ϕ ⟨v, N − Nϕ⟩ L⟨v, N − Nϕ⟩ dAµ ≤ |∇ϕ| ⟨v, N − Nϕ⟩ dAµ. (3.3.3) Σ Σ

1  1 2 Now, we use that L⟨v, N⟩ = 2 ⟨v, N⟩ and L⟨v, Nϕ⟩ = 2 + |A| ⟨v, Nϕ⟩ to get that

 2 ϕ ⟨v, N − Nϕ⟩ L⟨v, N − Nϕ⟩ dAµ Σ

1   = ϕ2⟨v, N − N ⟩2 dA − ϕ2|A|2⟨v, N ⟩⟨v, N − N ⟩ dA . (3.3.4) 2 ϕ µ ϕ ϕ µ Σ Σ

We apply a Cauchy inequality to (3.2.7) to get

 1   ⟨v, N ⟩ |A|2ϕ2⟨v, N⟩ ≤ ϕ2⟨v, N ⟩2|A|2 dA + 2 |∇ϕ|2|vT |2 dA . (3.3.5) ϕ 2 ϕ µ µ Σ Σ Σ

Using (3.3.3), (3.3.4), and (3.3.5) we get

  2 2 2 2 2 ϕ ⟨v, N − Nϕ⟩ dAµ + ⟨v, Nϕ⟩ ϕ |A| dAµ Σ Σ

25   2 2 2 T 2 ≤ 2 |∇ϕ| ⟨v, N − Nϕ⟩ dAµ + 4 |∇ϕ| |v | dAµ. (3.3.6) Σ Σ

Now, summing over v for an orthonormal frame of Rn+1, we get

  2 2 2 2 2 ϕ |N − Nϕ| dAµ + |Nϕ| ϕ |A| dAµ Σ Σ

  2 2 2 ≤ 2 |∇ϕ| |N − Nϕ| dAµ + 4n |∇ϕ| dAµ. (3.3.7) Σ Σ

2 Using that |N − Nϕ| ≤ 4, we then get the lemma.

3.3.1 Integral Curvature Estimates

Now, we will use our modified version of the stability inequality (3.3.2) to derive integral decay- estimates for |A|2. These estimates depend on being in a large enough ball and on reasonable euclidean area growth conditions for the hyper-surface.

During the course of proving our estimates, we will need to make use of arbitrary constants.

We will often need to take constants of a larger size to match the statement of another lemma or proposition. Therefore, everytime we make use of arbitrary constants, we write our statements in such a way as to guarantee their validity if we increase the size of those constants.

n+1 Proposition 2. Let Σ ⊂ BR(0) ⊂ R be a critical hyper-surface to the gaussian isoperimetric 1 problem with H = 2 ⟨x, N⟩ + C. Also let 0 < θ < 1. Then, there exists a constant Dn such that if 2 2 2 −1 −DnOC n 2 2 θ R /4 θR > DnOC and the euclidean area AE(Σ ∩ BR(0)) < Dn e OC (1 − θ) R e then

 2 2 D O2 −n AE(Σ) −2 −θR2/4 |A| dA ≤ D e n C O R e . (3.3.8) µ n C (1 − θ)2 BθR(0)

Proof. For brevity and clarity of notation, throughout this proof we will use AE(Σ) ≡ AE(Σ∩BR(0)) to the be euclidean area of Σ inside BR(0). Again, we make use of a linear cut-off function:

  1 r ≤ θR   ϕ(r) = 1 − r−θR θR ≤ r ≤ R (3.3.9)  (1−θ)R   0 r ≥ R.

26 1 Note that |∇ϕ| ≤ (1−θ)R . Using the modified stability inequality (3.3.2), we see that

       2 2 2  2  2 ϕ dAµ − 2  ϕ dAµ |Nϕ| +  ϕ |A| + 1 dAµ |Nϕ| Σ Σ Σ

A (Σ) 2 2 ≤ B E e−θ R /4. (3.3.10) n (1 − θ)2R2

We see that the left hand side of (3.3.10) is quadratic in |Nϕ|. Therefore, since any quadratic

2 2 b2 au + bu + c with a > 0 satisfies au + bu + c ≥ c − 4a , we get that

 2  2  ϕ dAµ 2 Σ AE(Σ) −θ2R2/4 ϕ dAµ −  2 2 ≤ Bn 2 e . (3.3.11) ϕ (|A| + 1) dAµ (1 − θ)R Σ Σ

Combining terms on the left hand side of (3.3.11), we get

     2  2 2 ϕ dAµ ϕ |A| dAµ Σ Σ AE(Σ) −θ2R2/4  2 2 ≤ Bn 2 2 e . (3.3.12) ϕ (1 + |A| ) dAµ (1 − θ) R Σ

Cross multiplying in (3.3.12) and grouping like terms we obtain

   A (Σ) 2 2  ϕ2 dA − B E e−θ R /4 ϕ2|A|2 dA  µ n (1 − θ)2R2  µ Σ Σ

A (Σ) 2 2  ≤ B E e−θ R /4 ϕ2 dA , (3.3.13) n (1 − θ)2R2 µ Σ 2 A (Σ) 2 2 ≤ B E e−θ R /4. (3.3.14) n (1 − θ)2R2

 2 2  2 To turn (3.3.14) into an estimate for Σ ϕ |A| dAµ, we need a lower bound for Σ ϕ dAµ. To create such a bound, it is sufficient to put a lower bound on the area around the point on Σwhere the minimum for |x| is achieved. By (A.0.82), it is sufficient to provide control over |H| in a region around the point where the minimum of |x| is achieved.

Let p ∈ Σ be a point where the minimum of |x| is achieved. Note that at such a point L|x|2 ≥ 0.

27 Therefore, from (3.2.8), we have at p that

0 ≤ 2n − |p|2 − 2C⟨p, N⟩, (3.3.15)

≤ 2n − |p|2 + 2|C||p|. (3.3.16)

Therefore, we have that

 2 |p| ≤ |C| + C + 2n ≤ Dn,1OC . (3.3.17)

So, for any radius R0 such that Dn,1OC ≤ R0 ≤ 2Dn,1OC , we have that Σ ∩ BR0 (0) ̸= ∅. Since H = 1 2 ⟨x, N⟩+C, we have that |H| ≤ Dn,2OC on BR0 (0). So, by (A.0.82), we have that there is a constant 2 −1 −Dn,3OC n Dn,3 such that if Dn,3OC ≤ R0 ≤ 2Dn,3OC , then we have that A(Σ ∩ BR0 (0)) ≥ Dn,3e OC .

2  2 −1 −Dn,4OC n Therefore, there is a constant Dn,4 such that if θR > Dn,4OC , then Σ ϕ dAµ ≥ Dn,4e OC .

So, if we require θR > Dn,4OC , then (3.3.14) becomes

   −1 −D O2 n AE(Σ) −θ2R2/4 2 2 D e n,4 C O − B e ϕ |A| dA , n,4 C n (1 − θ)2R2 µ Σ

2 A (Σ) 2 2 ≤ B E e−θ R /4. (3.3.18) n (1 − θ)2R2

2 2 2 1 −1 −Dn,4OC n 2 2 θ R /4 Now, if we also require that AE(Σ) ≤ 2 Dn,4e OC (1 − θ) R e , then we obtain that

 2 −1 −D O2 n 2 2 AE(Σ) −θ2R2/4 D e n,4 C O ϕ |A| dA ≤ D e . (3.3.19) n,4 C µ n,4 (1 − θ)2R2 Σ

From (3.3.19) we have the lemma.

