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Sobolev stability of the Positive Mass Theorem and Riemannian Penrose Inequality using inverse mean curvature flow

  • Brian AllenEmail author
Editor’s Choice (Research Article)

Abstract

We study the Sobolev stability of the Positive Mass Theorem and the Riemannian Penrose Inequality in the case where a region of a sequence of manifolds \(M^3_i\) can be foliated by a smooth solution of inverse mean curvature flow (IMCF) which is uniformly controlled for time \(t \in [0,T]\). In particular, we consider a sequence of regions of manifolds \(U_T^i\subset M_i^3\), foliated by a IMCF, \(\Sigma _t\), such that if \(\partial U_T^i = \Sigma _0^i \cup \Sigma _T^i\) and \(m_H(\Sigma _T^i) \rightarrow 0\) then \(U_T^i\) converges in \(W^{1,2}\) to a flat annulus or in the hyperbolic setting it converges to a annulus portion of hyperbolic space. If instead \(m_H(\Sigma _T^i)-m_H(\Sigma _0^i) \rightarrow 0\) and \(m_H(\Sigma _T^i) \rightarrow m >0\) then we show that \(U_T^i\) converges in \(W^{1,2}\) to a topological annulus portion of the Schwarzschild metric or in the Hyperbolic case to a topological annulus portion of the Anti-de Sitter Schwarzschild metric.

Keywords

Stability Positive mass theorem Riemannian Penrose Inequality Inverse mean curvature flow Hawking mass Sobolev spaces 

1 Introduction

If we consider a complete, asymptotically flat manifold with nonnegative scalar curvature then the Positive Mass Theorem (PMT) says that \(M^3\) has positive ADM mass. This was proved by Schoen and Yau [41] using minimal surface techniques. The rigidity statement says that if \(m_{ADM}(M) = 0\) then M is isometric to Euclidean space. Similarly, the Riemannian Penrose Inequality (RPI) says that if \(\partial M\) consists of an outermost minimal surface \(\Sigma _0\) then
$$\begin{aligned} m_{ADM}(M) \ge \sqrt{\frac{|\Sigma _0|}{16 \pi }} \end{aligned}$$
(1)
where \(|\Sigma _0|\) is the area of \(\Sigma _0\). In the case of equality, i.e. \(m_{ADM}(M) = \sqrt{\frac{|\Sigma _0|}{16 \pi }}\), then M is isometric to the Schwarzschild metric (6). This was first proved by Geroch [24] in the rotationally symmetric case using inverse mean curvature flow (IMCF) and the Geroch monotonicity of the Hawking mass
$$\begin{aligned} m_H(\Sigma ) = \sqrt{\frac{|\Sigma |}{(16\pi )^3}} \left( 16 \pi - \int _{\Sigma } H^2 d \mu \right) . \end{aligned}$$
(2)
Huisken and Ilmanen [29] then extended these ideas to general asymptotically flat manifolds with a connected horizon using novel weak solutions to IMCF. Soon after Bray [9] proved the general case of the RPI using a conformal flow method.
In the asymptotically hyperbolic case the notion of mass was defined mathematically and explored by Chruściel and Herszlich [14] and Wang [43]. Earlier explorations of mass in this context were carried out by Abbott and Deser [1], Ashtekar and Magnon [8], and Gibbons et al. [25]. The PMT in this context for manifolds with scalar curvature greater than or equal to \(-6\) has been proved by Wang [43], Chruściel and Herszlich [14], Andersson et al. [7], and Sakovich in various different cases. The notion of Hawking mass in this context is defined as
$$\begin{aligned} m_H^{{\mathbb {H}}}(\Sigma ) = \sqrt{\frac{|\Sigma |}{(16\pi )^3}} \left( 16 \pi - \int _{\Sigma } H^2-4 d \mu \right) . \end{aligned}$$
(3)
The RPI conjecture in the case of asymptotically hyperbolic manifolds satisfying the scalar curvature bound says that the appropriate mass for this context satisfies (1). In the case of equality the manifold is isometric to the Anti-de Sitter Scharzschild metric (8). Neves [35] observed that the method of using IMCF to prove the RPI in the asymptotically hyperbolic case is not sufficient. Later, Hung and Wang [28] discuss this issue further in a note on IMCF in hyperbolic space. This conjecture is still open but special cases and related estimates have been obtained by Dahl et al. [16], de Lima and Girão [18], and Brendle et al. [11].

