Evolution of Non-Compact Hypersurfaces by Inverse Mean Curvature

Evolution of non-compact hypersurfaces by inverse mean curvature

Beomjun Choi∗and Panagiota Daskalopoulos†

Abstract

We study the evolution of complete non-compact convex hypersurfaces in Rn+1 by the inverse mean curvature flow. We establish the long time existence of solutions and provide the characterization of the maximal time of existence in terms of the tangent cone at infinity of the initial hypersurface. Our proof is based on an a’priori pointwise estimate on the mean curvature of the solution from below in terms of the aperture of a supporting cone at infinity. The strict convexity of convex solutions is shown by means of viscosity solutions. Our methods also give an alternative proof of the result by Huisken and Ilmanen in [30] on compact star-shaped solutions, based on maximum principle argument.

1 Introduction

n n+1 A one-parameter family of immersions F : M × [0,T ] → R is a smooth complete solution to n+1 n the inverse mean curvature ﬂow (IMCF) in R if each Mt := F (·, t)(M ) is a smooth strictly mean convex complete hypersurface satisfying ∂ F (p, t) = H−1(p, t) ν(p, t) (1.1) ∂t

where H(p, t) > 0 and ν(p, t) denote the mean curvature and outward unit normal of Mt, pointing opposite to the mean curvature vector. This flow has been extensively studied for compact hypersurfaces. Gerhardt [20] and Urbas [42] showed compact smooth star-shaped strictly mean convex hypersurface admits a unique smooth solution for all times t ≥ 0. Moreover, the solution approaches to a homothetically expanding sphere as t → ∞. arXiv:1811.04594v3 [math.DG] 22 Oct 2020 For non-starshaped initial data it is well known that singularities may develop (See [27] [39]). This happens when the mean curvature vanishes in some regions which makes the classical flow undefined. However, in [27, 28] Huisken and Ilmanen developed a level set approach to weak variational solutions of the flow which allows the solutions to jump outwards in possible regions where H = 0. Using the weak formulation, they gave the first proof of the Riemannian Penrose inequality in General Relativity. One key observation in [28] was the fact the Hawking mass of surface in 3-manifold of nonnegative scalar curvature is monotone under the weak flow, which was first discovered for classical solutions by Geroch [23]. Note that the Riemannian Penrose inequality was shown in more general settings by Bray [3] and Bray-Lee [4] by different methods. Using

∗B. Choi thanks the NSF for support in DMS-1600658 †P. Daskalopoulos thanks the NSF for support in DMS-1600658 and DMS-1900702.

1 similar techniques, the IMCF has been used to show geometric inequalities in various settings. For instance, see [24, 6] for Minkowski type inequalities, [32] for Penrose inequalities and [34, 14, 19] for Alexandrov-Fenchel type inequalities among other results. Note another important application of the flow by Bray and Neves in [2]. In [30] Huisken and Ilmanen studied the IMCF running from compact star-shaped weakly mean convex initial data. Using star-shapedness and the ultra-fast diffusion character of the flow, they derive a bound from above on H−1 for t > 0 which is independent of the initial curvature assumption. This follows by a Stampacchia iteration argument and utilizes the Michael-Simon Sobolev inequality. The C∞ regularity of solutions for t > 0 easily follows from the bound on H−1. The estimate in [30] is local in time, but necessarily global in space as it depends on the area of the initial hypersurface M0 and uses global integration on Mt. As a consequence, the techniques in [30] cannot be applied directly to the non-compact setting. Note that [33] and [45] provide similar estimates for the IMCF in some negatively curved ambient spaces. This work addresses the long time existence of non-compact smooth convex solutions to the n+1 IMCF embedded in Euclidean space R . While extrinsic geometric flows have been extensively studied in the case of compact hypersurfaces, much remains to be investigated for non-compact cases. The important works by K. Ecker and G. Huisken [16, 17] address the evolution of entire graphs by mean curvature flow and establish a surprising result: existence for all times with the only assumption that the initial data M0 is a locally Lipschitz entire graph and no assumption of the growth at infinity of M0. This result is based on priori estimates which are localized in space. 2 In addition, the main local bound on the second fundamental form |A| of Mt is achieved without 2 any bound assumption on |A| on M0. An open question between experts in the field has been whether the techniques of Ecker and Huisken in [16, 17] can be extended to the fully-nonlinear setting, in particular on entire convex graphs evolving by the α-Gauss curvature flow (powers Kα of the Gaussian curvature) and the inverse mean curvature flow. In [10] the second author, jointly with Kyeongsu Choi, Lami Kim and Kiahm Lee, established the long time existence of the α-Gauss curvature flow on any strictly convex complete non-compact hypersurface and for any α > 0. They showed similar estimates as in [16, 17] which are localized in space can be obtained for this flow, however the method is more involved due to the degenerate and fully-nonlinear character of the Monge-Ampére type of equation involved. However, such localized results are not expected to hold for the inverse mean curvature flow where the ultra-fast diffusion tends to cause instant propagation from spatial infinity. In fact, one sees certain similarities between n the latter two flows and the well known quasilinear models of diffusion on R

m−1 ut = div u ∇u . (1.2)

Exponents m > 1 correspond to degenerate diffusion while exponents m < 0 to ultra-fast diffusion. We will see in the sequel that under the IMCF the mean curvature H satisfies an equation which is similar to (1.2) with m = −1. Our goal is to study this phenomenon and establish the long time existence, an analogue of the results in [16, 17] and [10]. Let us remark existing results on the IMCF of hypersurfaces other than closed ones. In [1], B. Allen investigated non-compact solutions in the hyperbolic space which are graphs on the horo- sphere. One key estimate in [1] was to show that uniform upper and lower bounds on the mean curvature persist under some initial assumptions. The second author and Huisken [13] studied n+1 non-compact solutions in R under some initial conditions and we will discuss this result later this section. The flow with free boundary, i.e. solutions with Neumann-type boundary condition,

2 C0 = CΓ0

Γ0

O M0

Figure 1: Definition 1.1 has quite extensive literature. We refer the reader to [40, 41],[18] and citations there-in for the mean curvature flow and [35, 36],[31] for the IMCF. We will next state our main results. The following observation motivates the formulation of our theorem. n Example 1.1 (Conical solutions to IMCF). For a solution Γt to the IMCF in S , the family of cones generated by Γt n+1 CΓt := {rx ∈ R : r ≥ 0, x ∈ Γt} n+1 is a solution to the IMCF in R which is smooth except from the origin. When Γ0 is a compact smooth strictly convex, Gerhardt [22] and Makowski-Scheuer [34] showed the unique existence of solution for time t ∈ [0,T ) with T < ∞ and the convergence of solution to an equator as t → T . n−1 Moreover, we have explicit formula T = ln |S | − ln |Γ0| by the exponential growth of area in n+1 time, (2) in Lemma 2.5. Note also that CΓt, restricted to the unit ball in R , moves by the IMCF n with free boundary on S in the sense of [31]. From Example 1.1 and the ultra-fast diffusive character of the equation, it is reasonable to guess that the behavior of non-compact convex solution and its maximal time of existence is governed by the asymptotics at infinity. For a non-compact convex set Mˆ 0 and the associated hypersurface M0 = ∂Mˆ 0, we recall the definition of the blow-down, so called the tangent cone at infinity. ˆ n+1 Definition 1.1 (Tangent cone at infinity). Let M0 ⊂ R be a non-compact closed convex set. For a point p ∈ Mˆ 0, we denote the tangent cone of Mˆ 0 at infinity by

Cˆ0 := ∩λ>0λ(Mˆ 0 − p).

The deﬁnition is independent of p ∈ Mˆ 0. C0 := ∂Cˆ0 is called the tangent cone of M0 = ∂Mˆ 0 ˆ ˆ n n ˆ at inﬁnity. Γ0 := C0 ∩ S and Γ0 := C0 ∩ S are called the links of tangent cones C0 and C0, respectively.

In this work, we say M0 is convex hypersurface if it is the boundary of a closed convex set with non-empty interior. See Deﬁnition 2.1 and subsequent discussion for more details. For convex

3 n+1 k hypersurface M0 in R , Lemma 2.3 shows M0 = N0 × R for some convex hypersurface N0 in n+1−k n−k n−k R which is homeomorphic to either S or R . In the ﬁrst case, the existence of compact k IMCF, say Nt, running from N0 is known in [30] and thus Nt × R becomes a solution with initial n data M0. Therefore, the essential remaining case is when M0 is homeomorphic to R . We state our existence result.

ˆ n+1 Theorem 1.2. Let M0 in R with n ≥ 2 be a non-compact convex set with interior whose 1,1 n boundary M0 is Cloc and homeomorphic to R , and T = T (M0) be a number deﬁned by n−1 ˆ T = ln |S | − ln P (Γ0) ∈ [0, ∞]. (1.3)

n−1 Here, Γˆ0 is the link of tangent cone of Mˆ 0 at inﬁnity, | · | := H (·), and P (Γ)ˆ is the perimeter of n Γˆ in S deﬁned by ( ˆ n |Γ0| if Γ0 has non-empty interior in S P (Γˆ ) = (1.4) 0 ˆ n 2|Γ0| if Γ0 has empty interior in S .

If T > 0, a smooth convex solution to the IMCF, say Mt = ∂Mˆ t, exists for 0 < t < T . M0 is the initial data in the sense that Mt converges to M0 locally uniformly as t → 0. Mt is strictly convex if and only if Mˆ 0 contains no inﬁnite straight line inside.

Remark 1.2.

ˆ n (i) Γ0 can be an arbitrary convex set in S which may possibly have empty interior. (See Defini- n tion 2.1 and Lemma 2.2 regarding the definition of convexity in S .) In that case the perimeter ˆ n P (Γ0) is the limit of outside areas of decreasing sequence of convex sets with interior in S which approximate Γ.ˆ (See Lemma A.10 and Lemma A.11.) According to (1.3), T = ∞ when P (Γˆ0) = 0 and this happens if only if Γˆ has Hausdorff dimension less than n − 1. Note that the definition of P (·) is not related with the notion of perimeter used in geometric measure theory.

n (ii) The tangent cone of Mt at inﬁnity, say Γt, also evolves by IMCF in S in some generalized sense (Lemma 4.5), and becomes ﬂat as t → T − when T < ∞. In Remark 4.2 we further discuss this in connection with the asymptotic behavior of Mt as t → T . ˆ n−1 ˆ n−1 (iii) According to (1.3), T = 0 when P (Γ0) = |S |. In [9], it was shown that P (Γ0) = |S | if and only if Γˆ0 is either a hemisphere or a wedge

ˆ n n−1 Wθ0 = S ∩ {(r sin θ, r cos θ): θ ∈ [0, θ0], and r > 0} × R for some θ0 ∈ [0, π) (1.5)

n up to an isometry of S . We show in Theorem 1.3 no solution exists from such a M0.

Remark 1.3. Let us emphasize Theorem 1.2 allows H = 0 on a possibly non-compact region of M0. Even in that case, H becomes strictly positive for t > 0 and this is due to Theorem 1.4. A similar phenomenon was observed for solutions to the Cauchy problem of the ultra-fast diﬀusion equation n (1.2) with m < 0 on R . See Remark 4.3 for more details.

Next result asserts that T = T (M0) in Theorem 1.2 is the maximal time of existence. The result holds not only for the solutions constructed in Theorem 1.2, but applies to arbitrary solutions.

4 Theorem 1.3. Let M0 = ∂Mˆ 0 satisfy the same assumptions as in Theorem 1.2 and T = T (M0) be given by (1.3). If T < ∞, there is no smooth solution Nt which is the boundary of Nˆt, ∩t>0Nˆt = Mˆ 0, and existing 0 < t < T + τ some τ > 0. In particular, no solution exists if T = 0.

n+1 Non-compact IMCF in R was ﬁrst considered by the second author and G. Huisken in [13], where they established the existence and uniqueness of smooth solution to the IMCF, under the 2 0 assumption that the initial hypersurface M0 is an entire C graph, xn+1 = u0(x ) with H > 0, in the following two cases:

(i) M0 has super linear growth at inﬁnity and it is strictly star-shaped, that is HhF −x0, νi ≥ δ > 0 n+1 holds, for some x0 ∈ R ;

n+1 (ii) M0 a convex graph satisfying 0 < c0 ≤ H hF − x0, en+1i ≤ C0 < +∞, for some x0 ∈ R and lies between two round cones of the same aperture, that is

0 0 0 α0|x | ≤ u0(x ) ≤ α0|x | + k, α0 > 0, k > 0. (1.6)

In the first case, a unique smooth solution exists up to time T = ∞, while in the second case a unique smooth convex solution Mt exists for t ∈ [0,T ) where T > 0 is the time when the round 0 conical solution from {xn+1 = α0|x |} becomes flat. In the latter case, the solution Mt lies between two evolving round cones and becomes flat as t → T . To derive a local lower bound of H, a parabolic Moser’s iteration argument was used along with a variant of Hardy’s inequality, which plays a similar role as the Micheal-Simon Sobolev inequality used in [30]. Theorem 1.2 and the results in [13] show that convex surfaces with linear growth at infinity have critical behavior in the sense that in this case the maximal time of existence is finite and it depends on the behavior at infinity of the initial data. However, while the techniques in [13] only treat this critical linear case under the condition (1.6), Theorem 1.2 allows any behavior at infinity. Moreover, the techniques in [13] require to assume that H is globally controlled from below at initial time, namely that HhF − x0, νi ≥ δ > 0 in the case of super-linear growth and H hF − x0, en+1i ≥ c > 0 in the case of linear growth. In this work we depart from the techniques in [13],[30], and establish a priori bound on H−1 which is local in time. For this, we develop a new method based on the maximum principle rather than the integrations used in [13],[30]. Our key estimate roughly says that a convex solution has a global bound on (H|F |)−1 as long as a nontrivial convex cone is supporting the solution from outside.

n n+1 Theorem 1.4. Let F : M ×[0,T ] → R , n ≥ 2 and T > 0, be a compact smooth convex solution to the IMCF and suppose there is θ1 ∈ (0, π/2) for which

n hF, en+1i ≥ sin θ1 |F | on M × [0,T ]. (1.7)

Then 1 1 ≤ C 1 + on M n × [0,T ] (1.8) H|F | t1/2 for a constant C = C(θ1) > 0.