3.3.2 Pointwise Curvature Estimates for Three-dimensional Case

First we need to make use of a Simons’ type inequality that is true for critical hyper-surfaces in any dimension.

Lemma 6. Let Σ ⊂ Rn+1 be any critical hyper-surface of the gaussian isoperimetric problem with 1 H = 2 ⟨x, N⟩ + C. Then, LA = A + CA2, (3.3.20) and we have the estimate (at least weakly)

 C2  L|A| ≥ − 1 + |A|3. (3.3.21) 2

28 Proof. We first show (3.3.20). Consider a point p ∈ Σ and geodesic normal coordinates centered at p. From the symmetry of ∇A and the Leibniz rule for the curvature tensor R, we have that

2 2 △Ajk = ∇iiAjk = ∇ijAik, (3.3.22)

2 = ∇jiAik + RijAik, (3.3.23)

2 l l = ∇jiAik − Riji Alk − Rijk Ail. (3.3.24)

Next, using Gauss’ equation and the symmetries of ∇A, we have that

2 l l l l △Ajk = ∇jiAik + (AiiAj − AijAi )Alk + (AikAj − AjkAi )Ail, (3.3.25)

2 2 2 = ∇jkH + HAjk − |A| Ajk. (3.3.26)

⟨x,N⟩ Now, we use that H = 2 + C to get that

1 ∇ H = A(∂ , xT ). (3.3.27) k 2 k

So, using a Leibniz formula and the symmetries of ∇A, we see that

1 1 1 ∇2 H = ∇ A(∂ , xT ) + A − ⟨x, N⟩A2 , (3.3.28) jk 2 j k 2 jk 2 jk 1 1 1 2 = ∇ T A(∂ , ∂ ) + A − ⟨x, N⟩A . (3.3.29) 2 x j k 2 jk 2 jk

Using (3.3.26) and (3.3.29) we have that

⟨x, N⟩ L A = A + HA2 − A2 , (3.3.30) jk jk jk 2 jk 2 = Ajk + CAjk. (3.3.31)

Therefore, we have (3.3.20)

Next, we show (3.3.21). Note that by (3.3.20), we have that

1 LA = A + CA2 − |A|2A. (3.3.32) 2

Now, note for any function f, by a straightforward calculation we have that

Lf |∇f|2 Lf = √ − . (3.3.33) 2 f 4f 3/2

29 So, we see that

L|A|2 |∇|A|2|2 L|A| = − , (3.3.34) 2|A| 4|A|3 L|A|2 |∇|A||2 = − , (3.3.35) 2|A| |A| ⟨A, LA⟩ + |∇A|2 − |∇|A||2 = . (3.3.36) |A|

From (3.3.32), we get that

1 |A|2 + C Tr A3 − |A|4 + |∇A|2 − |∇|A||2 L|A| = 2 , (3.3.37) |A| 1 C |∇A|2 − |∇|A||2 = |A| + Tr A3 − |A|3 + , (3.3.38) 2 |A| |A| 1 ≥ |A| − |C||A|2 − |A|3. (3.3.39) 2

Then, applying a Cauchy inequality we get

 C2  L|A| ≥ − 1 + |A|3, (3.3.40) 2 which is (3.3.21).

Now, instead of using a Choi-Schoen [8] type argument as in McGonagle-Ross [27], we will obtain point-wise decay estimates for |A| based on a de Giorgi-Moser-Nash type iteration argument

[3,4, 19]. The main technical point of our argument is that we wish to create estimates that give us exponential decay on any sub-ball BθR(0) for 0 < θ < 1. To do so, we must be careful with where we put our exponential terms in each step of the proof of the estimates.

Next, we put the Michael-Simon Sobolev inequality for hyper-surfaces [28] into a form suitable

2 3 for working with the weighted area measure dAµ for Σ ⊂ R .

Lemma 7. Let Σ2 ⊂ R3 be any critical hyper-surface of the gaussian isoperimetric problem, and let 1 f ∈ C0 (Σ ∩ Br(0)). Then, we have a Sobolev-type inequality:

 2  2  2   2 r 2 2 4        f dAµ ≤ De  |∇f| dAµ + 1 + r + C  |f| dAµ  . (3.3.41)

Br (0) Br (0) Br (0)

Here, D is a constant independent of both f and Σ.

Proof. The Michael-Simon Sobolev inequality[28] for hyper-surfaces Σ2 ⊂ R3 states that for any

30 1 g ∈ C0 (Σ), we have that

1   2       2     g dA ≤ D1  |∇g| dA + |gH| dA . (3.3.42)

Br (0) Br (0) Br (0)

1 −|x|2/8 Now, for any f ∈ C0 (Σ ∩ Br(0)), consider the function g = fe . Putting g into (3.3.42) we get

1   2       2  |x| 2  f 2 dA  ≤ D  |∇f|e|x| /8 dA + |f| + |H| e|x| /8 dA  , (3.3.43)  µ 1  µ 4 µ Br (0) Br (0) Br (0)  

r2   8   ≤ D2e  |∇f| dAµ + (1 + r + |C|) |f| dAµ . (3.3.44)

Br (0) Br (0)

Squaring both sides and using another estimate gives us inequality (3.3.41).

Now we may use the Simons inequality (3.3.21) and integration by parts to give integral estimates for |∇ϕ|A|k|2. For clarity of notation, we will use u ≡ |A|.

For clarity of keeping track of orders of terms, we will also make use of the following notation:

Or ≡ 1 + r, (3.3.45)

OC ≡ 1 + |C|, (3.3.46)

Oϕ ≡ 1 + ∥ϕ∥C1 . (3.3.47)

Here, ∥ϕ∥C1 ≡ sup |ϕ| + sup |∇ϕ|.

Lemma 8. Let Σ2 ⊂ R3 be a critical hyper-surface to the gaussian isoperimetric problem with ⟨x,N⟩ 1 H = 2 + C. Also, let ϕ ∈ C0 (Σ), k ≥ 1, and u ≡ |A|. Then we have that

   C2   |∇(ϕuk)|2 dA ≤ |∇ϕ|2u2k dA + k 1 + ϕ2u2k+2 dA . (3.3.48) µ µ 2 µ Σ Σ Σ

Proof. First, we note that

   k 2 2 2k 2 2 2k−2 2 |∇(ϕu )| dAµ = |∇ϕ| u dAµ + k ϕ u |∇u| dAµ Σ Σ Σ

 2k−1 + 2k ϕu ⟨∇ϕ, ∇u⟩ dAµ. (3.3.49) Σ

31 Next, we note that

⟨∇(ϕ2u2k−1), ∇u⟩ = 2ϕu2k−1⟨∇ϕ, ∇u⟩ + (2k − 1)ϕ2u2k−2|∇u|2. (3.3.50)

Using (3.3.49) and (3.3.50), we get that

   k 2 2 2k  2 2 2k−2 2 |∇(ϕu )| dAµ = |∇ϕ| u dAµ + k − k ϕ u |∇u| dAµ Σ Σ Σ

 2 2k−1 + k ⟨∇(ϕ u ), ∇u⟩ dAµ. (3.3.51) Σ

Now, for integers k ≥ 0, we have k − k2 ≤ 0. So using this, integration by parts, and the

Simons-type inequality (3.3.21), we have that (3.3.51) becomes

   C2   |∇(ϕuk)|2 dA ≤ |∇ϕ|2u2k dA + k 1 + ϕ2u2k+2 dA . (3.3.52) µ µ 2 µ Σ Σ Σ

Now, we may use the Sobolev inequality (3.3.41) and the inequality (3.3.48) from Lemma8 to

p get better estimates. Here, we will find it convenient to use ∥f∥p to be the weighted L norm

1   p  p ∥f∥p =  f dAµ , (3.3.53) Σ and on a ball Br(0), 1   p   p  ∥f∥p,r =  f dAµ . (3.3.54)

Σ∩Br (0)

Lemma 9. Let Σ2 ⊂ R3 be a critical hyper-surface to the gaussian isoperimetric problem with ⟨x,N⟩ 1 H = 2 + C. Also, let u ≡ |A| and ϕ ∈ C0 (Σ ∩ Br(0)). We have that

r2 k 4 4 k 2  k 2 2 2 k 2 2 k+1 2 ∥ϕu ∥4 ≤ De ∥ϕu ∥2 ∥u ∇ϕ∥2 + Or OC ∥ϕu ∥2 + OC k∥ϕu ∥2 . (3.3.55)

Here, D is independent of Σ and ϕ.