In this paper we are concerned with the stability of these four rigidity statements. Lee and Sormani [32] show that one cannot obtain smooth stability of the PMT even in the asymptotically flat, spherically symmetric setting. In that setting they prove Sormani-Wenger intrinsic flat (SWIF) convergence stability using Geroch monotonicity. LeFloch and Sormani [34] prove Sobolev stability using Geroch monotonicity but only in the asymptotically flat, sphereically symmetric setting. Additional related work will be mentioned below. The main goal of this paper is to improve upon the author’s previous results on \(L^2\) stability [2, 3] in order to show \(W^{1,2}\) stability.

Huisken and Ilmanen [29] show how to use weak solutions of IMCF in order to prove the PMT for asymptotically flat Riemanian manifolds as well as the RPI in the case of a connected boundary. The weak solutions defined by Huisken–Ilmanen jump over gravity wells and hence do not produce a complete foliation of the ambient manifold. This is not a problem for Huisken–Ilmanen since they are able to show that the Geroch monotonicity of the Hawking mass is preserved through these jumps. For our purposes, we need the IMCF to foliate the ambient manifold and hence we focus on regions of manifolds which can be foliated by smooth solutions of IMCF which are uniformly controlled. For a glimpse of long time existence and asymptotic analysis results for smooth IMCF in various ambient manifolds see the work of the author [4], Ding [19], Gerhardt [22, 23], Scheuer [39, 40], Urbas [42], and Zhou [44].

Definition 1.1

If we have \(\Sigma ^2\) a surface in a Riemannian manifold, \(M^3\), we will denote the induced metric, mean curvature, second fundamental form, principal curvatures, Gauss curvature, area, Hawking mass and Neummann isoperimetric constant as g, H, A, \(\lambda _i\), K, \(|\Sigma |\), \(m_H(\Sigma )\), \(IN_1(\Sigma )\), respectively. We will denote the Riemann curvature, Ricci curvature, scalar curvature, sectional curvature tangent to \(\Sigma \), and ADM mass as Rm, Rc, R, \(K_{12}\), \(m_{ADM}(M)\), respectively.

Define the two classes of manifolds with boundary foliated by IMCF as follows
$$\begin{aligned} {\mathcal {M}}_{{\mathbb {H}},r_0,H_0,I_0}^{T,H_1,A_1}:=\{&M \text { a Riemannian manifold, } U_T \subset M, R \ge -6|\\&\exists \Sigma \subset M \text {compact, connected surface, }\\&IN_1(\Sigma ) \ge I_0, m_H^{{\mathbb {H}}}(\Sigma ) \ge 0 \text {,and } |\Sigma |=4\pi r_0^2. \\&\exists \Sigma _t \text { smooth solution to IMCF, such that }\Sigma _0=\Sigma ,\\&H_0 \le H(x,t) \le H_1 < \infty , \Vert A\Vert _{W^{2,2}(\Sigma \times [0,T])} \le A_1,\\&\text {and } U_T = \{x\in \Sigma _t: t \in [0,T]\} \} \end{aligned}$$
and
$$\begin{aligned} {\mathcal {M}}_{r_0,H_0,I_0}^{T,H_1,A_1}:=\{&U_T \in {\mathcal {M}}_{{\mathbb {H}},r_0,H_0,I_0}^{T,H_1,A_1}| R \ge 0, m_H(\Sigma ) \ge 0 \} \end{aligned}$$
where \(0< H_0< H_1 < \infty \), \(0< I_0,A_1,A_2, r_0 < \infty \) and \(0< T < \infty \).

Remark 1.2

All norms in this paper are defined on \(\Sigma \times [0,T]\) with respect to the Euclidean metric \(\delta \) which is given in IMCF coordinates below. The diffeomorphism we use to impose coordinates on \(U_T\) is discussed in Sect. 2.1 before Proposition 2.7.