The compactness assumption on Mt above will only be used to apply maximum principle and will not aﬀect the application of the estimate in proving of our non-compact result, Theorem 1.2, as

5 we will approximate non-compact solutions by compact ones. Note the estimate does not depend on initial bound on (H|F |)−1, which will allow initial data with flat regions as described in Remark 1.3. Moreover, the new method developed while showing Theorem 1.4 leads us to a new proof of Theorem 1.1 in [30], the H−1 estimate for compact, star-shaped solutions. This is included in Theorem A.5 in the appendix. In fact, one expects that similar estimates as in Theorem A.5 can be possibly derived for the IMCF in other ambient spaces, including some positively curved spaces or asymptotically flat spaces, using this new method and this generalizes the results of [30, 33, 45]. See in [29] for a consequence of such an estimate when this is shown in asymptotically flat ambient spaces. Remark 1.4. Recently, the first author and P.-K. Hung in [9] addressed the IMCF on convex solutions allowing singularities on M0. Using Theorem 1.4 as a key ingredient, [9] shows the tangent cone obtained after blowing-up at a singularity point evolves by the IMCF. As a corollary, one can consider an arbitrary non-compact convex hypersurface M0 in Theorem 1.2 and obtain the following necessary and sufficient condition for the existence of a smooth solution: for an arbitrary non-compact convex M0 with T (M0) > 0, there is a smooth solution if and only if M0 has density |Br(p)∩M0| one everywhere. i.e. Θ (p) = lim n = 1 for all p ∈ M . See [9] for more details. 0 r→0 ωnr 0 A brief outline of this paper is as follows: In Section 2, we introduce basic notation, evolution equations of basic geometric quantities, and prove some useful identities. Section 3 is devoted to the proof of main a priori estimate Theorem 1.4. In Section 4, we prove the long time existence of solution (Theorem 1.2 and Theorem 1.3) via an approximation argument. Here, the passage to a limit relies on the estimate Theorem 1.4. In Appendix A.1, we prove the convexity of solution is preserved and show the solution becomes strictly convex immediately unless the lowest principle curvature λ1 is zero everywhere. This will be shown for the solutions to the IMCF in space forms as this adds no difficulty in the proof but could be useful in other application. In Appendix A.2, we give an alternative proof of H−1 estimate shown in [30] using a maximum principle argument, showing how the star-shapedness condition can be incorporated in our method. Finally, in Appendix A.3 we n+1 n show the approximation theorems of convex hypersurfaces in R and S that are used throughout the paper. Acknowledgement: The authors wish to express their gratitude to Gerhard Huisken and Pei- Ken Hung for stimulating and useful discussions on the inverse mean curvature flow.

2 Preliminaries

In this section we present some basic preliminary results. Let us begin by clarifying some notions and simple facts from convex geometry. Convex hypersurfaces are studied in both convex geometry and differential geometry of submanifolds. As a result there are different notions of convexity preferred in different subjects. In this paper, we use the following definition. n+1 n Definition 2.1 (Convex hypersurfaces in R and S ). n+1 ˆ (i) A hypersurface M0 ⊂ R is called convex if it is the boundary of a convex set M0 of ˆ n+1 non-empty interior and M0 6= R . ˆ n ˆ (ii) A set Γ0 ⊂ S is convex if for all p and q in Γ0 at least one minimal geodesic connecting the ˆ n two points is contained in Γ0.Γ0 is a convex hypersurface in S if it is the boundary of a ˆ ˆ n convex set Γ0 with non-empty interior and Γ0 6= S .

6 (iii)A C2 convex hypersurface is strictly convex at a given point if the second fundamental form with respect to the inner the normal is positive deﬁnite.

n There is a useful characterization of convexity in S which is immediate from Definition 2.1. ˆ n ˆ n+1 Lemma 2.2. A set Γ0 ⊂ S is convex if and only if it is connected and CΓ0 := {rx ∈ R : r ≥ ˆ n+1 0, x ∈ Γ0} is convex in R . We prefer Definition 2.1 to the other one that defines convexity through certain properties of the embedding or immersion since we will deal with convex hypersurfaces of low regularity. These n+1 two notions are, however, equivalent under suitable assumptions. For example, if M0 ⊂ R n (or Γ0 ⊂ S ) is a convex hypersurface, then it is a complete connected embedded submanifold. 2 Furthermore, if M0 (or Γ0) is C , then the second fundamental form with respect to the inner normal is nonnegative definite. The converse question, namely under what conditions an immersed or embedded hypersurface of nonnegative sectional curvature (or semi-definite second fundamental form) bounds a convex set, has a long history and has been answered, for instance, by Hadamard [25], Sacksteder-van Heijenoort [38, 26], H. Wu [44], do Carmo-Warner [8], Makowski-Scheuer [34] under different assumptions. We refer the reader to the results and references cited in these papers. The following simple observation will be used throughout this paper. ˆ ˆ n+1 Lemma 2.3. Let M0 = ∂M0 be the boundary of a closed convex set with interior M0 ⊂ R , n+1 n k that is M0 is a convex hypersurface in R . Then either M0 = R or M0 = R × N0, for some ˆ ˆ n+1−k 0 ≤ k < n and N0 = ∂N0 where N0 ⊂ R is a closed convex set with interior which contains n−k n−k no infinite line. Moreover, such N0 is either homeomorphic to S or R .

Proof. If Mˆ 0 contains an infinite straight line, then Mˆ 0 splits off in the direction of the line by n+1 ˆ the following elementary argument: suppose a line {ten+1 ∈ R : t ∈ R} is contained in M0 ˆ ˆ and let us denote its cross-sections Ωl × {l} := M0 ∩ {xn+1 = l} for l ∈ R. By convexity, for any 0 ≤ t < t the set t2−t1 Ωˆ × {t } is contained in Mˆ ∩ {x = t }. By taking t → ∞ while 1 2 t2 0 1 0 n+1 1 2 fixing t1, we have Ωˆ 0 × {t1} ⊂ Mˆ 0 ∩ {xn+1 = t1}. We can do a similar argument for t2 < t1 ≤ 0 ˆ ˆ ˆ and thus Ω0 × R ⊂ M0. Similarly, we can do the same argument for all other sections Ωl × {l} and ˆ ˆ ˆ ˆ ˆ obtain that Ωl × R ⊂ M0. Therefore Ωl1 = Ωl2 for all l1 6= l2 and M0 = Ω0 × R. By repeating this ˆ ˆ k ˆ splitting, we conclude that M0 = N0 × R , for some k ≥ 0 where N0 does not contain any infinite lines. In this case, a classical simple result in convex geometry (see for instance Lemma 1 in [44]), ˆ n−k n−k implies that ∂N0 is homeomorphic to either S or R .

g(t) We next derive some properties of smooth solutions to IMCF. Let ∇ := ∇ and ∆ := ∆g(t) n ∂F ∂F denote the connection and Laplacian on M with respect to the induced metric gij(t) = h ∂xi , ∂xj i. Recall that on a local system of coordinates {xi} on M n, ∂2F ∂F ∂F ∂ν = −h ν + Γk and , = h (2.1) ∂xi∂xj ij ij ∂xk ∂xj ∂xi ij where ν denotes the outer unit normal. We also deﬁne the operator 1 := ∂ − ∆ t H2 and use it frequently as this is the linearized operator of the IMCF. The IMCF or generally curvature ﬂows of homogeneous degree −1, have the following scaling property which we will frequently use:

7 n n+1 ˜ n n Lemma 2.4 (Scaling of IMCF). If Mt ⊂ R is a solution to the IMCF, then Mt = λ Mt is again a solution for λ > 0.

n+1 Lemma 2.5 (Huisken, Ilmanen [30]). Any smooth solution of the IMCF (1.1) in R satisﬁes 2 (1) ∂tgij = H hij

(2) ∂tdµ = dµ, where dµ is the volume form induced from gij

−1 1 (3) ∂tν = −∇H = H2 ∇H

1 2 |A|2 (4) ∂t − H2 ∆ hij = − H3 ∇iH∇jH + H2 hij

1 |A|2 1 2 2 |A|2 (5) ∂tH = ∇i H2 ∇iH − H = H2 ∆H − H3 |∇H| − H

1 −1 |A|2 −1 (6) ∂t − H2 ∆ H = H2 H

1 |A|2 (7) ∂t − H2 ∆ hF − x0, νi = H2 hF − x0, νi. n+1 Remark 2.1. If the ambient space is not R , then the evolution equations of gij, dµ, and ν remain n+1 the same as in R , but the evolution of curvature hij is diﬀerent and complicated. On a space form of sectional curvature K, the formula hugely simpliﬁes becoming

1 |A|2 2 nKh ∂ h = ∆h + h − ∇ H∇ H − ij (2.2) t ij H2 ij H2 ij H3 i j H2 (See Chapter 2 in [21].) In this paper we will mostly focus on the ﬂow in Euclidean space and we will only use (2.2) in Appendix A.1. Using Lemma 2.5 one can easily deduce the following formulas.

n+1 n+1 Lemma 2.6. For a ﬁxed vector ω in R , the smooth solutions to the IMCF (1.1) in R satisfy 1 2 2n 4 (1) ∂t − H2 ∆ |F − x0| = − H2 + H hF − x0, νi

1 |A|2 (2) ∂t − H2 ∆ hω, νi = H2 hω, νi 1 2 (3) ∂t − H2 ∆ hω, F − x0i = H hω, νi. Proof. By (2.1) we have

ij 2 k ij k k ∆F = g (∂ijF − ΓijFk) = g (−hijν + ΓijFk − ΓijFk) = −H ν.

−1 which combined with ∂tF = H ν implies (3). Next,

2 ∆|F − x0| = 2h∆F,F − x0i + 2h∇F, ∇F i = −2Hhν, F − x0i + 2n implies (1). Finally,

ij 2 k ij k k l ∆ν = g (∂ijν − Γij∂kν) = g (∂j(hi Fk) − ΓijhkFl) ij k k l k k l = g ((∂jhi )Fk − hi hjkν + Γjkhi Fl − ΓijhkFl) 2 ij k 2 = −|A| ν + g ∇jhi Fk = −|A| ν + ∇H where we used the Codazzi identity in the last equation. This implies (2).

8 The following simple lemma, which commonly appears in Pogorelov type computations, will be useful in the sequel when we compute the evolution of products.

2 Lemma 2.7. For any C functions fi(p, t), i = 1, . . . , m, denote

α1 α2 αm w := f1 f2 . . . fm .

Then on the region where w 6= 0, we have

−2 2 m −2 2 1 (∂t − H ∆)w 1 |∇w| X (∂t − H ∆)fi 1 |∇fi| ∂ − ∆ ln |w| = + = α + . (2.3) t H2 w H2 w2 i f H2 f 2 i=1 i i Proof. The lemma simply follows from

1 (∂ − H−2∆)f 1 |∇f|2 ∂ − ∆ ln |f| = t + . (2.4) t H2 f H2 f 2

Next two lemmas are straightforward computations and we leave their proofs for readers.

2 n 2 Lemma 2.8. For any two C functions f, g defined on M ×(0,T ) and any C function ψ : R → R, 2 (fg) = ( f)g + f( g) − h∇f, ∇gi H2 and ψ00(f) ψ(f) = ψ0(f) f − |∇f|2 H2 −2 where := (∂t − H ∆). 2 Lemma 2.9. If a C function f is defined on a solution Mt of the IMCF and satisfies

∂ 1 |A|2 − ∆ f = f ∂t H2 H2 then for any ﬁxed β 6= 0 we have

∂ 1 |A|2 β(β − 1) |∇f β|2 − ∆ f β = β f β − . ∂t H2 H2 β2 H2f β −1 For instance, H , hω, νi and hF − x0, νi are examples of such a function f.

We ﬁnish with the following local estimate which is an easy consequence of Proposition 2.11 in n+1 [13]. Here Br denotes an extrinsic ball of radius r > 0 in R .