32 Proof. We use f = ϕ2u2k in the Sobolev inequality (3.3.41) to get that

 2  2 2   k 4 r k k 2 2 2 2k 4      ∥ϕu ∥4 ≤ D1e  |ϕ|u |∇(ϕu )| dAµ + (1 + r + C )  ϕ u   . (3.3.56)

Br (0) Br (0)

Using Holder’s Inequality, we have that

r2 k 4 4 k 2  k 2 2 2 k 2 ∥ϕu ∥4 ≤ D1e ∥ϕu ∥2 ∥∇(ϕu )∥2 + Or OC ∥ϕu ∥2 . (3.3.57)

Then, using (3.3.48), we get that

r2 k 4 4 k 2  k 2 2 k+1 2 2 2 k 2 ∥ϕu ∥4 ≤ D2e ∥ϕu ∥2 ∥u ∇ϕ∥2 + OC k∥ϕu ∥2 + Or OC ∥ϕu ∥2 . (3.3.58)

p Now, we establish a recursion relation for some of the Lµ(Σ) norms on decreasing balls.

2 3 Lemma 10. Let Σ ⊂ BR(0) ⊂ R be a critical hyper-surface to the gaussian isoperimetric problem ⟨x,N⟩ with H = 2 + C and ∂Σ ⊂ ∂BR(0). Also, let u ≡ |A|.

There exists a constant D such that if Br(0) ⊂ Br+h(0) ⊂ BR(0), then we have that

1 1 1 3 2   2k  2  4k 1 (r+h) 1 3 (r+h) 3 ∥u∥ ≤ D 4k O 2k O 4k e 16k 1 + 1 + k 2 e 8 ∥u∥ ∥u∥ . (3.3.59) 4k,r r C h 6,r+h 2k,r+h

2 (r+h) 3 Remark 4. Note that in (3.3.59), we have left an exponential order e 8 with ∥u∥6,r+h. The 3 importance of this is that later we will show that ∥u∥6,r+h may be estimated using ∥u∥2,R, and this 2 (r+h) 3 estimate will cancel some of the exponential order in e 8 ∥u∥6,r+h. This order cancellation is necessary to obtain a final point-wise estimate with exponential decay for BθR(0) and any 0 < θ < 1.

Proof. First, consider a cut-off function ϕ with ϕ ≡ 1 on Br(0), ϕ ≡ 0 on BR(0) \ Br+h(0), and

2 |∇ϕ| ≤ h . Using Holder’s inequality, we may estimate

1   3  k+1 2 k 2  6  ∥ϕu ∥2 ≤ ∥ϕu ∥3  u dAµ . (3.3.60)

Br+h(0)

33 Using log-concavity interpolation for Lp spaces, we get that

2 4 k+12 ≤ 2 k 3 k 3 φu 2 u 6,r+h φu 2 φu 4 . (3.3.61)

3 3 ≤ 2 2 a 2 b3 So using equation (3.3.61) and a Cauchy -inequality of the form ab 3 + 33 ,wehavethat

2 2 (r+h) k k 3(r+h) 3 k 1 k D e 4 φu 2O2 kφu +12 ≤ D e 8 O3 k 2 u3 φu 4 + φu 4. (3.3.62) 1 2 C 2 2 C 6,r+h 2 2 4

So, then (3.3.55)givesusthat

2 2 k (r+h) 3 (r+h) k 4 ≤ O2O3 O2 4 2 8 3 4 φu 4 D3 r C φe 1+k e u 6,r+h φu 2. (3.3.63)

1 Then raising to the 4k power we get (3.3.59).

We may now use iteration to prove an L∞ bound.

2 3 Proposition 3. Let Σ ⊂ BR(0) ⊂ R be a critical hyper-surface to the gaussian isoperimetric x,N ⊂ ≡| | problem with H = 2 + C and ∂Σ ∂BR(0). Also, let u A .

There exists a constant D such that for any Br(0) ⊂ BR(0), we have that

3 1 1 2 2 2 R2 3 2 1 r + R 2 sup |A|≤DORO 1+ e 8 ( 3 3 ) 1+e 16 u u ,R. (3.3.64) C − 6,R 2 Br (0) R r

Proof. We deﬁne a sequence ri, ki,andhi for i =0, 1, 2, ... in order to create a decreasing sequence

−i−1 of balls between Br(0) and BR(0). We let r0 = R and ri+1 = ri − 2 (R − r). Note that r

−i−1 i hi =2 (r − R). Finally, take ki =2. See Figure 3.1 below.

i BR ki =2

Br r2 r1 r0

h2 h1

Figure 3.1: Set up of the iteration

34 We then have that (3.3.59) gives us that

∞  1     j 1 3  R2 3  2 2 4 1 4 j=0 ∥u∥ i+2 ≤ ∥u∥2,R D1O O 1 + √ 1 + e 32 ∥u∥ 2 ,r R C R − r 6,R

 −i 2  ∞ 1 (r+2 (R−r)) 1  16 2i i+1 × e (1 + 2 ) 2i . (3.3.65) i=0 We have that

 −i 2  ∞ 1 (r+2 (R−r)) 1 10 2 8 8 2  16 2i r + rR+ R e = e 16 ( 21 21 7 ), (3.3.66) i=0

1 1 r2+ 2 R2 ≤ e 8 ( 3 3 ). (3.3.67)

Also, note that   ∞ ∞ i+1  i+1 1  log(1 + 2 ) log (1 + 2 ) 2i = , (3.3.68)   2i j=0 j=0

log(1+2i+2) log(1+2x) which converges by a simple ratio test. For this note that lim i+1 = lim = 1. i→∞ log(1+2 ) x→∞ log(1+x) Therefore, we get that

  3 1 1 1 2 2 2  R2 3  2 ( r + R ) 2 ∥u∥ i+2 ≤ D O O 1 + e 8 3 3 1 + e 16 ∥u∥ ∥u∥ . (3.3.69) 2 ,r 2 R C R − r 6,R 2,R

Taking i → ∞ we obtain an L∞ norm.

6 4 To get complete estimates, we will need to estimate the Lµ and Lµ norms of u. Note that 3 4 6 plugging k = 1 and k = 2 into (3.3.55) won’t give us direct estimates for Lµ and Lµ in terms of 2 lower powers. To get around this, we make an assumption on the smallness of the Lµ norm of u. 4 First, we find an estimate for Lµ.