Remark 1.3

The reader should make note that the difference between the class of IMCF’s considered in this paper, as opposed to the author’s previous paper on \(L^2\) stability [2, 3], is the addition of a \(W^{2,2}\) bound on A which is asking for some uniform higher regularity of the family of IMCF’s. Notice by Morrey’s inequality this implies a \(C^{0,\alpha }\) bound on |A| as in the previous paper [2, 3].

Observe that the Riemannian metric, \({\hat{g}}_i\), on these manifolds can now be expressed using a gauge defined on \(\Sigma \times [0,T]\) by IMCF as,
$$\begin{aligned} {\hat{g}}^i&=\frac{1}{H(x,t)^2}dt^2 + g^i(x,t), \end{aligned}$$
(4)
where \(g^i(x,t)\) is the metric on \(\Sigma _t^i\). On Euclidean space, concentric spheres flow according to IMCF and so the Euclidean metric \(\delta \) can be expressed using a gauge defined on \(\Sigma \times [0,T]\) by IMCF as,
$$\begin{aligned} \delta&= \frac{r_0^2}{4}dt^2 + r_0^2e^t \sigma , \end{aligned}$$
(5)
where \(\sigma \) is the round metric on \(\Sigma \).
Similarly, one can express the Schwarzschild, hyperbolic, and Anti-de Sitter Schwarzschild metrics
$$\begin{aligned} g_S&= \frac{r_0^2}{4}\left( 1 - \frac{2}{r_0} m e^{-t/2} \right) ^{-1} e^tdt^2 + r_0^2e^t \sigma \end{aligned}$$
(6)
$$\begin{aligned} g_{{\mathbb {H}}}&= \frac{1}{4}\left( 1+ \frac{e^{-t}}{r_0^2} \right) ^{-1}dt^2 + r_0^2e^t \sigma , \end{aligned}$$
(7)
$$\begin{aligned} g_{AdSS}&= \frac{1}{4}\left( 1+\frac{e^{-t}}{r_0^2} - \frac{2}{r_0^3} m e^{-3t/2} \right) ^{-1} dt^2 + r_0^2e^t \sigma . \end{aligned}$$
(8)
In our first theorem we prove stability of the PMT: that when the Hawking mass of the outer boundary converges to 0 the regions converge to annular regions in Euclidean space. Prior work in this direction under a variety of hypotheses was conducted by Bray and Finster [10], Finster and Kath [21], Corvino [15], Finster [20], Lee [30], Lee and Sormani [32], Huang et al. [27], the author [2, 5], and Bryden [12].

Theorem 1.4

Let \(U_{T}^i \subset M_i^3\) be a sequence such that \(U_{T}^i\subset {\mathcal {M}}_{r_0,H_0,I_0}^{T,H_1,A_1}\) and
$$\begin{aligned} m_H(\Sigma _{T}^i) \rightarrow 0 \text { as }i \rightarrow \infty . \end{aligned}$$
(9)
If we assume,
$$\begin{aligned}&\Vert Rc^i(\nu ,\nu )\Vert _{W^{1,2}(\Sigma \times [0,T])} \le C, \end{aligned}$$
(10)
$$\begin{aligned}&\Vert R^i\Vert _{L^2(\Sigma \times [0,T])} \le C, \end{aligned}$$
(11)
$$\begin{aligned}&diam(\Sigma _t^i) \le D \text { } \forall \text { } i, t \in [0,T], \end{aligned}$$
(12)
$$\begin{aligned}&|K^i| \le C \text { on } \Sigma _T, \end{aligned}$$
(13)
where \(W^{1,2}(\Sigma \times [0,T])\) is defined with respect to \(\delta \), then
$$\begin{aligned} {\hat{g}}^i \rightarrow \delta \end{aligned}$$
(14)
in \(W^{1,2}\) with respect to \(\delta \) and thus volumes converge.

In our second theorem we prove stability of the RPI: that when the Hawking mass of the outer and inner boundary converge to the same value m the regions converge to annular regions in the Schwarzschild manifold. Prior work in this direction has been done by Lee and Sormani [31] in the rotationally symmetric case and by the author [2] using IMCF.