Proposition 2.10 (Proposition 2.11 [13]). For a solution Mt, t ∈ [0,T ], to the IMCF, there is a constant Cn > 0 such that −1 sup H ≤ Cn max ( sup H, r ). Mt∩Br/2 M0∩Br

9 Proof. Although this proposition is proven in [13] we include below its proof for completeness. For 2 2 2 fixed r > 0, let η := (r − |F | )+ be a cut-off function defined in the ambient space. Using Lemma 2.6 and 2.8,

∂ 1 ∂ 1 2 − ∆ η = −2(r2 − |F |2) − ∆ |F |2 − |∇(r2 − |F |2) |2 ∂t H2 + ∂t H2 H2 + 2n 4 8 = −2η1/2 − + hF, νi − |F T |2 H2 H H2 and

∂ 1 4nη1/2 8 |∇H|2 |A|2 2 − ∆ ηH = − 8η1/2hF, νi − |F T |2 + η −2 − − h∇η, ∇Hi ∂t H2 H H H3 H H2 4nη1/2 2 ηH ≤ + 8η1/2r − h∇(ηH), ∇Hi − . H H3 n In the last inequality, we used |A|2 ≥ n−1H2 and |hF, νi| ≤ |F | ≤ r. Let m(t) be the maximum of ηH on Mt. Then the above inequality implies

kηk3/2 m(t) r6 1 ∂ m(t) ≤ 4n ∞ + 8kηk1/2r − ≤ 4n + 8r3 − m(t). t m(t) ∞ n m(t) n

Thus m(t) will decrease if

4n m(t) √ r6 − + 8r3 ≤ 0 ⇐⇒ m2(t) − 8nr3m(t) − 4n2r6 ≥ 0 ⇐⇒ m(t) ≥ (4 + 2 5)n r3. m(t) n √ Therefore, m(t) ≤ max ( m(0), (4 + 2 5)n r3 ). The proposition is implied since √ sup (r2 − (r/2)2)2H ≤ sup ηH ≤ max ( m(0), (4 + 2 5)n r3 ) Mt∩Br/2 √ ≤ max ( sup r4H, (4 + 2 5)n r3 ). M0∩Br

3 L∞ bound of (H|F |)−1

Before giving the proof of Theorem 1.4, lets us introduce some notations. We consider spherical n+1 coordinates with respect to the origin in R , namely

n−1 x = (x1, . . . , xn+1) = (rω sin θ, r cos θ) with r ≥ 0, ω ∈ S , and θ ∈ [0, π] which are smoothly well-deﬁned away from xn+1-axis. We will also denote by ∇¯ and ∇ metric- n+1 n ∗ induced connections on (R , geuc) and (M ,F geuc), respectively. Before the proof, we need the evolution equation of θ, deﬁned in the ambient space as follows:

Deﬁnition 3.1. We deﬁne hx, e i θ : n+1 \{0} → [0, π] by θ(x) := arccos n+1 (3.1) R |x|

10 and n+1 r : R → [0, ∞) by r(x) := |x|. Moreover, we define smooth unit orthogonal vector fields 1 ∂ x0 cos θ e (x) = e (x0, x ) := = , − sin θ on n+1 \{x − axis} θ θ n+1 |x| ∂θ |x| sin θ R n+1 and ∂ x e (x) := = on n+1 \{0}. r ∂r |x| R 2 Though θ is not smooth at the points on the xn+1-axis, note that θ , cos θ, and sec θ are all smooth on {xn+1 > 0}. Note θ represents a scaled distance from the north pole measured in the sphere. The first negative term on the right hand side of (3.2) will justify the use of θ in the estimate. Compare (3.2) with (1) in Lemma 2.6.

Lemma 3.2. On the region {θ 6= 0, π} ∩ {|x|= 6 0},

1 1 n − |∇r|2 1 |∇θ|2 2 ∇r 2 ∂ − ∆θ = − + + h , ∇θi + hν, ∇¯ θi. (3.2) t H2 H2r2 tan θ H2 tan θ H2 r H Proof. Consider a spherical coordinate chart

α α n−1 (r, θ, (w )α=1...n−1) with r > 0, θ ∈ (0, π), (w ) ∈ S

n+1 α n−1 around a point {θ 6= 0, π} ∩ {|x|= 6 0} in R , where w is a coordinate chart of S . On this chart, 2 2 2 2 2 α β geuc = dr + r dθ + r sin θσαβ dw dw . (3.3) Also note that 1 ∂ 1 ∂ grad θ = = e and grad r = = e on ( n+1, g ). (3.4) r2 ∂θ r θ ∂r r R euc At a given p ∈ M n with {θ 6= 0, π} ∩ {|x|= 6 0}, let us choose a geodesic normal coordinate of n i n M , say {y }i=1. In this coordinate at this point, X X ∂ ∂ X ∂ 1 ∂ ∂ ∆θ = ∂ ∂ θ = dθ = h , i i i ∂yi ∂yi ∂yi r2 ∂θ ∂yi i i i X 2 ∂ ∂ 1 ∂ ∂ 1 ∂ = − h , e ih , e i + h∇¯ , i + h , −h νi. r2 ∂yi θ ∂yi r r2 ∂i ∂θ ∂yi r2 ∂θ ii i n ∂ n+1 Since i , ν constitutes an orthonormal basis of TF (p)R , ∂y i=1 X ∂ ∂ h , e ih , e i + hν, e ihν, e i = he , e i = 0. (3.5) ∂yi θ ∂yi r r θ r θ i Therefore, H 2 1 X ∂ ∂ ∆θ = − hν, e i + hν, e ihν.e i + h∇¯ , i. (3.6) r θ r2 r θ r2 ∂i ∂θ ∂yi i

11 Claim 3.1. n X ∂ ∂ cos θ h∇¯ , i = n − (1 − hν, e i2) − (1 − hν, e i2) . (3.7) ∂i ∂θ ∂yi sin θ r θ i=1 Proof of Claim 3.1 . By computing the Christoﬀel symbols from the metric (3.3), we get:

∂ ∂ ∂ 1 ∂ ∂ cos θ ∂ ∇¯ ∂ = −r , ∇¯ ∂ = , ∇¯ ∂ = . (3.8) ∂θ ∂θ ∂r ∂r ∂θ r ∂θ ∂wα ∂θ sin θ ∂wα

∂ a cos θ P α ¯ r P α Suppose ∂i = ∂yi = aθ∂θ + ar∂r + α aα∂w . Then ∇∂i ∂θ = −raθ∂r + r ∂θ + α aα sin θ ∂w and hence ∂ ∂ cos θ h∇¯ , i = −ra a + ra a + r2 sin2 θ a a σαβ ∂i ∂θ ∂yi θ r θ r sin θ α β " 2 2 2# cos θ ∂ ∂ ∂ = − , er − , eθ . sin θ ∂yi ∂yi ∂yi The claim follows by summing this over i. 1 hν, e i Now ∂ θ = dθ(∂ F ) = hν, grad θi = θ , (3.6) and (3.7) imply t t H rH 1 2hν, e i 1 cos θ ∂ − ∆θ = θ − [n − (1 − hν, e i2) − (1 − hν, e i2)] + 2hν, e ihν, e i . t H2 rH (rH)2 sin θ r θ r θ

n ∂ Hence, the lemma follows by using (3.4) and the orthonormality of i , ν in the equation ∂y i=1 above.

Proof of Theorem 1.4. Using the deﬁnition (3.1), our condition (1.7) can be written as θ(p, t) ≤ π − θ 1 π θ1 π π/2 − θ1. Setting c := > 1, we have c θ ≤ 2 − 2 < 2 and sec(cθ) ≤ 2 sec θ for θ = θ(p, t) π − 2θ1 on t ∈ [0,T ]. By lemma 2.8, 1 sec(cθ) = c sec(cθ) tan(cθ) θ − c2[sec(cθ) tan2(cθ) + sec3(cθ)]|∇θ|2] H2 c2 = sec(cθ) c tan(cθ) θ − (2 tan2(cθ) + 1)|∇θ|2 . H2

∇ϕ After deﬁning ϕ := sec(cθ), Lemma 3.2 and ϕ = c tan(cθ)∇θ imply

ϕ c tan(cθ) 2 ∇r ∇ϕ 2 ∇¯ ϕ = − (n − |∇r|2 − r2|∇θ|2) + , + ν, ϕ H2r2 tan θ H2 r ϕ H ϕ 2 |∇ϕ|2 1 − − c2|∇θ|2 H2 ϕ2 H2 tan(cθ) (since n − |∇r|2 − r2|∇θ|2 = n − 2 ≥ 0 and ≥ c) tan θ c2 2 |∇ϕ|2 2 ∇r ∇ϕ 2 ∇¯ ϕ ≤ − (n − |∇r|2) − + , + ν, . H2r2 H2 ϕ2 H2 r ϕ H ϕ

12 sec(cθ)t Note that this inequality holds on {x > 0}, where our solution M is located. Let w := = ϕψr−1t n+1 t Hr where ψ := H−1. Then by Lemma 2.7 and the previous inequality w 1 |∇w|2 + w H2 w2 ϕ 1 |∇ϕ|2 |A|2 1 |∇ψ|2 1 r2 1 |∇r2|2 1 = + + + − + + ϕ H2 ϕ2 H2 H2 ψ2 2 r2 H2 r4 t ϕ 1 |∇ϕ|2 |A|2 1 |∇ψ|2 n 2 ∇¯ r 2 |∇r|2 1 = + + + + − hν, i − + ϕ H2 ϕ2 H2 H2 ψ2 H2r2 H r H2 r2 t (3.9) |A|2 1 2 ∇¯ ϕ 2 ∇¯ r ≤ + + ν, − ν, H2 t H ϕ H r 1 |∇ψ|2 n |∇r|2 n − |∇r|2 |∇ϕ|2 ∇r ∇ϕ + + − 2 − c2 − + 2 , H2 ψ2 r2 r2 r2 ϕ2 r ϕ = : (1) + (2). n Suppose a nonzero maximum of w(p, t) on M × [0, t1] is achieved at (p0, t0) with t0 ∈ (0, t1]. At this point, ∇w ∇ψ ∇ϕ ∇r 0 = = + − w ψ ϕ r and therefore 2 2 2 2 |∇ψ| ∇r ∇ϕ |∇r| |∇ϕ| ∇r ∇ϕ = − = + − 2 , . ψ2 r ϕ r2 ϕ2 r ϕ n − |∇r|2 At the maximum point, by plugging this into (2) in (3.9), (2) = −(c2 − 1) . Therefore at H2r2 the maximum point, n − |∇r|2 0 ≤ (1) − (c2 − 1) . H2r2 Let us estimate terms in (1). Note that by our choice of c > 1,

∇¯ ϕ c 1 2 2 1 C = |c tan(cθ)∇¯ θ| ≤ tan(cθ) = sin(cθ) sec(cθ) ≤ ≤ = ϕ r r r cos θ r sin θ1 r |A|2 ¯ for some C = C(θ1). Next, H2 ≤ 1 from convexity and |∇r| ≤ 1 imply at (p0, t0),

2 c − 1 C 1 0 ≤ −(n − |∇r| ) 2 2 + + 1 + H r Hr t0 c − 1 C 1 2 ≤ − 2 2 + + 1 + (since |∇r| ≤ 1, n ≥ 2) H r Hr t0 c − 1 C 1 ≤ − 2 2 + + . 2H r 2(c − 1) t0 2 Note that 0 < t0 ≤ t1 and 1 ≤ ϕ ≤ C on M × [0, t1]. Multiplication of (ϕ(p0, t0)t0) implies ϕt 2 w2(p , t ) = 0 ≤ Ct (t + 1). 0 0 Hr 1 1

On any other point p ∈ M at t = t1, 2 2 1 w(p, t1) w (p0, t0) 1 2 2 (p, t1) = ≤ 2 2 ≤ C 1 + . H r t1ϕ(p, t1) t1ϕ (p, t1) t1 We used ϕ ≥ 1 in the last inequality. This ﬁnishes the proof of Theorem 1.4.

13 Remark 3.1. If we deﬁnew ¯ := ϕψr−1 and follow the rest similarly, we get an estimate which includes the initial bound 1 1 ≤ C max sup , 1 . H|F | M0 H|F | After the preprint of this work has been posted, M.N. Ivaki pointed that ϕ(sec θ) with ϕ of form (A.12) could replace the use of sec(cθ) in the previous proof. If ϕ(sec θ) is used, one may avoid computing the evolution of θ as the evolution of sec θ = hF, en+1i/|F | can be derived from Lemma 2.6.

4 Long time existence of non-compact solutions

Let us provide a sketch on how we prove Theorem 1.2. Since Theorem 1.4 was shown for compact solutions, we first construct a family of compact convex approximating solutions Mi,t = ∂Mˆ i,t which is monotone increasing in i. Each compact expanding solution Mi,t exists for all time by [20][42], thus we may define the limit Mˆ t = limi→+∞ Mˆ i,t as a set. We will see, however, that the limit Mˆ t ˆ n+1 is non-trivial only up to time t < T and the proof of Theorem 1.3 will show Mt = R for t > T , meaning Mt = ∂Mˆ t is empty. Let us briefly explain where the connection between our solution Mt in Euclidean space and solutions on the sphere is revealed. Recall Γˆ0 is the link of the tangent cone of Mˆ 0. For each δ > 0, δ n ˆ ˆδ n δ there is a smooth strictly convex Γ0 ⊂ S such that Γ0 ⊂⊂ Γ0 and Tδ := ln |S | − ln |Γ0| > T − δ. ˆ ˆ ˆ δ Here and later, we use A ⊂⊂ B to denote A ⊂ int(B). As explained in Example 1.1, Γ0 admits a δ n δ smooth Γt in S which exists up to time Tδ and we will make use of CΓt as a barrier which contains δ Mi,t. Indeed, by moving its vertex far away from M0 initially, we can make CΓt (after the initial translation) contain Mi,t up to time Tδ, implying that each Mi,t satisfies condition (1.7) in Theorem 1.4 up to time T −δ for all δ > 0. Theorem 1.4 then leads to an upper bound on (|F | H)−1 showing that the IMCF on Mi,t is locally uniformly parabolic and the rest is straightforward.