Lemma 11. Let Σ2 ⊂ R3 be a critical hyper-surface to the gaussian isoperimetric problem with ⟨x,N⟩ 1 H = 2 +C. Also let R ≥ 0 and ϕ ∈ C0 (Σ∩BR(0)). Then, there exists a constant D (independent of Σ, C, and R) such that if

R2  2 4 2 DOC e u dAµ ≤ 1, (3.3.70)

BR(0)

35 then we have the estimates

  4 4 2 2 2 2 ϕ u dAµ ≤ OϕOR ϕ u dAµ, (3.3.71)

BR(0) BR(0)   2 2 2 2 2 2 |∇(ϕ u)| dAµ ≤ 5OϕOROC u dAµ. (3.3.72)

BR(0) BR(0)

Remark 5. One of the purposes of the sufficient condition (3.3.70) is to cancel orders of exponential

2 2 2 R − θ R terms. However, note that the order of e 4 is of opposite order of e 4 in (3.3.8) as θ → 1. It is for this reason, that we can not increase the order of the exponential term in (3.3.70) if we want reasonable sufficient conditions in our theorems for any 0 < θ < 1.

Proof. Put f = ϕ2u2 into the Sobolev inequality (3.3.41) to get

 2  2 2   4 R 2 2 2 2 4      ∥ϕu∥4 ≤ D1e  ϕu|∇(ϕu)| dAµ + OROC  ϕ u dAµ  . (3.3.73)

BR(0) BR(0)

When we use (3.3.48) to estimate our derivative terms, we will inevitably need to deal with u4 appearing on the right hand side of our estimate. In order to absorb the u4 term into the left hand side, we will need to make sure the power of ϕ matches. So, we use that ϕu|∇(ϕu)| ≤ u|∇(ϕ2u)| + ϕ|∇ϕ|u2. Therefore, by Holder’s inequality we have

    2    4 R 2 2 2 2 2 2 2 2 4     ∥ϕu∥4 ≤ D2e  u dAµ  |∇(ϕ u)| dAµ + OϕOROC ϕ u dAµ . (3.3.74)

BR(0) BR(0) BR(0)

Now, from (3.3.48), we have that

   2   2 2 2 2 2 C 4 4 |∇(ϕ u)| dA ≤ 4∥ϕ∥ 1 ϕ u dA + 1 + ϕ u dA . (3.3.75) µ C µ 2 µ BR(0) BR(0) BR(0)

Therefore, we get that

    2    4 2 R 2 4 4 2 2 2 2 4     ∥ϕu∥4 ≤ D3OC e  u dAµ  ϕ u dAµ + OϕOR ϕ u dAµ . (3.3.76)

BR(0) BR(0) BR(0)

36 Therefore, we see that there is a constant D4 such that if

R2  2 4 2 −1 OC e u dAµ ≤ D4 , (3.3.77)

BR(0) then we have that  4 2 2 2 2 ∥ϕu∥4 ≤ OϕOR ϕ u dAµ. (3.3.78)

BR(0)

This gives us (3.3.71).

Then, using (3.3.75), we get that

  2 2 2 2 2 2 2 |∇(ϕ u)| dAµ ≤ 5OϕOROC ϕ u dAµ. (3.3.79)

BR(0) BR(0)

This gives us (3.3.72).

6 8 4 Next, we have an estimate that compares Lµ and Lµ norms to Lµ norms.

Lemma 12. Let Σ2 ⊂ R3 be a critical hyper-surface to the gaussian isoperimetric problem with ⟨x,N⟩ 1 H = 2 + C. Also let R ≥ 0 and ϕ ∈ C0 (Σ ∩ BR(0)). There exists a constant D such that we have the estimates

 R2  R2  4 6 2 6 8 8 2 ϕ u dAµ ≤ DOC ∥u∥4,Re OϕOR + e ∥u∥4,R , (3.3.80)

BR(0)

 R2  R2  4 8 4 4 8 2 2 4 4 ϕ u dAµ ≤ DOC e ∥u∥4,R OϕOR + e ∥u∥4,R . (3.3.81)

BR(0)

Proof. Using Holder’s inequality we have that

1 1   2   2    4 6  4 4   4 8  ϕ u dAµ ≤  ϕ u dAµ  ϕ u dAµ , (3.3.82)

BR(0) BR(0) BR(0)

2 2 2 = ∥ϕu∥4∥ϕu ∥4. (3.3.83)

Now, we use (3.3.55) to get

R2 2 4 2 4 2 2  2 2 2 2 3 2 ∥ϕu ∥4 ≤ D1OC e ∥ϕu ∥2 OϕOR∥u ∥2,R + ∥ϕu ∥2 . (3.3.84)

37 3 2 2 2 2 Using Holder’s inequality, we have that ∥ϕu ∥2 ≤ ∥u∥4,R∥ϕu ∥4. From a Cauchy ϵ-inequality of a2 b2 the form ab ≤ ϵ 2 + 2ϵ we obtain that

2 2 2 R 2 2 2 2 2 4 R 2 4 4 1 2 4 D O e 4 ∥ϕu ∥ ∥u∥ ∥ϕu ∥ ≤ D O e 2 ∥ϕu ∥ ∥u∥ + ∥ϕu ∥ . (3.3.85) 1 C 2 4,R 4 2 C 2 4,R 2 4

Therefore, from (3.3.84) we get that

R2  R2  2 4 4 8 4 2 2 4 4 ∥ϕu ∥4 ≤ D3OC ∥u∥4,Re OϕOR + e ∥u∥4,R . (3.3.86)

This gives us (3.3.81).

Then, using (3.3.83), we have that

 R2  R2  4 6 2 6 8 8 2 ϕ u dAµ ≤ D4OC ∥u∥4,Re OϕOR + e ∥u∥4,R . (3.3.87)

BR(0)

Now we may combine Proposition2 and Proposition3 to obtain pointwise curvature estimates for |A|.

3 Theorem 3. Let Σ ⊂ BR(0) ⊂ R be a smooth stable critical hyper-surface of the gaussian isoperi- 1 metric problem with H = 2 ⟨x, N⟩ + C and ∂Σ ⊂ ∂BR(0). Consider any 0 < θ < 1. We have that there exists a constant D (independent of Σ, θ, and R) such that if θ(1 − θ)R > DOC and 2 2 (1−θ)R −DOC n 2 2 AE(Σ) < De OC (1 − θ) R e 16 , then

1 2 1 1 2 2 2 D4OC −4 4 − 32 (1−θ )R sup |A| ≤ DOC e (1 − θ) AE(1 + AE)ORe . (3.3.88) BθR(0)

−1 −1 −1 Proof. First, we restrict ourselves to R ≥ D1 so that R ≤ D1OR .

θ+1 θi+1 Consider θ1 = 2 , and similarly recursively define θi+1 = 2 . Note that θ ≤ θ1 ≤ θ2 ≤ ... ≤ 1. −i −i−1 We also have that 1 − θi = 2 (1 − θ), and θi+1 − θi = 2 (1 − θ). See Figure 3.2 below.