Theorem 1.5

Let \(U_{T}^i \subset M_i^3\) be a sequence such that \(U_{T}^i\subset {\mathcal {M}}_{r_0,H_0,I_0}^{T,H_1,A_1}\),
$$\begin{aligned} m_H(\Sigma _{T}^i)- m_H(\Sigma _0^i) \rightarrow 0\text {, and }m_H(\Sigma _0) \rightarrow m > 0 \text { as }i \rightarrow \infty . \end{aligned}$$
(15)
If we assume (10), (11), (12), and (13) then
$$\begin{aligned} {\hat{g}}^i \rightarrow g_S \end{aligned}$$
(16)
in \(W^{1,2}\) with respect to \(\delta \) and thus volumes converge.

In our third theorem we prove stability of the PMT in the asymptotically hyperbolic case: that when the Hawking mass of the outer boundary converges to 0 the regions converge to annular regions in the hyperbolic space. Prior work in this direction was conducted by Dahl et al. [17], Sakovich and Sormani [38], the author [3], and Cabrera Pacheco [13].

Theorem 1.6

Let \(U_{T}^i \subset M_i^3\) be a sequence such that \(U_{T}^i\subset {\mathcal {M}}_{{\mathbb {H}}, r_0,H_0,I_0}^{T,H_1,A_1}\) and
$$\begin{aligned} m_H^{{\mathbb {H}}}(\Sigma _{T}^i) \rightarrow 0 \text { as }i \rightarrow \infty . \end{aligned}$$
(17)
If we assume (10), (11), (12), and (13) then
$$\begin{aligned} {\hat{g}}^i \rightarrow g_{{\mathbb {H}}} \end{aligned}$$
(18)
in \(W^{1,2}\) with respect to \(\delta \) and thus volumes converge.

In our fourth theorem we prove stability of the RPI in the asymptotically hyperbolic case: that when the Hawking mass of the outer and inner boundary converge to the same value m the regions converge to annular regions in the Anti-de Sitter Schwarzschild manifold. Prior work in this direction has been conducted by the author [3] using IMCF.

Theorem 1.7

Let \(U_{T}^i \subset M_i^3\) be a sequence such that \(U_{T}^i\subset {\mathcal {M}}_{{\mathbb {H}}, r_0,H_0,I_0}^{T,H_1,A_1}\),
$$\begin{aligned} m_H^{{\mathbb {H}}}(\Sigma _{T}^i)- m_H^{{\mathbb {H}}}(\Sigma _0^i) \rightarrow 0\text { and }m_H^{{\mathbb {H}}}(\Sigma _0) \rightarrow m > 0 \text { as }i \rightarrow \infty . \end{aligned}$$
(19)
If we assume (10), (11), (12), and (13) then
$$\begin{aligned} {\hat{g}}^i \rightarrow g_{ADSS} \end{aligned}$$
(20)
in \(W^{1,2}\) with respect to \(\delta \) and thus volumes converge.

Remark 1.8

One should not expect \(W^{1,2}\) convergence to imply SWIF convergence since the author and Sormani [6] have shown that \(L^2\) convergence does not agree with GH and/or SWIF convergence (see example 3.4 in [6]) since valleys can form on sets of measure zero. By a similar example, one can see that \(W^{1,2}\) convergence in dimension three will not imply SWIF convergence either. By what the main theorem of the author and Sormani [6], one expects to need to combine \(L^p\) convergence with \(C^0\) convergence from below in order to be able to conclude SWIF convergence which is what the author carries out in [5] for the PMT under various assumptions.

Since \(W^{1,2}\) convergence provides additional convergence information about the geometry of the sequence, as shown by LeFloch and Mardare [33], it is useful to show both SWIF and \(W^{1,2}\) convergence when appropriate. Also, notice that the curvature hypotheses of the main theorems will clearly hold in the ends of asymptotically flat or asymptotically hyperbolic manifolds whose asymptotic decay rates are uniformly controlled.

In Sect. 2 we will use IMCF to get important higher order estimates of the metric \({\hat{g}}^i\) on the foliated region \(U_T^i\subset M_i\) which build upon the estimates of the previous papers [2, 3]. We also review some key estimates obtained in [2, 3] that are needed in this paper.