The proof of Theorem 1.2 consists of four steps. First, we deﬁne our solution Mt = ∂Mˆ t as a limit of compact approximating solutions. Second, we show that Mt is a smooth non-trivial solution for t ∈ (0,T ) using the idea mentioned above. Third, we show that Mt locally converges to M0, as t → 0. Finally, we show the strict convexity assertion of Theorem 1.2. Since the proof is long, we will address some proofs of technical lemmas in Appendix.

Proof of Theorem 1.2. Suppose T = T (M0) satisﬁes 0 < T ≤ ∞ following the assumption of theorem. We may also assume without loss of generality that Mˆ 0 does not contain any inﬁnite k straight lines. Let us justify this claim. By Lemma 2.3, M0 = R × N0 for some non-compact n+1−k n−k n convex hypersurface N0 ⊂ R and it is homeomorphic to R as M0 is homeomorphic to R . Note 0 ≤ k ≤ n − 2 since k = n − 1 or n would imply Γˆ is a wedge in (1.5) (when k = n − 1) 0,Mˆ 0 or a hemisphere (when k = n), and T = 0 in both cases. If Γˆ ⊂ n and Γˆ ⊂ n−k are the 0,Mˆ 0 S 0,Nˆ0 S links of the tangent cones of Mˆ 0 and Nˆ0, then we have

T (M ) = ln | n−1| − ln P (Γˆ ) = ln | n−k−1| − ln P (Γˆ ) = T (N ). (4.1) 0 S 0,Mˆ 0 S 0,Nˆ0 0 n−k ˆ (n−k)+1 In conclusion, N0 = ∂N0 is a non-compact convex hypersurface in R (with n − k ≥ 2) n−k ˆ homeomorphic to R , i.e. N0 satisﬁes the assumption of Theorem 1.2 and N0 has no inﬁnite n−k+1 straight line inside. Since the existence of a solution Nt of IMCF in R with initial data N0 k n+1 implies that Mt := R × Nt is a solution of IMCF in R with initial data M0, we conclude that it

14 M3,0

M2,0 M0

M1,0

Figure 2: Approximation of M0

would be suﬃcient, as we claimed, to assume that Mˆ 0 does not contain any inﬁnite straight lines. In this case, the link Γˆ0 of the tangent cone of Mˆ 0 does not contain any antipodal points and is n n compactly contained in some open hemisphere H(v0) := {p ∈ S : hp, v0i > 0} for some v0 ∈ S (see Lemma 3.8 in [34]).

Next, we create hypersurfaces Mi,0 = ∂Mˆ i,0 with certain properties which approximate M0 from inside. A sequence of sets Aî is called strictly increasing if Aî ⊂ int(Aî+1) (and we write Ai ⊂⊂ Ai+1). Let us denote the ball of radius r centered at p by Br(p). Setting Σˆ i,0 := Bi(0)∩Mˆ 0, Σˆ i,0 is a (weakly) increasing sequence of convex sets. By Lemma A.6, each Σi,0 admits a strictly increasing approximation by compact smooth strictly convex hypersurfaces Σi,j = ∂Σˆ i,j. Furthermore, we −1 may assume dH (Σˆ i,0, Σˆ i,j) ≤ j , where dH is the Hausdorff metric (A.17). Then a diagonal ˆ ˆ ˆ argument gives a sequence ni → ∞ so that Mi,0 := Σi,ni strictly increases to M0. By [42] and [22], each Mi,0 admits a unique smooth IMCF Mi,t = ∂Mˆ i,t, which exists for all t ∈ [0, ∞). Note Mi,t is smooth strictly convex hypersurface (see Remark A.1) which is strictly monotone increasing in i by the comparison principle. We use the sequence of solutions Mi,t to define next the notion of innermost candidate solution from Mˆ 0.

Deﬁnition 4.1. For a convex set Mˆ 0 with non-empty interior, let Mˆ i,0 be a sequence of compact sets with smooth strictly convex boundary which strictly increases to Mˆ 0. We deﬁne the innermost candidate solution from Mˆ 0 as

ˆ ∞ ˆ Mt := ∪i=1Mi,t, for t ∈ [0, ∞) (4.2) where Mi,t = ∂Mˆ i,t is a sequence of compact smooth strictly convex solutions with initial data Mi,0. Mˆ t is convex by deﬁnition and the deﬁnition does not depend on Mˆ i,t (See Remark 4.1.)

It remains to show that Mt := ∂Mˆ t defines a non-empty strictly convex smooth solution to + the flow, for t ∈ (0,T (M0)), and converges to M0 locally uniformly as t → 0 . We need an approximation lemma for Γˆ0, the link of the tangent cone of Mˆ 0 at infinity. ˆ n Claim 4.1. Let Γ0 ⊂ S be a closed convex set contained in an open hemisphere. Then there is a j ˆj sequence of smooth, strictly convex hypersurfaces Γ0 = ∂Γ0 in the open hemisphere which strictly ˆj ˆ j ˆj ˆ decreases and ∩jΓ0 = Γ0. For every such sequence, |Γ0| = P (Γ0) → P (Γ0). Proof of Claim 4.1. This is a direct consequence of Lemma A.9 and Lemma A.10. Since their proofs require some other results from convex geometry, we prove them separately in the appendix.

15 Mt0−τ

Mt0 r1 − Lτ

r1en+1

x0 = 0 {xn+1 = 0}

Figure 3: Outside hyperplane and inside sphere barriers around x0 ∈ Mt0

j n−1 Now fix an arbitrary time t0 ∈ (0,T ). By the claim, we may find a j such that T := ln |S | − j ˆ ˆj 0 n+1 ln |Γ0| > t0. Since Γ0 is contained in the interior of Γ0, we may find a vector vj ∈ R so that ˆ ˆj 0 ˆ ˆ ˆ ˆj 0 M0 ⊂ CΓ0 + vj. By the definition of M0, it then follows that Mi,0 ⊂ M0 ⊂ CΓ0 + vj, for all i. j n Theorem 1.4 [34] guarantees the existence of a smooth strictly convex IMCF solution Γt in S j j j with initial data Γ0, for t ∈ [0,T ). Note CΓt is an IMCF which is smooth away from the origin. j 2 Unless the convex cone CΓt is flat, the origin can not be touched from inside by a C hypersurface. ˆ ˆj 0 Therefore a version of comparison principle can be justified and we obtain Mi,t ⊂ CΓt + vj for j j t ∈ [0,T ). Since Γt is a strictly convex solution which converges to an equator, we may find a n direction ω0 ∈ S and small δ0 > 0 such that

0 0 hF − vj, ω0i ≥ (sin δ0) |F − vj|, for t ∈ [0, t0] on Mi,t. By Theorem 1.4, we have a uniform bound

0 −1 −1/2 (H|F − vj|) ≤ C(1 + t ) for Mi,t on t ∈ (0, t0]. (4.3)

ˆj 0 ˆ n+1 The barrier CΓt + vj also shows Mt 6= R and hence Mt is non-empty for t ∈ [0, t0]. Let us next prove that Mt, for t ∈ (0,T (M0)), is a smooth solution of IMCF. First, note that 1,1 Mˆ 0 ⊂ int(Mˆ t) for t > 0: indeed, since M0 is locally in C , for every point p ∈ M0, there is an 1,1 inscribed sphere at p whose largest radius depends on the local C norm of M0. By the comparison principle between the sphere solution running from this inscribed sphere and Mi,t, we conclude that p ∈ int(Mˆ t) for all t > 0. (In practice, if Nt = ∂Nˆt is a smooth compact solution containing the origin and Nˆ0 ⊂ Mˆ τ some τ ≥ 0, then the comparison and Lemma 2.4 imply (1 − )Nˆt ⊂ Mˆ i,τ+t for i ≥ i, showing (1 − )Nˆt ⊂ Mˆ τ+t. We then take → 0 to conclude Nˆt ⊂ Mˆ τ+t.) n+1 Next goal is to show, for each (x0, t0) ∈ R × (0,T (M0)) with x0 ∈ Mt0 , there is a spacetime neighborhood, say U ×[t0 −τ, t0], around (x0, t0) such that the portions of Mi,t in this neighborhood can be represented as graphs over a ﬁxed hyperplane with uniformly bounded C1 norm. We ˆ may assume that x0 = 0 and that {xn+1 = 0} is a supporting hyperplane of Mt0 satisfying Mˆ i,t ⊂ {xn+1 ≥ 0} for t ≤ t0. For the discussion below we refer the reader to Figure 3. The observation in the previous paragraph says 2r0 := dist (Mˆ 0, 0) > 0. Note that the estimate on H shown in Proposition 2.10 holds even when M0 ∩ Br(x0) is empty. Thus, applying this proposition −1 gives that H is bounded by cnr0 on Mi,t ∩ Br0 (0) for all i and t > 0. Recall that for smooth convex hypersurfaces, one has |A|2 ≤ H2. Since a (local) uniform limit of smooth functions with

16 2 1,1 1,1 bounded C norm is in C , Mt0 has to be C around 0 and hence there is some r1 such that ˆ Br1 (r1en+1) ⊂ Mt0 . Let us choose r1 suﬃciently small so that Br1 (r1en+1) ⊂ Br0 (0). The uniform 0 −1 −1 bound on (H|F − vj|) in (4.3) implies that there is L such that H ≤ L on Br0 (0) ∩ Mi,t, for t ∈ [t0/2, t0]. Using this speed bound, we obtain that

ˆ t0 r1 Br1−Lτ (r1en+1) ⊂ Mt0−τ for all τ ∈ [0, min( 2 , L )].

t0 To prove this, let us deﬁne, for − 2 ≤ s ≤ 0,

d(s) := dist (r1en+1,Mt0+s) and di(s) := dist (r1en+1,Mi,t0+s).

−1 Note that di(s) is a Lipschitz function and the bound on H gives

˙ t0 ˆ 0 ≤ di(s) ≤ L if s ∈ [− 2 , 0] and r1en+1 ∈ Mi,t0+s. (4.4)

Since d(s) = limi→∞ di(s), d(s) is Lipschitz and (4.4) holds for d(s) as well. Since d(0) = r1 and ˆ r1en+1 ∈ Mt0+0, we may integrate (4.4) to conclude

ˆ t0 r1 d(s) ≥ r1 + Ls and r1en+1 ∈ Mt0+s for all s ∈ [− min( 2 , L ), 0].

In summary, we have shown that there are positive r0, h0, and τ 0 such that for i > i0 large, we have 0 ˆ 0 Br0 (h en+1) ⊂ Mi,t ⊂ {xn+1 ≥ 0} for all t ∈ [t0 − τ , t0]. 0 n 0 0 0 0 Therefore: if Dr0 := {x ∈ R : |x | ≤ r }, we can write Mi,t ∩ (Dr0 × [−h , h ]) as a graph (i) 0 0 xn+1 = u (x , t) on Dr0 × [t0 − τ , t0]. Note we have a uniform bound of |Dxu| on Dr0/2 by the ball and hyperplane barriers above and below. Since Mi,t are solutions to IMCF, the graphs (i) 0 xn+1 = u (x , t) evolve by the fully nonlinear parabolic equation

(1 + |Du|2)1/2 Du −1 ∂ u = − = −(1 + |Du|2)1/2 div (4.5) t H (1 + |Du|2)1/2 and the equation is uniformly parabolic if |Du|, H, H−1 are bounded. Therefore, our estimates (i) 0 above show that u are solutions to a uniformly parabolic equation on Dr0/2 × [t0 − τ , t0] and moreover they are uniformly bounded, since |u(i)| ≤ h0. Standard parabolic regularity theory implies i 0 a smooth subsequential convergence u → u on Dr0/4 × [t0 − τ /2, t0]. Since the sequence of surfaces 0 Mi,t is monotone in i, this proves that xn+1 = u(x , t) is a smooth graphical parametrization of Mt which is a solution to (4.5). Our argument holds in a neighborhood around any point x0 ∈ Mt0 and for any t0 ∈ (0,T (M0)), therefore showing that Mt is a smooth solution to the IMCF for t ∈ (0,T ).

We will next obtain the local uniform convergence of Mt → M0, as t → 0, by showing that ˆ ˆ ˆ ˆ ∩t>0Mt = M0. Arguing by contradiction, suppose that M0 ( ∩t>0Mt, that is there exists a point p ∈ int(∩t>0Mˆ t) \ Mˆ 0. This means for each t > 0, there is it such that Mˆ i,t contains p if i > it. Let us deﬁne di(t) := dist (p, Mˆ i,t). Note that di(0) > 0 and di(t) is nonnegative decreasing function. In view of the bound (4.3), by choosing t0 = T/2, we conclude that there is C = C(p, M0) > 0 such that if 0 < t < T/2, then −1/2 d˙i(t) ≥ −C t . (4.6)

Furtermore, the function di(t) is Lipschitz continuous and hence the above inequality holds a.e. One obtains this inequality by considering those points attaining the distance at each ﬁxed time

17 and estimate the rate of changes in the distances between those points and p using the bound on −1 1/2 H . Integrating (4.6) from 0 to t, we get di(t) − di(0) ≥ −2C t , for 0 < t < T/2. Note that di(0) ≥ dist (p, Mˆ 0) > 0. There is t1 > 0 such that di(t) > dist (p, Mˆ 0)/2 for all i and t ∈ (0, t1). This is a contradiction to the assumption which says di(t) = 0 for large i > it.