We first apply Proposition3 to the balls BθR(0) ⊂ Bθ1R(0). We have that

3 θ2R2 3 −1 1 1 θ2+ 2 θ2 R2 1 sup |A| ≤ D O O 2 (1 − θ) e 8 ( 3 3 1 ) (1 + e 16 ∥u∥ 2 )∥u∥ (3.3.89) 2 R C 6,θ1R 2,θ1R BθR(0)

38 BR

BθR

θ1R

θ2R θ3R

Figure 3.2: Set up of θiR

3 2 We need to estimate u 6,θ1R. From Lemma 12,wehavethat ) * 1 2 4 2 3 1 3 θ2R 1 1 θ2R 1 u 2 ≤ D O 2 u 2 e 32 1+ O 4 + e 32 u 2 . (3.3.90) 6,θ1R 3 C 4,θ2R 1 − θ R 4,θ2R

1 2 Now, we want to use Lemma 11 to estimate u 4,θ2R. To do so, we need to guarantee the suﬃcient condition, 2 2 θ3 R 2 4 2 D4OC e u dAμ ≤ 1, (3.3.91) B θ3R(0)

2 of Lemma 11.Toguarantee(3.3.91) is true, we want to estimate u 2,θ4R using Proposition

2. To apply Proposition 2, it is suﬃcient that (1 − θ4)θ4R>D5OC and AE(Σ ∩ BR(0)) < 2 2 2 − −D O2 (θ4 −θ3 )R 1 5 C O2 − 2 2 16 D5 e C (1 θ) R e . This gives us conditions to impose on Σ. So, from Proposition 2,wehavethat(3.3.91) becomes

2 2 2 2 θ3 R θ3 R 4 2 4 2 D6OC e u dAμ ≤OC e u dAμ, (3.3.92) B B θ3R(0) θ4R(0) 2 2 2 2 ∩ 2 (θ −θ )R D O − AE(Σ Bθ3R(0)) − − 3 4 ≤ D e 6 C O 1 θ 2R 2e 4 , (3.3.93) 6 C (1 − θ)2 4 2 2 2 2 (θ3 −θ4 )R D7Oc 2 2 8 ≤ D7e OC (1 − θ) R e , (3.3.94)

≤ 1. (3.3.95)

2 − 2 ≤−29 − 2 For the last inequality, we used that θ3 θ4 16 (1 θ) and the conditions we impose on Σ with 2 a choice of larger constant D7. Note that we have used Proposition 2 to estimate u 2,θ4R instead 2 of u 2,θ3R so that we get a negative exponential in (3.3.94).

39 1 −1 1 1 So, now we may apply Lemma 11 to get ∥u∥ 2 ≤ D (1 − θ) 4 O 4 ∥u∥ 4 . Hence, we have 4,θ2R 8 R 2,θ3R that (3.3.90) becomes

3 1 θ2R2 3  1 θ2R2  −1 2 2 ∥u∥ 2 ≤ D O 2 (1 − θ) O e 32 ∥u∥ 4 1 + ∥u∥ 4 e 32 . (3.3.96) 6,θ1R 9 C R 2,θ3R 2,θ3R

Thus far, the conditions we have imposed on Σ are sufficient to allow us to apply Proposition2 to 2 2 1 2 −1 −1 1 −1 −θ R 4 D O 4 4 3 get the estimate ∥u∥ ≤ D e 10 C O (1 − θ) 4 A R 4 e 32 . 2,θ3R 10 C E So, using (3.3.96), we then have that

3 1 −θ2R2 3 1 D O2 −2 3   ∥u∥ 2 ≤ D e 11 C (1 − θ) O 4 e 16 A 4 1 + A 4 . (3.3.97) 6,θ1R 11 R E E

So,

θ2R2 3 1 3 1 1 D O2 −2   e 16 ∥u∥ 2 ≤ D e 12 C (1 − θ) O 4 A 4 1 + A 4 . (3.3.98) 6,θ1R 12 R E E

Therefore, (3.3.89) becomes

3 2 5 1 1 2 2 2 2 2 D13OC −3 4 8 ( 3 θ + 3 θ1 )R sup |A| ≤ D13OC e (1 − θ) OR(1 + AE)e ∥u∥2,θ1R. (3.3.99) BθR(0)

Now, note that ∥u∥2,θ1R ≤ ∥u∥2,θ3R, and so we have that

1 2 1 1 1 2 2 2 2 2 2 D14OC −4 4 8 ( 3 θ + 3 θ1 −θ3 )R sup |A| ≤ D14OC e (1 − θ) AE(1 + AE)ORe , (3.3.100) BθR(0)

1 2 1 1 2 2 2 2 D14OC −4 4 24 (θ −θ3 )R ≤ D14OC e (1 − θ) AE(1 + AE)ORe . (3.3.101)

θ2−θ3 R2 ( 4 2 ) (1−θ)R2 1 16 16 Now, note that θ4(1 − θ4) > 16 θ(1 − θ), and that e > e . So therefore, if we meet the sufficient conditions of the theorem, then we meet the sufficient conditions needed toapplythe lemmas that we have used throughout the course of the proof. Also, for simplification, note that

1  2 2 1 2 24 θ − θ3 ≤ − 32 (1 − θ ).

40 4

Gaussian Harmonic One Forms

Now, we consider results by the author [26] concerning the consequences of the construction of

Gaussian Harmonic one-forms on a self-shrinker M 2 ⊂ Rn. Gaussian Harmonic One-Forms are  2 closed one-forms satisying the Euler Lagrange equation for minimizing the weighted norm |ω| dAµ Σ in a cohomology class.

Definition 10. A one-form ω is a Gaussian Harmonic One-Form if

• ω is closed (∇ω is symmetric; i.e. ∇iωj = ∇jωi).

1 T • The trace of ∇ω satisfies ∇iωi = 2 ω(x ) where x is the position vector in euclidean space.

2 n 2 For M ⊂ R , we will use Hµ(M) to denote the space of Lµ(M) gaussian harmonic one-forms on M.

Much like the case of harmonic one-forms on Riemann surfaces, we have that the genus of a

2 surface Σ gives us a lower bound on the dimension of the Lµ Gaussian Harmonic One-forms that may be constructed.

Lemma 13. Let M 2 ⊂ Rn be a sub-manifold with polynomial volume growth and genus g. Then dim Hµ(M) ≥ g.

Proof. The proof is similar to that of the euclidean case [15, 23]. First consider g disjoint non- separating Jordan curves γi ⊂ M. There exists dual forms ηi to each γi such that their supports   are in small neighborhoods of each γi, and for any one-form ω we have that ω = ηi ∧ ω. Note γi Σ that these integrals are just integrals of differential forms and do not involve dAµ.

Since the γi are non-separating, we may find Jordan curves αi such that αi intersects γi exactly once, and we also have that αi does not intersect γj for i ̸= j. One may use the dual forms τi to αi

41 ∞ to show that the ηi are linearly independent. Furthermore, for any f ∈ C0 (M) we have that ηi + df are linearly independent.

Now, we consider the corresponding weak versions of the definition for ω being a Gaussian

2 ∞ ⊥ Harmonic One-Form. Let A ⊂ Lµ(M) be the closed subspace A = Span{df : f ∈ C0 (M)} (A is 1 T 2 the space of forms that weakly satisfy ∇iωi = 2 ω(x )), and let B ⊂ Lµ(M) be the closed subspace 1 T ∞ B = Span{δψ + 2 Inn(x )ψ : ψ is a C0 2-form on M} (B is the space of weakly closed forms). Note 1 T 2 that δ + 2 Inn(x ) is the adjoint of d for Lµ(M). Hence, we see that A and B are orthogonal in 2 ⊥ 2 Lµ(M). Let H0 = (A ⊕ B) . We see that H0 is the space of Lω(M) forms that are weakly Gaussian Harmonic.