In Sect. 3 we use the estimates of the previous section to show convergence of \({\hat{g}}\) to the appropriate prototype space \(\delta , g_S, g_{{\mathbb {H}}},\) or \(g_{ADSS}\). This is done by showing convergence of \({\hat{g}}\) to simpler metrics, successively, until we get to \(\delta , g_S, g_{{\mathbb {H}}},\) or \(g_{ADSS}\), and combining this chain of estimates by the triangle inequality.

2 Higher order estimates for manifolds foliated by IMCF

In this section we expand upon the estimates found in the previous papers of the author on \(L^2\) convergence [2, 3]. For the readers convenience we will repeat some of the estimates obtained in [2] since we will also need them here but a majority of the estimates will be new.

We remember that IMCF is defined for surfaces \(\Sigma ^n \subset M^{n+1}\) evolving through a one parameter family of embeddings \(F: \Sigma \times [0,T] \rightarrow M\), F satisfying inverse mean curvature flow
$$\begin{aligned} {\left\{ \begin{array}{ll} \frac{\partial F}{\partial t}(p,t) = \frac{\nu (p,t)}{H(p,t)} &{}\text { for } (p,t) \in \Sigma \times [0,T) \\ F(p,0) = \Sigma _0 &{}\text { for } p \in \Sigma \end{array}\right. } \end{aligned}$$
(21)
where H is the mean curvature of \(\Sigma _t := F_t(\Sigma )\) and \(\nu \) is the outward pointing normal vector. The outward pointing normal vector will be well defined in our case since we will be considering regions of asymptotically flat or asymptotically hyperbolic manifolds with one end.
We also remind the reader of the definition of the Hawking mass in the asymptotically Euclidean setting,
$$\begin{aligned} m_H(\Sigma ) = \sqrt{\frac{|\Sigma |}{(16\pi )^3}} \left( 16 \pi - \int _{\Sigma } H^2 d \mu \right) , \end{aligned}$$
(22)
as well as the Hawking mass in the asymptotically hyperbolic setting,
$$\begin{aligned} m_H^{{\mathbb {H}}}(\Sigma ) = \sqrt{\frac{|\Sigma |}{(16\pi )^3}} \left( 16 \pi - \int _{\Sigma } H^2-4 d \mu \right) . \end{aligned}$$
(23)

2.1 Previous estimates for \(L^2\) convergence

As a notational convenience we will use \({\mathcal {H}}^2 =H^2, H^2 - 4\) depending on whether we are considering the Euclidean or hyperbolic setting when it is clear from the context which model we have in mind. All of the results in this subsection are from the author’s previous papers on \(L^2\) stability [2, 3] so the reader is directed to [2, 3] for proofs of the following results.

We begin by noting some simple consequences of the assumptions on the Hawking mass.

Lemma 2.1

Let \(\Sigma ^2 \subset M^3\) be a hypersurface and \(\Sigma _t\) it’s corresponding solution of IMCF. If
$$\begin{aligned} m_1 \le&m_H(\Sigma _t) \le m_2, \end{aligned}$$
(24)
$$\begin{aligned} 0< H_0 \le&H(x,t) \le H_1 < \infty \end{aligned}$$
(25)
thenwhere \(|\Sigma _t|\) is the n-dimensional area of \(\Sigma \).
Hence if
$$\begin{aligned} m_{H}(\Sigma _T) \rightarrow 0 \end{aligned}$$
(29)
thenfor every \(t\in [0,T]\).
If
$$\begin{aligned}&m_{H}(\Sigma _T)-m_{H}(\Sigma _0) \rightarrow 0\text { and }m_{H}(\Sigma _0) \rightarrow m > 0 \end{aligned}$$
(31)
thenfor every \(t\in [0,T]\).

Remark 2.2

The corresponding Lemma holds in the hyperbolic setting for \(m_H^{{\mathbb {H}}}\) and \({\mathcal {H}}^2\).

By rearranging the Geroch monotonicity calculation we arrive at the following result.