Finally, we prove the strict convexity assertion in Theorem 1.2 using Appendix A.1. If Mˆ 0 contains an infinite line, a solution at later time Mˆ t also contains the same line and hence Mt it is not strictly convex by Lemma 2.3. Now suppose Mˆ 0 has no infinite straight. In view of Corollary n A.4, it suffices to show H (ν[Mt]) > 0 for all t ∈ (0,T ). Let us fix an arbitrary t0 ∈ (0,T ). In the ˆ construction of Mt, we have shown that Mi,t0 (and hence Mt0 ) are contained in some round cone n+1 Cˆ := {x ∈ R : hx − v, ωi ≥ (sin δ)|x − v|}. Observe that the outward normal of each supporting ˆ hyperplane of C should belong to ν[Mt0 ] as we may translate the hyperplane to make it support ˆ ˆ Mt0 somewhere. We directly compute the set of outward normals of supporting hyperplanes of C n ˆ0 n n ˆ0 as {v ∈ S : hv, −ωi ≥ cos δ} =: Γ . This shows H (ν[Mt0 ]) ≥ H (Γ ) > 0, finishing the proof.

Remark 4.1. The deﬁnition of innermost candidate in (4.2) does not depend on the choice of approximation Mˆ : if Mˆ and Mˆ 0 are two approximations of Mˆ , we have Mˆ ⊂⊂ Mˆ 0 for i,0 i,0 i,0 0 i,0 ni,0 ˆ ˆ 0 large ni, showing that Mi,t ⊂ ∪jMj,t and vice versa. By the same argument, the comparison principle holds between two innermost candidates if one contains the other at initial time. The solution is innermost in the sense described in Lemma 4.2. This fact will be used in the remaining of the section.

Lemma 4.2. Let Nt = ∂Nˆt for t > 0 be a smooth solution to the IMCF with initial data Nˆ0 := ∩t>0Nˆt and Mˆ t be the innermost candidate from Mˆ 0 by Deﬁnition 4.1. If Mˆ 0 ⊂ Nˆ0, then Mˆ t ⊂ Nˆt as long as Nt = ∂Nˆt exists.

Proof. Mˆ i,0 ⊂⊂ Nˆ0 implies Mˆ i,t ⊂ Nˆt by the comparison principle, showing Mˆ t ⊂ Nˆt.

Next lemma shows conical solutions can be used as barriers from inside. ˆ n n Lemma 4.3. Let Γ0 = ∂Γ0 ⊂ S be a smooth strictly convex hypersurface in S and Γt be the unique solution to the IMCF obtained by [34] and [22]. Let Nˆt the innermost candidate from Nˆ0 ˆ ˆ ˆ ˆ n−1 by Definition 4.1. If CΓ0 ⊂ N0, then CΓt ⊂ Nt for t ∈ [0, ln |S | − ln |Γ0|). ˆ n Proof. After a rotation, we may assume that en+1 is in the interior of Γ0 in S . Then CΓ0 can be n written as a graph of an entire homogeneous function xn+1 = f(x), x ∈ R , which is uniformly Lipschitz. Since the graph is a cone, we have f(λx) = λf(x). Let η be a usual smooth radially symmetric mollifier supported on B1(0), and define Z −1 −n f(x) := f ∗ η(x) = f(x + w)η( w) dw. Rn

The convexity of this molliﬁed function f can be easily checked: Z −1 −n f(λx + (1 − λ)y) = f(λx + (1 − λ)y + w)η( w) dw Z ≥ [λf(x + w) + (1 − λ)f(y + w)]η(−1w)−n dw (4.7)

= λf(x) + (1 − λ)f(y).

18 Moreover, f ≥ f since Z η(−1w) + η(−−1w) f (x) = f(x + w) −ndw 2 (4.8) Z f(x + w) + f(x − w) Z = η(−1w)−ndw ≥ f(x)η(−1w)−ndw = f(x), 2 the uniform Lipschitz condition of f implies that kf −fk∞ < ∞ for all > 0, and that kf −fk∞ → 0, as → 0. Next, observe that Z Z Z f1(λx) f(λx + w) w n = η(w)dw = f(x + )η(w)dw = f(x + y)η(λy)λ dy = f −1 (x). (4.9) λ λ λ λ

Let M0 = ∂Mˆ 0 be the convex hypersurface {(x, xn+1): xn+1 = f1(x)}. Then (4.9) implies that the tangent cone of M0 at infinity is CΓ0. Theorem 1.2 shows the existence of a smooth solution n−1 Mt, for t ∈ [0, ln |S | − ln |Γ0|) with initial data M0. n−1 ∞ We will show next that: for t ∈ [0, ln |S | − ln |Γ0|), Mt converges to CΓt in Lloc. Assuming this, let us first finish the proof of the lemma: for each > 0, (4.9) implies M0 = {xn+1 = f(x)} and thus (4.8) implies Mˆ 0 is contained in Nˆ0. Mˆ t is an innermost candidate as Mˆ t is. Thus Mˆ t ⊂ Nˆt by the comparison in Remark 4.1 and, by taking → 0, we conclude CΓˆt ⊂ Nˆt.

We are left to prove the convergence of Mt to CΓt. Following the construction in Theorem 1.2, let Mi,t be compact convex solutions which approximate Mt from inside. Since Mˆ i,0 is contained in CΓˆ0, the comparison principle implies Mˆ i,t ⊂ CΓˆt, showing Mˆ t ⊂ CΓˆt. Let us express Mt by the entire graphs xn+1 = f1(x, t) and CΓt by xn+1 = f(x, t). Observe that the gradients |Df1| n n−1 and |Df| are uniformly bounded on (x, t) ∈ R × [0, |S | − ln |Γ0|). This is because en+1 is an interior point of Γˆ0 and hence Mˆ t and CΓˆt contain a fixed round convex cone whose axis in ∞ n ∞ n positive en+1 direction. Next, note f(·, 0) ∈ Cloc(R \{0}) and f(·, 0) → f(·, 0) in Cloc(R \{0}) as → 0. This convergence and (4.9) imply that there is C > 0 such that H(|x| + 1) ≤ C for M0. Proposition 2.10 then implies that the mean curvature H(x, t) of Mt at (x, f1(x, t)) satisfies 0 the bound H(x, t)(|x| + 1) ≤ C . Next, since CΓt works as a conical barrier outside, Theorem 1.4 (see also Remark 3.1) can be applied to the approximating compact solutions Mi,t to conclude that −1 (H|F |) ≤ Cδ on Mt for t ∈ [0,T (M0) − δ]. All the bounds above imply that the solutions Mt when viewed as entire graphs, have uniform gradient bounds, locally uniform height bounds, and locally uniform bounds of H and H−1 on compact domains which do not contain the origin. In the previous statement, the uniformity of estimates holds both in > 0 and t ∈ [0,T (M0) − δ] for all fixed δ > 0. By the regularity estimates of uniformly parabolic equations, we may pass to a sub-sequential limit and obtain, as −1 → 0, f( x, t) converges to some f0(x, t) smoothly on (Bδ−1 \ Bδ) × [0,T − δ) for all δ > n 0. It follows that {xn+1 = f0(x, t)} is a smooth solution to IMCF on R \{0}. Meanwhile, {xn+1 = f0(x, t)} represents a convex cone as it is the blow-down of Mt. Combining these together, 0 0 n 0 {xn+1 = f0(x, t)} = CΓt for some smooth convex hypersurface Γt ⊂ S and Γt evolves by the IMCF. 0 0 Since Γt is the unique solution to IMCF with initial data Γ0 = Γ0, we conclude that Γt = Γt. This proves that f0(x, t) = f(x, t) and the convergence of Mt to CΓt.

Proposition 4.4. Let Mt = ∂Mˆ t be a convex non-compact solution obtained from Theorem 1.2 ˆ ˆ ˆ n n−1 ˆ and Γt be the link of the tangent cone of Mt. Suppose Γ0 has no interior in S but H (Γ0) > 0. Then Γˆt has interior for t > 0.

19 ˆ Γˆ0 Γt

Figure 4: Proposition 4.4

n−1 Proof. We may assume that H (Γˆ0 ∩ {xn+1 > 0}) > 0 and 0 ∈ int(Mˆ 0), by rotating and ˆ n+1 ˆ ˆ translating the coordinates respectively. Since CΓ0 is convex in R , CΓ0 ∩ {xn+1 = 1} =: Ω is n n n−1 convex in R . Moreover, Ωˆ has no interior in R and H (Ω)ˆ > 0 since the gnomonic projection n n (A.18) is (locally) bi-Lipschitz map between R and S ∩ {xn+1 > 0}. If a convex set in Euclidean space has no interior, then it should be contained in some hyperplane. Therefore, Ωˆ is contained in a (n − 1)-plane and Ωˆ has interior in that (n − 1)-plane as otherwise Ωˆ would be contained in a n−1 n (n−2)-plane, contradicting H (Ω)ˆ > 0. As a result, Ωˆ in R ×{1} contains a (n−1)-dimensional disk of some radius r0. The cone generated by this (n − 1)-disk is contained in CΓˆ0 and thus CΓˆ0 should contain a rotated image of the following n-dimensional cone for some 0 < r ≤ r0: q n+1 2 2 C := {(x1, x2, . . . , xn, 0) ∈ R : x1 + ··· + xn−1 ≤ rxn}. √ 2 n By letting c := r/ 1 + r , observe Bca(aen+1) × {0} is contained in C for all a > 0. Since we ˆ ˆ n assumed 0 ∈ int(M0), there are ~v ∈ Γ0 ⊂ S , > 0, and a rotation operator J such that the family n ˆ of expanding thin disks a~v + J(Bca(0) × (−, )) are contained in M0 for all a > 0. ˆ n n+1 ˆ Claim 4.2. Let DR, := BR(0) × (−, ) ⊂ R be a thin disk ∈ (0,R/100) and Nt = ∂Nt be a smooth IMCF. If Dˆ ⊂ Nˆ , then there is c > 0 such that Bn+1 (0) ⊂ Nˆ for t ∈ [0, c ]. R, 0 n cnRt t n

Proof of Claim. Let us smoothen out the edges of DR, := ∂DˆR, to obtain a smooth pancake like convex hypersurface ΣR, by a similar method of Lemma 4.3: consider the convex conical n+2 n+2 hypersurface in R generated by DR, × {1} ⊂ R from the origin. We can represent this cone −1 by an entire graph xn+2 = f(x) of a 1-homogeneous function f(x). Then f ({1}) = DR,. If we consider the regularization of f, say fδ, as constructed in Lemma 4.3, then fδ ≥ f and it is −1 smooth convex function. For sufficiently small δ, the level set fδ ({1}) =: ΣR, is a smooth convex n+1 ˆ hypersurface in R which is contained in DR,. Since the regularization of a linear function is n+1 the same as itself, DR, and ΣR, coincide on BR/2 (0) for small δ > 0. Observe that ΣR, has the same symmetry as DR,. i.e. O(n)-rotational symmetry and reflection symmetry with respect to {xn+1 = 0}. Thus the IMCF ΣR,(t) starting at ΣR, must contain two points (0, + c(t)) and (0, − − c(t)), for each t > 0, at which the normal vectors to ΣR,(t) are 0 en+1 and −en+1, respectively. In view of Lemma 2.10, c (t) > cR as long as + c(t) < R/2. Since n ˆ ΣR,(t) contains these two points and the disk BR/2 × {0}, convexity implies that ΣR,(t) includes our desired ball. This finishes the proof as Σˆ R,(t) ⊂ Nˆt by the comparison principle.

20 R cnRt

t = 0 t > 0

Figure 5: Claim 4.2

n+1 ˆ ˆ By the claim, a~v + Bcncat(0) ⊂ Mt for all 0 ≤ t ≤ cn and a ≥ a0 some a0 > 0. Γt has interior since Mˆ t contains a round cone.

We now prove Theorem 1.3. Note that the same proof works for both T = 0 and T ∈ (0, ∞).