Since, the ηi are closed, we see that ηi ∈ H0 ⊕ A. Let ωi be the projection of ηi onto H0. Since

∞   we have for any f ∈ C0 (M) that τi ∧ (ηj + df) = ηj + df = δij, we see that the ωi are linearly M αi independent. Note that since the suppt τi is compact, we have that the above relation still holds in

2 the Lµ(Σ) limit.

Therefore, Span{wi} is a g-dimensional space of weakly Gaussian Harmonic One-Forms. So,

∞ once we show that the ωi are actually C , then we are done.

∞ ∞ Since ωi ∈ H0, we have for any C0 function f and C0 two-form ψ that

 0 = ⟨ωi, df⟩ dAµ, (4.0.1) M  1 0 = ⟨ω , δψ + Inn(xT )ψ⟩ dA . (4.0.2) i 2 µ M (4.0.3)

|x|2 |x|2 Plugging in f → fe 4 in (4.0.1) and ψ → ψe 4 into (4.0.3), we may rewrite both equations as

 1 0 = ⟨ω , df + fd|x|2⟩ dA. (4.0.4) i 4 M  0 = ⟨ωi, δψ⟩ dA. (4.0.5) M

Now, we use local conformal flat coordinates (u, v). Note that (4.0.4) and (4.0.5) are conformally

∞ invariant. Let ωi = p du + q dv. For any ϕ ∈ C0 (M) supported on the coordinate chart, let f = ϕv

42 and ψ = ϕu du ∧ dv. Then (4.0.4) and (4.0.5) become

  ∂ |x|2 ∂ |x|2  0 = pϕ + qϕ + u pϕ + v qϕ du dv, (4.0.6) uv vv 4 v 4 v M  0 = (qϕuu − pϕuv) du dv. (4.0.7) M

Adding these two equations we obtain

  ∂ |x|2 ∂ |x|2  0 = q△ ϕ + u pϕ + v qϕ du dv (4.0.8) uv 4 v 4 v M

Now, using f = ϕu and ψ = ϕv du ∧ dv, we get

  ∂ |x|2 ∂ |x|2  0 = p△ ϕ + u pϕ + v qϕ du dv. (4.0.9) uv 4 u 4 u M

Equations (4.0.8) and (4.0.9) give us a weakly linear elliptic system for p and q. From standard linear elliptic system theory, we have that p and q must be C∞ on M [16, 13, 30].

Now, we consider some identities for Gaussian Harmonic Forms.

Lemma 14. Let ω be a Gaussian Harmonic Form on a self-shrinker M 2 ⊂ Rn with dual vector field W ∈ TM. For any v ∈ TpM, we have that

1 (LΣ ω)(v) = ω(v) − Aα(W, e )Aα(e , v). (4.0.10) 2 i i

Proof. We first use a Weitzenbock formula △M ω = −(dδ + δd)ω + Kω. Here, −(dδ + δd) is the

Hodge Laplacian on differential forms while K is the scalar curvature of M [24]. We have that δω =

1 T 1 T 1 Σ T − Div ω = − 2 ω(x ), and dω = 0. So the Hodge Laplacian −(dδ +δd)ω = 2 d[ω(x )] = 2 ∇ [ω(x )]. So, from a Leibniz rule, we get that

1 1 (△M ω)(v) = Kω(v) + ω(∇M xT ) + ∇M ω(xT , v). (4.0.11) 2 v 2

M T H For a self-shrinker, we have that ∇v x = v − 2A (v, ei)ei. So, we get

1 (LM ω)(v) = Kω(v) + ω(v) − AH (v, W ). (4.0.12) 2

43 Now, let {ηα} be an orthonormal frame for the normal bundle. Let καi be the eigenvalues of

ηα H α α A with orthonormal eigenvectors eαi. Note that A = HαA = (κα1 + κα2)A . Here we summed over different orthonormal frames in TpM for each α. Now, let Wαi be the components of W in the frame {eαi} for fixed α. Using the Gauss equation, we have that

 H  KW − A (W, ei)ei = [κα1κα2Wαieαi − (κα1 + κα2) καiWαieαi] , (4.0.13) i α,i

 2 = −καiWαieαi, (4.0.14) α,i

α α = −A (W, ei)A (ei, ej)ej. (4.0.15)

Note, that in (4.0.15), we are able to turn the sum over eαi into a sum over any frame ei, because the trace of a tensor is independent of the choice of orthonormal frame. From here, we have (4.0.10)

Now, we consider the differential operator LE acting on vector fields on M viewed as vector fields in Rn. During our calculations, we will often need to make use of vectors and coordinates in euclidean space. We will use indices such as a, b, ... to denote euclidean vectors or components.

n For example, the vectors {∂a} are an orthonormal basis in R , and for any vector v we may write a v = v ∂a.

Lemma 15. Let ω be a Gaussian Harmonic One-Form on a self-shrinker M 2 ⊂ Rn. Let W be the vector field dual to ω. We have that

1 LE W = −2⟨∇M ω, Aα⟩η + W − 2Aβ(W, e )Aβ(e , e )e . (4.0.16) α 2 j j k k

b Proof. We note that W = ⟨ω, dx ⟩∂b. So therefore we have that

E M b M M b M b L W = ⟨L ω, dx ⟩∂b + 2⟨∇ ω, ∇ dx ⟩∂b + ⟨ω, L dx ⟩∂b. (4.0.17)

We need to compute the three terms on the right hand side of (4.0.17). First, we examine the quantity

M M b M T M M b b α ⟨∇ ω, ∇ dx ⟩∂b. We see that ∇ ∂a = −⟨∂b, ηα⟩∇ ηα. Hence, we have that ∇ dx = −ηαA . So, we get

M M b M α ⟨∇ ω, ∇ dx ⟩∂b = −⟨∇ ω, A ⟩ηα. (4.0.18)

M b Next, we compute ⟨ω, L dx ⟩∂b. We fix a point p ∈ M. Then, we use a geodesic tangential frame

M N {ei} and normal frame {ηj} such that ∇ ei(p) = 0 and ∇ ηj(p) = 0. From the Codazzi Equation,

44 we have that

M b α T α  N b (△ dx )(ei) = −A (∂b , ej)A (ej, ei) − ∇j H , (4.0.19) ηb = −Aα(∂T , e )Aα(e , e ) − α Aα(xT ,W ). (4.0.20) b j j i 2

So, we see that we have that

M b α α ⟨ω, L dx ⟩∂b = −A (W, ej)A (ej, ek)ek. (4.0.21)

So, using equations (4.0.10), (4.0.18), and (4.0.21) we then get (4.0.16).

Now we prove a result that is like a rigidity theorem for the genus of M.

Theorem 4. If M 2 ⊂ RN is an orientable self-shrinker of polynomial volume growth with genus g ≥ 1, then 1 sup Aα(v, i)Aα(i, v) ≥ . (4.0.22) x∈M,v∈TxM,|v|=1 2

Proof. From Lemma 13, we have that there are g linearly independent gaussian harmonic forms in

2 Lµ(M). 2 Consider any Gaussian Harmonic Form ω ∈ Lµ(M) with dual (with respect to the euclidean ∞ metric on M) vector field W . Consider any ϕ ∈ C0 (M). From a calculation similar to the proof of (3.2.6) in Lemma4, we have that

 M 2 0 ≤ |∇ (ϕW )| dAµ, (4.0.23) M   2 2 M = |W | |∇ϕ| dAµ − ⟨W, L W ⟩ dAµ. (4.0.24) M M

Then, using (4.0.10), we get that

  1  0 ≤ |W |2|∇ϕ|2 dA + ϕ2Aα(W, i)Aα(i, W ) dA − ϕ2|W |2 dA . (4.0.25) µ µ 2 µ M M M

α α 2 For clarity of notation, let S = sup A (v, i)A (i, v). Since W ∈ Lµ(M), we may use x∈M,v∈TxM,|v|=1

45 standard cut-off functions exhausting M and with |∇M ϕ| ≤ 1 to show that (4.0.25) gives us that

 1  0 ≤ S − |W |2. (4.0.26) 2 M

2 As we remarked earlier, since the genus g ≥ 1, we have that the space of Lµ(M) gaussian harmonic 1 one-forms is non-zero. Therefore, we get that S ≥ 2 .