Lemma 2.3

For any solution of IMCF we have the following formula
$$\begin{aligned} \frac{d}{dt} \int _{\Sigma _t} H^2 d\mu = \frac{(16 \pi )^{3/2}}{|\Sigma _t|^{1/2}} \left( \frac{1}{2} m_H(\Sigma _t) - \frac{d}{dt}m_H(\Sigma _t) \right) \end{aligned}$$
(33)
So if we assume that
$$\begin{aligned} m_H(\Sigma _t^i) \rightarrow 0\text { as }i \rightarrow \infty \end{aligned}$$
(34)
then we have for a.e. \(t \in [0,T]\) that
$$\begin{aligned} \frac{d}{dt} \int _{\Sigma _t^i} H^2 d\mu \rightarrow 0 \end{aligned}$$
(35)
If we assume that
$$\begin{aligned} m_H(\Sigma _T^i) - m_H(\Sigma _0^i) \rightarrow 0\text { and }m_H(\Sigma _t^i)\rightarrow m > 0\text { as } i \rightarrow \infty \end{aligned}$$
(36)
then we have that
$$\begin{aligned} \frac{d}{dt} \int _{\Sigma _t^i} H^2 d\mu \rightarrow \frac{16 \pi }{r_0}me^{-t/2} \end{aligned}$$
(37)

Remark 2.4

The corresponding Lemma holds in the hyperbolic setting for \(m_H^{{\mathbb {H}}}\) and \({\mathcal {H}}^2\).

The two following Corollaries state the crucial estimates which get the rest of the results moving in the right direction. These convergence results follow from the Geroch monotonicity calculation.

Corollary 2.5

Let \(\Sigma ^i\subset M^i\) be a compact, connected surface with corresponding solution to IMCF \(\Sigma _t^i\). If
$$\begin{aligned} m_H(\Sigma _0)\ge 0 \text { and }m_H(\Sigma ^i_T) \rightarrow 0 \end{aligned}$$
(38)
then for almost every \(t \in [0,T]\),
$$\begin{aligned}&\int _{\Sigma _t^i} \frac{|\nabla H_i|^2}{H_i^2}d \mu \rightarrow 0, \int _{\Sigma _t^i} (\lambda _1^i-\lambda _2^i)^2d \mu \rightarrow 0, \int _{\Sigma _t^i} R^i d \mu \rightarrow 0,\end{aligned}$$
(39)
$$\begin{aligned}&\int _{\Sigma _t^i} Rc^i(\nu ,\nu )d \mu \rightarrow 0, \int _{\Sigma _t^i} K_{12}^id \mu \rightarrow 0, \int _{\Sigma _t^i} H_i^2 d\mu \rightarrow 16\pi , \end{aligned}$$
(40)
$$\begin{aligned}&\int _{\Sigma _t^i} |A|_i^2 d \mu \rightarrow 8 \pi , \int _{\Sigma _t^i} \lambda _1^i\lambda _2^i d \mu \rightarrow 4\pi , \chi (\Sigma _t^i) \rightarrow 2, \end{aligned}$$
(41)
as \(i \rightarrow \infty \) where \(K_{12}\) is the ambient sectional curvature tangent to \(\Sigma _t\). As a consequence \(\Sigma _t^i\) must eventually become topologically a sphere.
If
$$\begin{aligned} \left( m_H(\Sigma ^i_T)-m_H(\Sigma ^i_0) \right) \rightarrow 0\text { where } m_H(\Sigma _0) \rightarrow m > 0 \end{aligned}$$
(42)
then the first three integrals listed above \(\rightarrow 0\) and for almost every \(t \in [0,T]\)
$$\begin{aligned}&\int _{\Sigma _t^i} H_i^2 d\mu \rightarrow 16 \pi \left( 1 - \sqrt{\frac{16 \pi }{|\Sigma _0|}}me^{-t/2} \right) ,\end{aligned}$$
(43)
$$\begin{aligned}&\int _{\Sigma _t^i} |A|_i^2 d \mu \rightarrow 8 \pi \left( 1 - \sqrt{\frac{16 \pi }{|\Sigma _0|}}me^{-t/2} \right) , \end{aligned}$$
(44)
$$\begin{aligned}&\int _{\Sigma _t^i} \lambda _1^i\lambda _2^i d \mu \rightarrow 4 \pi \left( 1 - \sqrt{\frac{16 \pi }{|\Sigma _0|}}me^{-t/2} \right) , \end{aligned}$$
(45)
$$\begin{aligned}&\int _{\Sigma _t^i}Rc^i(\nu ,\nu )d\mu \rightarrow -\frac{8\pi }{r_0}m e^{-t/2}, \end{aligned}$$
(46)
$$\begin{aligned}&\int _{\Sigma _t^i}K_{12}^i d\mu \rightarrow -\frac{8\pi }{r_0}m e^{-t/2},\chi (\Sigma _t^i) \rightarrow 2. \end{aligned}$$
(47)
As a consequence \(\Sigma _t^i\) must eventually become topologically a sphere.