Proof of Theorem 1.3. Let Mˆ t, for t ≥ 0, be the innermost candidate solution from Mˆ 0 (see Def- ˆ ˆ ˆ n+1 inition 4.1). Since Mt ⊂ Nt (by Lemma 4.2), it suffices to show MT +τ = R for τ > 0. One ˆ ˆ n+1 useful observation is that if Mt0 contains a half space at t0 ≥ 0 then Mt = R for t > t0: suppose ˆ 0 ˆ 0 {xn+1 ≥ 0} ⊂ Mt0 . By the comparison with spherical solutions, ∂Breτ /n (ren+1) ⊂ Mt0+τ . Since n+1 0 0 ˆ 0 ˆ 0 Br(eτ /n−1)(0) ⊂ Breτ /n (ren+1) ⊂ Mt0+τ for all r > 0, we get Mt0+τ = R . ˆ ˆ ˆ0 We may assume 0 ∈ int(M0). Suppose Mt0 contains a cone CΓ0 with a smooth strictly convex 0 0 link Γ0 in an open hemisphere. Since the IMCF running from Γ0 converges to an equator as n−1 0 ˆ n+1 t → ln |S | − ln |Γ0| (by [22][34]), Lemma 4.3 and the observation above imply that Mt = R n−1 0 for t > t0 + ln |S | − ln |Γ0|. In view of the approximation in Lemma A.9, the same assertion ˆ0 holds when Γ0 is a convex set with interior and contained in an open hemisphere. In general, the link of initial tangent cone Γˆ0 is a convex set in a closed hemisphere. After a rotation, we may assume that en+1 ∈ Γˆ0 and represent Mˆ 0 using a convex function f on a convex n ˆ ˆ domain Ω0 ⊂ R by M0 = {(x, xn+1): xn+1 ≥ f(x), x ∈ Ω0}. Let us define M,0 := {(x, xn+1): p 2 1,1 xn+1 ≥ f(x)+ |x| + 1, x ∈ Ω0} and observe that it is still convex with Cloc boundary. The links of the tangent cones of Mˆ ,0, say Γˆ,0, are contained in a fixed open hemisphere and Γˆ,0 increases + to Γˆ0 as → 0 . P (Γˆ,0) is monotone in by Lemma A.11. Let us assume the following claim for the moment. + Claim 4.3. P (Γˆ,0) increases to P (Γˆ0) as → 0 . ˆ 0 n−1 ˆ 0 Choose M0,0 such that T := ln |S | − P (Γ0,0) < T + τ/2 and note T > 0 by Lemma 0 A.11. By Theorem 1.2, there is a smooth strictly convex solution M0,t for 0 < t < T . At 0 ˆ 0 ˆ 0 t0 := min(τ/4,T /2), the link of the tangent cone of M ,t0 , say Γ ,t0 , has interior by Proposition ˆ 0 0 4.4. Γ ,t0 is contained in an open hemisphere due to strict convexity of M ,t0 . By Lemma A.11, 00 n−1 n−1 0 ˆ 0 ˆ ˆ T := ln |S | − ln |Γ ,t0 | ≤ ln |S | − P (Γe ,0) < T + τ/2. M,t0 ⊂ Mt0 by Lemma 4.2. Because n+1 ˆ 0 ˆ ˆ CΓ ,t0 ⊂ Mt0 , we apply the assertion in the second paragraph to conclude that Mt = R for 00 00 t > t0 + T . Note that t0 + T < T + 3τ/4 finishing the proof.

21 Proof of Claim. Let us deﬁne a locally Lipschitz map

n+1 n+1 (x, xn+1 + |x|) ψ : R − {0} −→ R by ψ(x, xn+1) = , |(x, xn+1 + |x|)|

ˆ ˆ n and observe Γ,0 = ψ(Γ0). ψ induces a bijection between {xn+1 = α|x|} ∩ S and {xn+1 = n n (α + )|x|} ∩ S for all α ∈ R, hence ψ induces a bijection from and onto S . Let us define p+φ(p) φ(x, xn+1) := |x|en+1 so that ψ(p) = |p+φ(p)| . At each differentiable point of ψ, the differential of ψ, Dψ, represented by a (n + 1) × (n + 1) matrix is 1 p + φ Dψ (p) = (I + Dφ) − (p + φ)T (I + Dφ). |p + φ| n+1 |p + φ|3 n+1 n Using |p| = 1, |φ| ≤ 1 and |Dφ| ≤ 1 on p ∈ S , we may find positive C and 0 such that for n n n+1 0 < ≤ 0 at each p ∈ S − {±en+1} and V ∈ TpS = {W ∈ R : hW, pi = 0}

(1 − C)|V | ≤ |(dψ)p(V )| = |Dψ(p)V | ≤ (1 + C)|V |.

n−1 n−1 The shows that (1 − C) |Γ0| ≤ |Γ,0| ≤ (1 + C) |Γ0|.

In the remaining part of this section, we assume 0 < T (M0) < ∞ and discuss the behavior of − innermost solution Mˆ t as t → T . By Theorem 1.2, Mt is a smooth IMCF for 0 < t < T which is innermost by Lemma 4.2. Observe Γˆt, the link of the tangent cone of Mˆ t, is a kind of weak solution n to the IMCF on S . ˆ0 ˆ ˆ0 Lemma 4.5. Suppose Γ0 ⊂ Γ0 for some compact convex set Γ0 with smooth strictly convex boundary 0 0 ˆ0 ˆ ˆ ˆ0 ˆ0 n and let Γt be the IMCF from Γ0. Then, Γt ⊂ Γt. Similarly, if Γ0 ⊂ Γ0 for some Γ0 6= S with ˆ ˆ0 0 smooth strictly convex boundary, then Γt ⊂ Γt. The inequalities hold as long as Γt exists. ˆ0 ˆ n+1 ˆ0 ˆ Proof. For such a Γ0 ⊂ Γ0, we may ﬁnd a vector v ∈ R such that CΓ0 + v ⊂ M0. Then Lemma ˆ0 ˆ ˆ0 ˆ 4.3 implies that CΓt + v ⊂ Mt, hence Γt ⊂ Γt. ˆ ˆ0 In the other case, if we ﬁrst assume strict inclusion Γ0 ⊂⊂ Γ0, then the proof goes similarly except that we don’t use Lemma 4.3 and use the usual comparison principle between the approximating compact solutions Mi,t and the conical barrier outside (and then take i → ∞). For ˆ ˆ0 ˆ0 ˆ0 general Γ0 ⊂ Γ0, consider a strictly decreasing approximating sequence {Γi,0} of Γ0, obtained from ˆ ˆ0 Lemma A.9, and apply the result which assumes strict inclusion. This shows that Γt ⊂ ∩iΓi,t and 0 ˆ0 Γt ⊂ ∩iΓi,t. In view of Lemma A.9, we have the following equalities

ˆ0 0 t 0 t 0 0 |∂(∩iΓi,t)| = lim |Γi,t| = lim e |Γi,0| = e |Γ0| = |Γt| i→∞ i→∞

ˆ0 ˆ0 and the strict outer area minimizing property (the last assertion in Lemma A.9) implies Γt = ∩iΓi,t, ˆ ˆ0 and this shows Γt ⊂ Γt.

22 − Remark 4.2 (Asymptotic behavior of Mt, as t → T ). In the case when Γ0 admits a smooth strictly convex IMCF, the comparison in Lemma 4.5 implies that Γt must be this solution and, by the result 1,α − n−1 ˆ of [34] or [22], Γt converges to an equator in C , as t → T = ln |S |−ln P (Γ0). In other words, − the solution Mt becomes ﬂat, as t → T , in the sense that the tangent cone CΓt converges to a 1,α ˆ ˆ ˆ n+1 ˆ hyperplane in Cloc . Next, since ∪t

There exist, however, some u0(x) ≥ 0 such that 1 Z lim u0 = C > 0 R→∞ n−1 R BR but for which no solution exist with initial data u0, for any T > 0. Such solutions are characterized by a non-radial structure at spatial infinity. Indeed, for initial data which is bounded from below near infinity by positive radial functions there is a necessary and sufficient for existence as follows:

23 there is an explicit constant E∗ > 0 such that if the problem has a solution for t ∈ (0,T ), then Z R Z 1 ds ∗ 1/2 lim sup n−1 u0 dx ≥ E T . R→∞ R 0 wns Bs

Moreover, if u0 is radially symmetric and locally bounded, Z R Z 1 ds ∗ 1/2 lim inf u0 dx ≥ E T R→∞ n−1 R 0 wns Bs n guarantees an existence of a solution on R × (0,T ). For non-radial u0, there is a similar condition in Theorem 1.3. [12]. Every result mentioned here is in some sense sharp when explicit solutions p2(n − 1)(T − t) vT (x, t) = + |x| are considered. These results explain partial conditions for non-existence and existence of solutions, but a complete description was missing. For the convex IMCF, however, Theorem 1.2 and 1.3 depict a fairly complete picture. This was possible by the geometric estimate Theorem 1.4. Note that this lower bound has the same decay of vT (x, t) above. Instead of the integral operators used in [12], the asymptotic geometry of M0 is used to provide the lower bound on H in Theorem 1.4. It would be interesting to see if a similar idea could be implemented in the theory of ultra-fast diﬀusion equation (1.2), with m < 0.

A Appendix

A.1 Strict convexity of solutions in space forms Throughout this subsection, unless otherwise stated, we assume the solutions are smooth immersed n-dimensional possibly incomplete submanifods in (N n+1, g¯) which is a space form of sectional curvature K ∈ R. This ambient space, in particular, includes Euclidean space, the sphere, or hyperbolic space. Since smooth solutions are strictly mean convex (H > 0), this necessarily implies that the solutions are orientable. As before, we denote the outward unit normal which is opposite to the mean curvature vector by ν, the norm of mean curvature by H, and the second fundamental form with respect to −ν by hij. Moreover, we say a solution is convex if hij is nonnegative definite everywhere. Note that the convex solution in this subsection is weaker notion than the convex solution in other sections which uses Definition 2.1. For instance, a C2 hypersurface convex in the sense of Definition 2.1 should necessarily be complete and embedded. Our aim is to prove Theorem A.3, a strong minimum principle on λ1. However by looking at the evolution of the second fundamental form hij given in (2.2), it is not clear that the convexity is preserved. To do so we need to use a viscosity solution argument and we need the following lemma shown from [5].

Lemma A.1 (Lemma 5 in Section 4 [5]). Suppose that φ is a smooth function such that λ1 ≥ φ everywhere and λ1 = φ at x =p ¯ ∈ Ω. Let us choose an orthonormal frame so that

hij = λiδij at p¯ ∈ Ω with λ1 = λ2 = ... = λµ < λµ+1 ≤ ... ≤ λn.

We denote µ ≥ 1 by the multiplicity of λ1. Then at p¯, ∇ihkl = δkl∇iφ for 1 ≤ k, l ≤ µ. Moreover, X −1 2 ∇i∇iφ ≤ ∇i∇ih11 − 2 (λj − λ1) (∇ih1j) . j>µ

24 Proposition A.2. For n ≥ 1, let F :Ω × (0,T ) → (N n+1, g¯) be a smooth convex solution to the i IMCF where (N, g¯) is a space form. Let λ1 denote the lowest eigenvalue of hj. Then u := λ1/H is a viscosity supersolution to the equation ∂ 1 1 W u − ∆u + hV, ∇ui + u ≥ 0 (A.1) ∂t H2 H3 H4 where V is a vector field, and W is a scalar function such that |W |, |V | ≤ C(|∇H|, n) at each point. i Proof. Using equation (2.2) in Remark 2.1, we compute the evolution of hj/H: 1 hi |A|2 hi hikh 2 ∂ − ∆ j = 2 j − 2 kj + (∇ H∇mhi − ∇iH∇ H). (A.2) t H2 H H2 H H2 H4 m j j Suppose a smooth function of space time, namely φ/H, touches λ1/H from below at (¯p, t¯). At time t¯ aroundp ¯, let us fix a time independent frame {ei} using the metric g(t¯) as in Lemma A.1. 1 ¯ 1 ¯ ¯ Since φ ≤ λ1 ≤ h1 and they coincide at (¯p, t), ∂tφ ≥ ∂th1 at (¯p, t). At this point (¯p, t) with the frame {ei}, we use Lemma A.1, equation (A.2), and the Codazzi identity ∇ihjk = ∇jhik to obtain φ ∂ h1 1 h1 2 X X ≥ 1 − ∆ 1 + (λ − λ )−1|∇ h |2 H ∂t H H2 H H3 j 1 i 1j i j>µ h1 2 X = 1 + (λ − λ )−1|∇ h |2 H H3 j 1 1 ij i≥1,j>µ   P 2 P 2 (A.3) 2 j λj − 2λ1 j λj λ1 2 2 X |∇1hij| = 2 + 4 ∇mH∇mh11 − |∇1H| + H  H H H λj − λ1 i≥1,j>µ   2 2 2 X |∇1hij| ≥ 4 ∇mH∇mφ − |∇1H| + H  . H λj − λ1 i≥1,j>µ P P 2 2 The last inequality uses λ1 j λj ≤ j λj = |A| , which holds on convex (or more generally on mean convex) hypersurfaces. Next, note that X 2 2 X ∇1H∇1H = ∇1hii∇1hjj = 2µ∇1H∇1φ − µ |∇1φ| + ∇1hii∇1hjj. (A.4) i,j i>µ,j>µ φ φ Since H ∇ = ∇φ − ∇H, we have the following for each fixed unit direction e H H m φ φ ∇ H∇ φ = H∇ H∇ + |∇ H|2. (A.5) m m m m H H m We first plug (A.5) with m = 1 into (A.4) and then plug that into the last line of (A.3) to obtain φ 2 X φ 2(1 − 2µ) φ ≥ |∇ H|2 + |∇ H|2 H H4 m H H4 1 H m>1 2 2 2 X φ 2(1 − µ) φ 2µ |∇1φ| + ∇ H∇ + ∇ H∇ + H3 m m H H3 1 1 H H3 H (A.6) m>1   2 X X + H (λ − λ )−1|∇ h |2 − ∇ h ∇ h . H4  j 1 1 ij 1 ii 1 jj i≥1,j>µ i>µ,j>µ

25 We now use the convexity, λ1 ≥ 0, in the proof of the following claim. h P −1 2 P i Claim A.1. H i≥1,j>µ(λj − λ1) |∇1hij| − i>µ,j>µ ∇1hii∇1hjj ≥ 0 on {λ1 ≥ 0}. 2µ2 |∇ φ|2 Assuming that the claim is true, then by taking away the good term 1 in (A.6), we H3 H easily conclude that (A.1) holds by choosing a vector ﬁled V and a scalar function W as a function of ∇H accordingly. Thus it remains to show the claim. P P Proof of Claim A.1. Since λ1 ≥ 0, H = l≥1 λl ≥ l>µ λl, the claim follows by:

X −1 2 X X −1 2 X −1 2 H (λj − λ1) |∇1hij| ≥ λl λi |∇1hii| = λjλi |∇1hii| i≥1,j>µ l>µ i>µ i>µ,j>µ −1 2 −1 2 X λjλi |∇1hii| + λiλj |∇1hjj| = (A.7) 2 i>µ,j>µ X ≥ ∇1hii∇1hjj. i>µ,j>µ

n+1 Now, let Mt ⊂ N be a smooth complete convex solution for t > 0, which could be either compact or non-compact. One expects Mt to be strictly convex, that is to have λ1 > 0 for t > 0. Indeed, this follows easily by Proposition A.2 and the strong minimum principle for nonnegative supersolutions which is a consequence of the weak Harnack inequality for nonnegative viscosity super solutions to (locally) uniformly parabolic equations. (See Chapter 4 in [43]).