Now, using methods of Ros [33] and Urbano [37], we show that if the principal curvatures of M are not too far from each other in absolute value, then we have a lower bound for the index of L

∞ acting on C0 functions on M.

Proof. We may assume IndexM (L) = J < ∞. From Fischer-Colbrie [17], we know that there exist

2 ∞  Lµ(M) functions ψ1, ..., ψJ such that if f ∈ C0 (M) and fψi dAµ = 0 for all i, then we have that M  − fLf dAµ ≥ 0. M For now, let us consider the case that g < ∞. From Lemma 13, we have that there are g linearly

2 independent Lµ(M) gaussian harmonic one-forms ωi on M. Let Wi be the dual vector field of ωi with respect to the euclidean metric, and let V = Span{Wi}. Let W ∈ V . From a calculation similar to that of (3.2.6), we have that

   E 2 2 2 E − ⟨ϕW, L (ϕW )⟩ dAµ = |W | |∇ϕ| − ϕ ⟨W, L W ⟩ dAµ. (4.0.28) M M M

So, then (4.0.16) gives us that

   2 E 2 2 2 2 2 ϕ ⟨W, L W ⟩ dAµ = ϕ |W | dAµ + ϕ |A| |W | dAµ M M M

 2 α α − 2 ϕ A (W, ei)A (ei,W ) dAµ. (4.0.29) M

46 Now, note that

  α α 2 2   2 2 2 2 2 2 2A (W, ei)A (ei,W ) − |A| |W | = 2 καiWαi − (κα1 + κα2)(Wα1 + Wα2), (4.0.30) α i

  2 2 2 = (καi − καj)Wαi, (4.0.31) α i̸=j

 2 2 2 ≤ |κα1 − κα2||W | . (4.0.32) α

So, (4.0.29) becomes

   2 E 2 2 2 2  2 2 ϕ ⟨W, L W ⟩ dAµ ≥ ϕ |W | dAµ − ϕ |W | |κα1 − κα2| dAµ. (4.0.33) α M M M

Now, at every point x ∈ M, one is allowed to choose to sum over any frame ηα for the normal bundle. So, one gets

   2 E 2 2 2 2  2 2 ϕ ⟨W, L W ⟩ dAµ ≥ ϕ |W | dAµ − ϕ |W | inf |κα1 − κα2| dAµ, (4.0.34) {η } α α M M M  2 2 ≥ (1 − δ) ϕ |W | dAµ. (4.0.35) M

2 Since Dim(V ) < ∞ and V ⊂ Lµ(M), an argument similar to that given in Proposition1 ∞ shows that there exists a cut-off function ϕ ∈ C0 (M) with 0 ≤ ϕ ≤ 1 and |∇ϕ| ≤ 1 such that  2 2 2  2 Dim(ϕV ) = Dim(V ). Furthermore, for any W ∈ V we have that M ϕ |W | dAµ ≥ 3 M |W | dAµ,  2 2 1−δ  2 ∞ and M |∇ϕ| |W | dAµ ≤ 3 M |W | dAµ. Then using (4.0.35) for this cut-off function ϕ ∈ C0 (Σ), we have that

   E 2 2 2 2 − ⟨ϕW, L ϕW ⟩ dAµ ≤ (δ − 1) ϕ |W | dAµ + |∇ϕ| |W | dAµ, (4.0.36) M M M δ − 1  ≤ |W |2 dA . (4.0.37) 3 µ M

 E So, − ⟨ϕW, L ϕW ⟩ dAµ < 0 if W ̸= 0. M Now, consider the linear map F : ϕV → RnJ defined by

    F (ϕW ) =  ϕψ1W dAµ, ..., ϕψJ W dAµ . (4.0.38) M M

47  E So, if ϕW ∈ Ker F , then we have that ⟨ϕW, L ϕW ⟩ dAµ ≥ 0. Therefore, ϕW ≡ 0. So, g = M Dim(V ) = Dim(ϕV ) ≤ nJ, and the theorem follows for g < ∞.

m For g = ∞, we need only take subspaces of finite dimension and work as above to get that J ≥ n for all m. This gives J = ∞.

48 Appendix

Consider M ⊂ N and a compact normal variation F (x, t) of M. For completeness, we first prove a  standard formula for ∇Ft Ht=0 [9].

Lemma 16.

 N,2  ¯ N ∇Ft Ht=0 = −∇ii Ft − ⟨Aij,Ft⟩Aij − R(Ft,Fi)Fi − ⟨Aij, ∇Fi Ft⟩Fi. (A.0.39)

Proof. Consider a point p ∈ M and geodesic coordinates on M centered at p. Note that ∇Fi Fj = ∇N F when t = 0. Also, take note that for all time t we have ∇ F = ∇ F and ∇ F = ∇ F . Fi j Fi j Fj i Ft i Fi t

Let gij(t) be the metric on F (·, t). Now, use that H = −gij∇N F = −gij ∇ F − ∇ F . Fi j Fi j Fi j ij il jm Using [Ft,Fi] = 0, note that ∇Ft (g ) = −2g g ⟨Alm,Ft⟩ = −2⟨Aij,Ft⟩ at p. We also get

¯ ∇Ft (∇Fi Fi) = R(Ft,Fi)Fi + ∇Fi (∇Ft Fi), (A.0.40) ¯ = R(Ft,Fi)Fi + ∇Fi (∇Fi Ft). (A.0.41)

Again using normal coordinates, we have at p that

jk −∇Ft (∇Fi Fi) = −∇Ft (⟨∇Fi Fi,Fj⟩g Fk), (A.0.42)

jk jk = −⟨∇Ft (∇Fi Fi),Fj⟩g Fk − ⟨∇Fi Fi, ∇Ft Fj⟩g Fk, (A.0.43)

 ¯ T  T jk = − R(Ft,Fi)Fi⟩ − ∇Fi (∇Fi Ft) + ⟨Aii, ∇Fj Ft⟩g Fk. (A.0.44)

(A.0.45)

49 Therefore, at p we get

N ∇ H = −2⟨A ,F ⟩A − R¯(F ,F )F  − ∇N (∇ F ) − ⟨A , ∇ F ⟩F . (A.0.46) Ft ij t ij t i i Fi Fi t ii Fj t j

Note that

∇N (∇ F ) = ∇N,2F + ∇N ⟨∇ F ,F ⟩F  , (A.0.47) Fi Fi t ii t Fi Fi t j j

= ∇N,2F + ⟨∇ F ,F ⟩∇N F , (A.0.48) ii t Fi t j Fi j

N,2 = ∇ii Ft − ⟨Aij,Ft⟩Aij. (A.0.49)

Therefore, at p

N,2  ¯ N ∇FT H = −∇ii Ft − ⟨Aij,Ft⟩Aij − R(Ft,Fi)Fi − ⟨Aij, ∇Fi Ft⟩Fi, (A.0.50) and the lemma follows.