Now a similar corollary in the hyperbolic setting.

Corollary 2.6

Let \(\Sigma ^i\subset M^i\) be a sequence of compact, connected surface with corresponding solution to IMCF \(\Sigma _t^i\) where \(R^i \ge -6\).

If
$$\begin{aligned} m_H^{{\mathbb {H}}}(\Sigma _0^i)\ge 0 \text { and }m_H^{{\mathbb {H}}}(\Sigma ^i_T) \rightarrow 0 \end{aligned}$$
(48)
then for almost every \(t \in [0,T]\),
$$\begin{aligned}&\int _{\Sigma _t^i} \frac{|\nabla H_i|^2}{H_i^2}d \mu \rightarrow 0, \int _{\Sigma _t^i} (\lambda _1^i-\lambda _2^i)^2d \mu \rightarrow 0, \int _{\Sigma _t^i} R^i+6 d \mu \rightarrow 0,\end{aligned}$$
(49)
$$\begin{aligned}&\int _{\Sigma _t^i} Rc^i(\nu ,\nu )+2d \mu \rightarrow 0, \int _{\Sigma _t^i} K_{12}^i+1d \mu \rightarrow 0, \int _{\Sigma _t^i} H_i^2-4 d\mu \rightarrow 16\pi , \end{aligned}$$
(50)
$$\begin{aligned}&\int _{\Sigma _t^i} |A_i|^2-2 d \mu \rightarrow 8 \pi , \int _{\Sigma _t^i} \lambda _1^i\lambda _2^i-1 d \mu \rightarrow 4\pi , \chi (\Sigma _t^i) \rightarrow 2, \end{aligned}$$
(51)
as \(i \rightarrow \infty \) where \(K_{12}\) is the ambient sectional curvature tangent to \(\Sigma _t\). As a consequence \(\Sigma _t^i\) must eventually become topologically a sphere.
If
$$\begin{aligned} \left( m_H^{{\mathbb {H}}}(\Sigma ^i_T)-m_H^{{\mathbb {H}}}(\Sigma ^i_0) \right) \rightarrow 0\text { where }m_H^{{\mathbb {H}}}(\Sigma _0) \rightarrow m > 0 \end{aligned}$$
(52)
then the first three integrals listed above tend to zero and for almost every \(t \in [0,T]\),
$$\begin{aligned}&\int _{\Sigma _t^i} H_i^2-4 d\mu \rightarrow 16 \pi \left( 1 - \sqrt{\frac{16 \pi }{|\Sigma _0|}}me^{-t/2} \right) , \end{aligned}$$
(53)
$$\begin{aligned}&\int _{\Sigma _t^i} |A_i|^2-2 d \mu \rightarrow 8 \pi \left( 1 - \sqrt{\frac{16 \pi }{|\Sigma _0|}}me^{-t/2} \right) , \end{aligned}$$
(54)
$$\begin{aligned}&\int _{\Sigma _t^i} \lambda _1^i\lambda _2^i-1 d \mu \rightarrow 4 \pi \left( 1 - \sqrt{\frac{16 \pi }{|\Sigma _0|}}me^{-t/2} \right) ,\end{aligned}$$
(55)
$$\begin{aligned}&\int _{\Sigma _t^i}Rc^i(\nu ,\nu )+2d\mu \rightarrow -\frac{8\pi }{r_0}m e^{-t/2},\end{aligned}$$
(56)
$$\begin{aligned}&\int _{\Sigma _t^i}K_{12}^i+1d\mu \rightarrow \frac{8\pi }{r_0}m e^{-t/2},\chi (\Sigma _t^i) \rightarrow 2. \end{aligned}$$
(57)
As a consequence \(\Sigma _t^i\) must eventually become topologically a sphere.
In the following proposition an important diffeomorphism from \(U_T^i\) to \(\Sigma \times [0,T]\) is defined which is used throughout the rest of the paper to define and show \(W^{1,2}\) convergence of \({\hat{g}}^i\) to the appropriate prototype space. We start by choosing an area preserving diffeomorphism
$$\begin{aligned} F_i:\Sigma _0^i \rightarrow S^2(r_0) \end{aligned}$$
(58)
which we know is well defined since we assume that
$$\begin{aligned} |\Sigma _0^i| = |S^2(r_0)| = 4 \pi r_0^2. \end{aligned}$$
(59)
Then by the evolution of area under IMCF, \(F_i\) automatically extends to a diffeomorphism
$$\begin{aligned} F_i(t): \Sigma _t \rightarrow S^2(r_0e^{t/2}) \end{aligned}$$
(60)
which defines a diffeomorphism from \(U_T^i\) to \(\Sigma \times [0,T]\) and is the coordinate system we will use throughout the rest of the paper.