Theorem A.3. Suppose F : M n × (0,T ) → (N n+1, g¯) is a smooth complete convex solution to n+1 the IMCF with H > 0 where (N , g¯) is a space form. If λ1(p0, t0) = 0 at some (p0, t0) with n 0 < t0 < T , then λ1 = 0 on M × (0, t0].

−1 Proof. Since solution is smooth, |H|, |∇H|, and |H | = |∂tF | are locally bounded. Therefore, λ1 is a nonnegative supersolution to equation (A.1) which is locally uniformly parabolic with bounded coeﬃcients. We can apply strong minimum principle on a sequence {Ωk} of expanding domains n containing (p0, t0) such that M = ∪kΩk and conclude that the theorem holds.

n n+1 Corollary A.4. Let F : M × (0,T ) → R be a smooth complete convex solution to the IMCF. n If H (ν[Mt0 ]) > 0 at t0 ∈ (0,T ), then Mt0 is strictly convex.

n n Proof. If it is not, Theorem A.3 implies λ1 ≡ 0 for all M × (0, t0]. The Gauss map ν : Mt0 → S is Lipschitz. Thus the area formula implies Z Z 0 −1 n n Kdµ = H (ν ({z}))dH (z) ≥ H (ν[Mt0 ]). (A.8) n Mt0 S Since K ≡ 0, this is a contradiction and proves the assertion.

26 Remark A.1 (Strict convexity of compact solutions). We may use Theorem A.3 to show the initial strict convexity of smooth compact solution is preserved. Let Mt be the smooth IMCF running from smooth compact immersed hypersurface M0 with positive hij. By considering the first time λ1 becomes zero at some point, Theorem A.3 implies that there is no such time as long as the n+1 smooth solution exists and this proves hij is positive for Mt. Furthermore, if Mt is a flow in R , then it is the boundary of a compact convex set with interior by Hadamard [25], showing Mt is convex in the sense of Definition 2.1.

A.2 Speed estimate for closed star-shaped solutions The goal is this section is to give an alternative proof of Theorem 1.1 in [30] which will be based on the maximum principle. The theorem holds in any dimension n ≥ 1.

n n+1 Theorem A.5 (Theorem 1.1 in [30]). Let F : M × [0,T ] → R be a smooth closed star-shaped n solution to (1.1) such that M0 := F0(M ) satisﬁes

0 < R1 ≤ hF, νi ≤ R2. (A.9)

Then, there is a constant Cn > 0 depending only on n such that 1 R2 1 t ≤ C 1 + R e n (A.10) n 1/2 2 H R1 t holds everywhere on M n × [0,T ].

Proof. Since M0 satisﬁes (A.9), by Proposition 1.3 in [30], we have

t t R1 ≤ R1 e n ≤ hF, νi ≤ |F | ≤ R2 e n (A.11) for all 0 < t < +∞. Let us denote w := hF, νi−1 and we will consider a function

ϕ1−(w) eγ|F |2 Q := H for some function ϕ := ϕ(w), constants γ > 0 and ∈ (0, 1) which will be chosen shortly. By (i) in Lemma 2.6,

1 2 1 2n 4 ∂ − ∆ ln e|F | = ∂ − ∆|F |2 = − + . t H2 t H2 H2 Hw Moreover, by (7) in Lemma 2.5 and Lemma 2.9 with β = −1,

1 |A|2 2 ∂ − ∆w = − w − |∇w|2 t H2 H2 wH2 and hence, on {ϕ 6= 0},

1 (∂ − H−2∆)ϕ 1 |∇ϕ|2 |A|2 ϕ0w |∇w|2 ϕ0 ϕ00 ϕ02 ∂ − ∆ ln ϕ = t + = − − 2 + − . t H2 ϕ H2 ϕ2 H2 ϕ H2 wϕ ϕ ϕ2 Inspired by the choice of ϕ in the well known interior curvature estimate by Ecker and Huisken in [17] (see also [7]), we deﬁne s ϕ(s) := −1 . (A.12) 2R1 − s

27 For this ϕ := ϕ(w), under the notation ϕ0 = ϕ0(w) and ϕ00 = ϕ00(w), a direct computation yields

ϕ0w 2 ϕ0 ϕ00 ϕ02 ϕ02 = − and 2 + − 2 = 2 . ϕ 2 − wR1 wϕ ϕ ϕ ϕ Lemma 2.7 and the computations above imply

1 |A|2 1 |∇H−1|2 4 2n |A|2 ϕ0w 1 |∇ϕ|2 ∂ − ∆ ln Q = + + γ − − (1 − ) + t H2 H2 H2 H−2 Hw H2 H2 ϕ H2 ϕ2 2 −1 2 2 wR1 − 2 |A| 4 2n − 4 γ |F | ∇ |F | = − 2 + γ − γ 2 2 − wR1 H Hw H " 2 # 1 |∇ϕ|2 |∇H−1|2 |∇e|F | |2 − (1 − ) − + −1γ2 . H2 ϕ2 H−2 |e|F |2 |2 (A.13) 2 −1γ2 ∇|F |2 Note that we have added and subtracted the term H2 in the last equality. At a nonzero critical point of Q, ∇Q ∇ϕ ∇e|F |2 ∇H−1 0 = = (1 − ) + γ + , Q ϕ e|F |2 H−1 and thus

2 2 * 2 + 2 2 ∇H−1 2 ∇ϕ ∇e|F | ∇ϕ 2 ∇ϕ ∇e|F | ∇e|F | = (1 − ) + γ = (1 − )2 + 2(1 − )γ , + γ2 −1 |F |2 |F |2 |F |2 H ϕ e ϕ ϕ e e 2 2 ∇ϕ 2 1 − ∇e|F | ≤ ((1 − )2 + (1 − )) + (1 + )γ2 |F |2 ϕ e |∇ϕ|2 |∇e|F |2 |2 = (1 − ) + −1γ2 . ϕ2 e2|F |2

R1 For a given T > 0, note that T ≤ wR1 ≤ 1. It remains to choose and γ. The choice R2e n R1 := T makes the ﬁrst term on RHS of the second equality in (A.13) nonpositive. Next, choose 2R2e n 1 −1 2 γ := T > 0 so that 4 γ|F | ≤ n on Mt for t ∈ [0,T ]. Combining the choices and 4n 2 (R2e n ) estimates, at a nonzero spatial critical point of Q,

1 (∂ − H−2∆)Q |∇Q|2 n 4 ∂ − ∆ ln Q = t + ≤ γ − + . (A.14) t H2 Q Q2 H2 H w

We will now apply the maximum principle on Qˆ := tQ. Suppose that nonzero maximum of Qˆ n on M × [0,T ] occurs at the point (p0, t0), which necessarily implies t0 > 0. At this point, (A.14) implies

1 n 4 1 n 8 T 1 ˆ 2 2 n 0 ≤ ∂t − 2 ∆ ln Q ≤ γ − 2 + + ≤ γ − 2 + R2e + (A.15) H H H w t0 2H n t0 where the second inequality comes from

4 8 n 8 2 2 T n ≤ + ≤ R e n + . H w nw2 2H2 n 2 2H2

28 The rest is a standard argument shown in the proof of Theorem 3.1 [17]. By the choices of , γ, bounds (A.11) and R 1 ≤ ϕ((R eT/n)−1) ≤ ϕ(w) ≤ ϕ(R−1) = 1, T/n 2 1 2R2e we proceed and obtain, for every (p, t) ∈ M n × (0,T ],

2− 1 R2 T T 1 n n 2 2 (p, t) ≤ Cn e (R2e ) 1 + . (A.16) H R1 t

Now for time t > 1, we can alway apply this estimate starting at time t − 1. Inequality (A.11) implies that the ratio between star-shapedness bounds from above and below remains unchanged T 2− 2− over time. This way we can replace (R2e n /R1) in the above estimate by (R2/R1) after 2− 2 possibly enlarging the constant Cn. Since (R2/R1) ≤ (R2/R1) , the theorem follows.

A.3 Smooth approximation of convex hypersurfaces In this appendix, unless it is stated otherwise, convergence of compact convex sets (or their boundary hypersurfaces) means the convergence in the Hausdorﬀ metric, deﬁned as

dH (A, B) := max(sup inf kx − yk, sup inf kx − yk). (A.17) x∈A y∈B x∈A y∈B

In case where A, B are compact convex sets, it is known that dH (A, B) = dH (∂A, ∂B) (Lemma n+1 δ 1.8.1 [37]). For a compact convex set Mˆ ⊂ R , Mˆ denotes the δ-envelope ˆ δ n+1 ˆ ˆ M := {x ∈ R : dist (x, M) ≤ δ} = M + δB0(1).

Here the dist (x, Mˆ ) is measured in Euclidean distance.

n+1 Lemma A.6. For n ≥ 1, let Mˆ ⊂ R be a compact convex set with non-empty interior. Sup- pose 0 ∈ int(Mˆ ). Then there is a sequence compact convex sets Mˆ k with smooth strictly convex boundaries such that

ˆ de −1 ˆ ˆ in −1 ˆ Mk := (1 + k )Mk is strictly decreasing, Mk := (1 − k )Mk is strictly increasing,

ˆ ˆ de ˆ in and both converge to M as k → ∞. Here, we say that Σk (Σk ) is strictly decreasing (strictly ˆ de ˆ de ˆ in ˆ in increasing) if Σk+1 ⊂ int (Σk ) (Σk ⊂ int (Σk+1), respectively). Proof. By Theorem 3.4.1 in [37] and its immediate following discussion, there is a sequence of compact convex sets Mˆ k with non-empty interior and smooth strictly convex boundaries such that −1 −1 dH (M,ˆ Mˆ k) → 0. Note that (1 + k )Mˆ and (1 − k )Mˆ are strictly monotone sets. Hence, using −1 −1 a diagonal argument, we may choose a subsequence of Mˆ k so that (1 + k )Mˆ k and (1 − k )Mˆ k are strictly monotone as well.

ˆ n n ˆ Let Γ be a compact convex set in the upper open hemisphere S ∩ {xn+1 > 0} =: S+. Since CΓ n+1 ˆ ˆ ˆ n is convex in R , if we deﬁne Ω × {1} := CΓ ∩ {xn+1 = 1}, then Ω is a compact convex set in R . ˆ ˆ ˆ ˆ n+1 n Γ and Ω are related by Γ = ϕ(Ω) using the gnomonic projection (x, 1) ∈ R 7→ ϕ(x) ∈ S+

1 n ϕ(x) = (x, 1) for x ∈ R . (A.18) (1 + |x|2)1/2

29 n ∗ n ˆ ˆ Since ( , ϕ g n ) and ( , g n ) are isometric, we have |∂Γ| = |∂Ω|ϕ∗g n . Here, the surface area R S S+ S S n measure |·|ϕ∗g n is (n−1)-Hausdorﬀ measure on metric space ( , dϕ∗g n ), induced from Riemannian S R S n ∗ ˆ structure (R , ϕ gSn ). Since ∂Ω is (n − 1)-rectiﬁable, we will use the area formula and avoid using ˆ the metric dϕ∗g n in actual computation of |∂Ω|ϕ∗g n . S S ˆ n+1 Lemma A.7. For n ≥ 1, assume that a sequence of compact convex sets Mk ⊂ R converges to a compact convex set Mˆ with non-empty interior. Then limk→∞ |Mk| = |M|. Similarly, if a sequence ˆ n ˆ n of compact convex sets Γk ⊂ S+ converges to a compact convex set with non-empty interior Γ ⊂ S+, then limk→∞ |Γk| = |Γ|.