Now we may prove Lemma1.

Proof of Lemma1. Note that for all time t we have

∂  A (U ) = ef ⟨∇f + H,F ⟩ dA. (A.0.51) ∂t f t t Ut

Since M is f-minimal, we have that

2   ∂  f   2 Af (Ut) = e ∇Ft (∇f + H),Ft dA. (A.0.52) ∂t t=0 U

Since F is a normal variation, we have from (A.0.39) that

  f f  N,2  f e ⟨∇Ft H,Ft⟩ dV = − e ∇ii Ft + ⟨Aij,Ft⟩Aij,Ft + e Ric(Ft,Ft) dA. (A.0.53) U U

Also, note that ⟨∇Ft ∇f, Ft⟩ = Hessf (Ft,Ft). Therefore, we get the lemma.

∂  For completeness, we prove a common formula for ∂t Nt=0 using the found in McGonagle-Ross [27].

50 n+1 Lemma 17. For any normal variation of a hyper-surface Σ ⊂ R with u ≡ Ft|t=0, we have that

 ∂  N = −∇u. (A.0.54) ∂t t=0

  n+1  Proof. Fix an orientation on R such that the n + 1-form N ∧ Fi is positive. We also fix a i choice of sign for the Hodge star ∗ operator such that ∗ω ∧ ω is positive for this orientation. Then, we have that  ∗ Fi i N =  . (A.0.55) | Fi| i Note that

  jk ∂t| Fi| = | Fi|g ⟨∂jFt, ∂k⟩, (A.0.56) i i  = | Fi|uH. (A.0.57) i

So, we have that

  j−1  (−1)  j N    ∂tN = −uHN +  ∗ (∂jFt) Fj + (∂jFt) N ∧ Fi , (A.0.58) | Fi| j  i̸=j  i   j−1 N  i  (−1) (∂jFt)  = − uH − (∂iFt) N +  ∗ N ∧ Fi . (A.0.59) | Fi| j i̸=j i

j−1   Since Fj ∧ (−1) ∧ N ∧ Fi = −N ∧ Fi, we may compute in local geodesic coordinates that i̸=j i

 i ij ∂tN = − uH − (∂iFt) N − g ⟨∂iFt,N⟩Fj. (A.0.60)

Now, take note that

ij ij ij −g ⟨∂iFt,N⟩Fj = −g (∂iu)Fj + g ⟨Ft, ∂iN⟩Fj, (A.0.61)

ij = −g (∂iu)Fj. (A.0.62)

Also, observe that

i uH − (∂iFt) = 0. (A.0.63)

Combining (A.0.60), (A.0.62), and (A.0.63), we get (A.0.54).

51 Now, we prove Lemma2.

|x|2 Proof of Lemma2. We may use (A.0.51) with f = − 4 to get that

∂  A (U ) = ⟨∇f, N⟩u + Hu dA . (A.0.64) ∂t f t µ Ut

So, using (A.0.39) we have that

2    ∂   2  2 Aµ(Ut) = − u △u + |A| u dAµ + u∂t⟨∇f, N⟩ dAµ (A.0.65) ∂t t=0 U U    + ⟨∇f, N⟩ + H ∂t(u dAµ). (A.0.66) U

Now, for a critical hyper-surface, we have that ⟨∇f, N⟩ + H = C a constant. Therefore, for a volume preserving variation, we have that

    ⟨∇f, N⟩ + H ∂t(u dAµ) = C ∂t(u dAµ), (A.0.67) U U    = C∂t  u dAµ , (A.0.68) U = 0. (A.0.69)

From (A.0.54), we see that

∂t⟨∇f, N⟩ = Hessf (N,N)u − ⟨∇f, ∇u⟩, (A.0.70) 1 1 = − u + ∇ T u. (A.0.71) 2 2 x

Therefore, using (A.0.66), (A.0.69), and (A.0.71), we get that

2     ∂  1 2 1 2 Aµ(Ut) = − u △u − ∇xT u + |A| u + u dAµ (A.0.72) ∂t t=0 2 2 U

Now, we prove a lemma relating bounds on the mean curvature of a hyper-surface to a mean- value inequality, as found in McGonagle-Ross[27]. The proof is based on the mean value inequality

52 for minimal surfaces [9].

Lemma 18. Let Σ ⊂ Rn+1 be a hyper-surface with |H| ≤ M. Also let f ∈ C2(Σ) satisfy f ≥ 0 and −2 △f ≥ −λt f on Bt(x) ⊂ BR(0) for some λ. Then, for s ≤ t we have that

 ( λ +M)s −n ωnf(x) ≤ e 2t s f dA. (A.0.73)

Bs(x)∩Σ

n Here ωn is the volume of the n-dimensional unit sphere in R .

Proof. For general Σn ⊂ Rn+1, we have that △|x|2 = 2n − 2⟨x, N⟩H. By translation, we consider

Σ ⊂ Bs(0). So, we then have that

   2n f dA = f∆|x|2 dA + 2 f⟨x, N⟩H dA, (A.0.74)

Bs∩Σ Bs∩Σ Bs∩Σ     = |x|2∆f dA + 2 f|xT | dA − s2 ∆f dA + 2 ⟨x, N⟩Hf dA.

Bs∩Σ ∂Bs∩Σ Bs∩Σ Bs∩Σ (A.0.75)

Now we let g(s) = s−n  f dA. Using the coarea formula and (A.0.75) , we have that Bs∩Σ

  |x| g′(s) = −ns−n−1 f dA + s−n f dA, (A.0.76) |xT | Bs∩Σ ∂Bs∩Σ   1    = −s−n−1 |x|2 − s2 ∆f dA + f|xT | dA + ⟨x, N⟩fH dA 2  Bs∩Σ ∂Bs∩Σ Bs∩Σ  |x| + s−n f dA, |xT | ∂Bs∩Σ 1   ≥ s−n+1 ∆f dA − s−n−1 ⟨x, N⟩fH dA. (A.0.77) 2 Bs∩Σ Bs∩Σ

Note that the final inequality uses that f ≥ 0.

−2 From the assumption that △f ≥ −λt f on Bt and the bound |H| ≤ M, we have for all s ≤ t that

λ   g′(s) ≥ − s1−n ft−2 dA − Ms−1−n sf dA, (A.0.78) 2 Bs∩Σ Bs∩Σ  λ  ≥ − + M g(s). (A.0.79) 2t

53 So, we get that

d  λ +M s g(s)e( 2t ) ≥ 0. (A.0.80) ds

Integrating this we get that  ( λ +M)s −n ωnf(p) ≤ e 2t s f dA. (A.0.81)

Bs∩Σ

Note that by taking f = 1 in (A.0.73), we have that if p ∈ Σ and |H| ≤ M on Bt(p) ∩ Σ, then for all s ≤ t, we have that

−Ms n AE(Bs(p) ∩ Σ) ≥ ωne s , (A.0.82)

n where ωn is the volume of the standard unit ball in R .

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58 Curriculum Vitae

Matthew McGonagle was born on August 21, 1985 in Crofton, Maryland. In 2008, he received dual

BS degrees in Mathematics and Physics from the University of Maryland at College Park. Also in

2008, he was accepted into the Ph.D. program in the Mathematics Department at Johns Hopkins

University. He received an M.A. degree in Mathematics from Johns Hopkins University in 2009. His dissertation was completed under the guidance of Professor William P. Minicozzi II and defended on May 27th, 2014.