Proof. Our proof is a modification of the proof of Theorem 4.2.3 in [37]. For the first part, we may assume 0 ∈ int(Mˆ k) for all k. Let ρ(Mˆ k, ·) be a spherical parametrization of Mk around 0, ˆ n ˆ meaning that ρ(Mk, y) y for y ∈ S is a point in Mk. Let ν(Mk, y) denote an arbitrary outer unit normal (in the sense of supporting normal) of Mˆ k at ρ(Mˆ k, y)y. The normal ν(Mˆ k, y) is n n ˆ ˆ unique for H -almost all y ∈ S (Theorem 2.2.5 in [37]). If Mk converges to M as k → ∞, then n ρ(Mˆ k, ·) → ρ(M,ˆ ·) everywhere. Moreover, ν(Mˆ k, ·) → ν(M,ˆ ·) H -almost everywhere, otherwise if ˆ 0 0 0 ν(Mik , y ) → ν for some ν as ik → ∞, this would implies that M has a supporting hyperplane {hx − ρ(M,ˆ y0)y0, ν0i = 0} at ρ(M,ˆ y0)y0, but on the other hand the outer normal of M uniquely n exists H -almost everywhere. Finally, since Mˆ k contains the origin in its interior, there is a uniform δ > 0 such that hy, ν(Mˆ k, y)i ≥ δ for all k and y. n n 1 n n Let ψ : U ⊂ R → S , z = (z , . . . , z ) 7→ ψ(z) = y, be a smooth local coordinate chart of S ˆ and gij be the metric gSn on U. Note that ρ(Mk, ψ(·)) is a Lipschitz function. At each point y where the function ρk(y) := ρ(Mˆ k, y) is differentiable, one can directly compute that

ρ y − gij ∂ρk ∂y ˆ k ∂zi ∂zj ν(Mk, y) = q . 2 2 ρ + kdρkkg n k S

Thus from the convergence of ν(Mˆ k, ·) and the lower bound hy, ν(Mˆ k, y)i ≥ δ, if ρ∞(y) := ρ(M,ˆ y), then we have ∂ρk ≤ C and ∂ρk → ∂ρ∞ almost everywhere for all i = 1, . . . , n. ∂zi δ ∂zi ∂zi n+1 ˆ Denote by fk and f∞ : U → R the functions fk(z) := ρ(Mk, ψ(z))ψ(z) and f∞(z) := ˆ R n ρ(M, ψ(z))ψ(z). By the area formula, we have |fk(U)| = U Jfk (z)dz , where s s ∂f ∂f ∂ρ ∂ρ J (z) = det k (z), k (z) = det k k + ρ2g fk ∂zi ∂zj ∂zi ∂zj k ij q q 2n−2 2 2 p n−1 2 2 p = ρ (ρ + kdρkkg n ) det gij = ρ ρ + kdρkkg n det gij k k S k k S holds almost everywhere. The Lebesgue dominated convergence theorem, then yields that Z Z q n n−1 2 2 p |fk(U)| = Jfk (z)dz → ρ∞ ρ∞ + kdρ∞kg n det gij = |f∞(U)|, as k → ∞. U U S

Using a standard partition of unity argument, we conclude that |Mk| → |M| as k → ∞, which concludes the proof of the ﬁrst assertion of the lemma.

We will now prove the second assertion of the lemma. Note that the preimages of Γˆk and Γˆ n n ∗ under ϕ are compact convex sets in R . On a given compact set, the metrics induced by (R , ϕ gSn ) n and (R , gRn ) are equivalent. i.e. one is less than a constant multiple of the other and the constant

30 n ˆ −1 ˆ depends on the compact set. We may assume that there is some p ∈ R such that Mk := ϕ (Γk)−p −1 and Mˆ := ϕ (Γ)ˆ − p contain the origin in their interiors and dH (Mˆ k,M) → 0 as k → ∞. n−1 n−1 n−1 Deﬁning ρk, ρ∞ : S → R, a smooth local chart ψ : U ⊂ R → S , and the functions n fk, f∞ : U → R similarly as in the previous case (note that the dimension is 1 less than the dimension in the previous case), we get the convergence of ρk to ρ∞ with positive uniform upper ∂ρk ∂ρ∞ and lower bounds and the convergence of ∂zi to ∂zi with uniform bound on their absolute value. R n Withϕ ˜(x) := ϕ(x + p), the area formula says that |ϕ(fk(U) + p)| = U Jϕ˜◦fk (z)dz , where s ∂ϕ˜ ∂f α ∂ϕ˜ ∂f α J (z) = det (f (z)) k (z), (f (z)) k (z) . ϕ˜◦fk ∂xα k ∂zi ∂xα k ∂zj

Note that ϕ is a smooth function, which in particular has bounded higher order derivatives on each compact domain. Therefore the Lebesgue dominated convergence theorem yields that

|ϕ(fk(U) + p)| → |ϕ(f∞(U) + p)|, as k → ∞ and a partition of unity can be used to show

|ϕ(Mk + p)| = |Γk| → |ϕ(M + p)| = |Γ|.

The following is a well known lemma and a stronger statement than this also holds, but we provide a proof of this simple version for the completeness of our work.

n+1 Lemma A.8. For n ≥ 1, a convex hypersurface M = ∂Mˆ in R has the outer area minimizing property among convex hypersurfaces, i.e. if Mˆ ⊂ Mˆ 0 then |M| ≤ |M 0|. Moreover, when M is smooth strictly convex, then the property is strict in the sense that equality holds if and only if M = M 0.

Proof. The proof uses a standard calibration argument. Assume 0 ∈ int(Mˆ ). Assume M is smooth and strictly convex. Then λM, λ ≥ 1, gives a foliation of smooth strictly convex hypersurfaces. Let Mˆ 0 be a set containing Mˆ . The foliation gives a smooth vector field consisting of the outer 0 0 normal vectors ν of {λM}λ≥1. If we denote the unit normal on ∂(Mˆ \ Mˆ ) pointing from Mˆ \ Mˆ by ν0, then the divergence theorem implies Z Z Z Z Z Z 0 ≤ H = div ν = hν, ν0idA + hν, ν0idA = hν, ν0idA − dA Mˆ 0\Mˆ Mˆ 0\Mˆ M 0 M M 0 M and hence Z Z Z |M| = dA ≤ hν, ν0idA ≤ dA = |M 0|. M M 0 M 0 (The divergence theorem can be applied for a set with rough boundary when the boundary consists of convex hypersurfaces. One could also avoid doing this by approximating Mˆ 0 from outside using Lemma A.6 and Lemma A.7). The strict outer area minimizing is a consequence from the fact H > 0 on Mˆ 0 \ Mˆ . ˆ ˆ in For a general convex M = ∂M, we consider smooth approximation from inside, say Mk which in 0 was shown to exist in Lemma A.6. By the first case, |Mk | ≤ |M |. Lemma A.7 implies that in |Mk | → |M| as k → ∞ and this finishes the proof.

31 n We do have a similar result for convex hypersurfaces in S . ˆ n n Lemma A.9. For n ≥ 2, let Γ ⊂ S ∩ {xn+1 > 0} = S+ be a compact convex set with non-empty interior. Then Γˆ can be approximated from inside (and outside) by a strictly monotone sequence of n compact sets in S+ with smooth strictly convex boundaries. For any sequence of compact convex sets Γˆk converging to Γˆ, we have |Γk| → |Γ| as k → ∞. Moreover, Γˆ satisﬁes the outer area minimizing n ˆ ˆ0 n ˆ0 0 property on S+. That is, if Γ ⊂ Γ ⊂ S+ and Γ is convex, then |Γ| ≤ |Γ |, and if Γ is smooth strictly convex, then |Γ| = |Γ0| if and only if Γ = Γ0.

−1 n Proof. Let Ωˆ := ϕ (Γ)ˆ ⊂ R . Then by Lemma A.6, Ωˆ could be approximated from inside (and outside) by strictly monotone sequence of compact sets with smooth strictly convex boundaries. The images of these sequences of sets under ϕ give the desired approximating sequences. The convergence of area is shown in Lemma A.7. The proof of the second part is similar to the proof of Lemma A.8. Suppose ﬁrst Γ = ∂Γˆ ˆ ˆ0 n ˆ −1 ˆ is smooth and Γ ⊂ Γ ⊂ S ∩ {xn+1 > 0}. Fix p ∈ Ω = ϕ (Γ) and consider the foliation {λ(Ω − p) + p}λ≥1. Then the image of this foliation under ϕ, that is

n ϕ(λ(Ω − p) + p) ⊂ S , for λ ≥ 1,

n ˆ n gives a foliation of the region S+ − int(Γ) by smooth convex hypersurfaces in S . By the same calibration argument, we obtain |Γ0| ≥ |Γ|. In the non-smooth case we approximate Γ from inside by smooth sets and apply Lemma A.7.

ˆ n The next approximation lemma concerns with the case where Γ ⊂ S+ has empty interior. ˆ n Lemma A.10. For n ≥ 2, suppose Γ in S+ is a compact convex set which has empty interior in n ˆ S . Then there is {Γk} a sequence of compact convex sets with non-empty interior and smooth strictly convex boundaries which strictly decreases to Γˆ. For any such sequence Γˆk, |Γk| = |∂Γˆk| decreases to P (Γ)ˆ as k → ∞. Here P (Γ)ˆ is deﬁned by (1.4).

ˆ −1 ˆ n ˆ δ ˆ n Proof. Ω = ϕ (Γ) has empty interior in R . Consider the set Ω := δ-envelope of Ω in (R , gRn ). Then Ωˆ δ is a compact convex set with non-empty interior, implying that ϕ(Ωˆ δ) is a closed convex n set with non-empty interior in S . By a diagonal argument applied to the approximations of ϕ(Ωˆ 1/k), which is similar to the proof of Lemma A.6, we obtain the existence of a strictly decreasing approximation. ˆ ˆ n Let us now prove the convergence |Γk| → P (Γ), as k → ∞. If a convex set Ω ⊂ R has empty n n ∗ interior, it is contained in a hyperplane of R . Since a rotation is an isometry of (R , ϕ gSn ), ˆ ˆ n−1 ˆ we may assume that Ω ⊂ {xn = l}, for some l > 0. Let Σ ⊂ R denote the projection of Ω n−1 ˆ δ ˆ n−1 to {xn = 0} = R and Σ denote the δ-envelope of Σ in R with respect to the standard δ δ δ δ Euclidean metric. Observe that Σˆ × {l} = Ωˆ ∩ {xn = l}. Moreover, Ωˆ ⊂ Σˆ × [l − δ, l + δ]. The outer area minimizing property (Lemma A.9) implies that

ˆ δ ˆ δ |∂Ω |ϕ∗g n ≤ |∂(Σ × [−δ, δ])|ϕ∗g n S S ˆ δ ˆ δ ˆ δ = |∂Σ × [l − δ, l + δ]|ϕ∗g n + |Σ × {l − δ}|ϕ∗g n + |Σ × {l + δ}|ϕ∗g n . S S S

It is clear that as δ → 0, the ﬁrst term in the last line is of order O(δ). Since Σˆ δ decreases to Σ,ˆ together with the smoothness of ϕ, we conclude that each of remaining two terms converges

32 ˆ ˆ δ ˆ ˆ δ ˆ to |Σ × {l}|ϕ∗g n . This shows lim sup |∂Ω |ϕ∗g n ≤ P (Γ). Next, ∂Ω contains Σ × {l − δ} and S δ→0 S Σˆ × {l + δ}, implying that

ˆ ˆ ˆ δ |Σ × {l − δ}|ϕ∗g n + |Σ × {l + δ}|ϕ∗g n ≤ |∂Ω |ϕ∗g n S S S and hence ˆ ˆ δ 2|Σ × {l}|ϕ∗g n ≤ lim inf |Ω |ϕ∗g n . S δ→0 S ˆ δ ˆ This proves that lim |Ω |ϕ∗g n = P (Γ). Now, for any decreasing approximation by convex sets δ→0 S with non-empty interior Γˆk, limk→∞ |Γk| exists as it is a decreasing sequence. For each k, we may δ1(k) δ2(k) ﬁnd δ1(k) and δ2(k) which converge to 0 as k → ∞ such that ϕ(Ωˆ ) ⊂ Γˆk ⊂ ϕ(Ωˆ ) and this, ˆ δ1(k) ˆ δ2(k) ˆ in particular, implies that |Ω |ϕ∗g n ≤ |Γ | ≤ |Ω |ϕ∗g n . We conclude that |Γ | → P (Γ), as S k S k k → ∞.

We conclude this appendix by the following generalized outer area minimizing property.

ˆ ˆ ˆ ˆ n Lemma A.11. For n ≥ 2, if Γ1, Γ2, are compact convex sets such that Γ1 ⊂ Γ2 ⊂ S+. Then, ˆ ˆ n−1 P (Γ1) ≤ P (Γ2)< |S |. Proof. By Lemma A.9 and A.10, there are approximating sequences of strictly decreasing compact ˆ ˆ n ˆ convex sets Γ1,k and Γ2,k in S+ with smooth strictly convex boundaries. |Γ1,k| → P (Γ1) and |Γ2,k| → P (Γˆ2) by the two lemmas say. By the strict monotonicity of the sequences, for fixed k there is l0 such that Γˆ1,l ⊂ Γˆ2,k for l > l0. Taking l → ∞, the outer area minimizing property in Lemma A.9 implies P (Γˆ1) ≤ |Γ2,k|. The first inequality now follows by letting k → ∞. The ˆ n second inequality is implied by Γ2 ⊂ S ∩ {xn+1 ≤ } for small > 0, the first inequality, and n n−1 |S ∩ {xn+1 = }| < |S |.

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