Proposition 2.7

If \(\Sigma _t^i\) is a sequence of IMCF solutions where
$$\begin{aligned}&\int _{\Sigma _t^i} \frac{|\nabla H|^2}{H^2}d \mu \rightarrow 0 \text { as }i \rightarrow \infty , \end{aligned}$$
(61)
$$\begin{aligned}&0< H_0 \le H(x,t) \le H_1 < \infty , \end{aligned}$$
(62)
$$\begin{aligned}&|A|(x,t) \le A_0 < \infty \end{aligned}$$
(63)
then
$$\begin{aligned} \int _{\Sigma _t^i} (H_i - {\bar{H}}_i)^2 d \mu \rightarrow 0 \end{aligned}$$
(64)
as \(i \rightarrow \infty \) for almost every \(t \in [0,T]\) where Open image in new window .
Let \(d\mu _t^i\) be the volume form on \(\Sigma \) w.r.t. \(g^i(\cdot ,t)\) then we can find a parameterization of \(\Sigma _t\) so that
$$\begin{aligned} d\mu _t^i = r_0^2 e^t d\sigma \end{aligned}$$
(65)
where \(d\sigma \) is the standard volume form on the unit sphere.
Then for almost every \(t \in [0,T]\) and almost every \(x \in \Sigma \), with respect to \(d\sigma \), we have that
$$\begin{aligned} H_i(x,t) - {\bar{H}}_i(t) \rightarrow 0, \end{aligned}$$
(66)
along a subsequence.

In Corollaries 2.5 and 2.14 we note that the Ricci curvature integrals are not so useful since we have not assumed anything about the sign of the Ricci curvature. In order to obtain useful estimates of the Ricci curvature we now turn to obtain weak convergence to the expected values.

Lemma 2.8

Let \(\Sigma ^i_0\subset M^3_i\) be a compact, connected surface with corresponding solution to IMCF \(\Sigma _t^i\). Then if \(\phi \in C_c^1(\Sigma \times (a,b))\) and \(0\le a <b\le T\) we can compute the estimate
$$\begin{aligned}&\int _a^b\int _{\Sigma _t^i} 2\phi Rc^i(\nu ,\nu )d\mu d t= \int _{\Sigma _a^i} \phi H_i^2 d\mu - \int _{\Sigma _b^i} \phi H_i^2 d\mu \end{aligned}$$
(67)
$$\begin{aligned}&+ \int _a^b\int _{\Sigma _t^i}2\phi \frac{|\nabla H_i|^2}{H_i^2}-2\frac{{\hat{g}}^j(\nabla \phi , \nabla H_i)}{H_i} +\phi (H_i^2-2|A|_i^2) d\mu \end{aligned}$$
(68)
If
$$\begin{aligned} m_H(\Sigma ^i_T) \rightarrow 0 \end{aligned}$$
(69)
and \(\Sigma _t\) satisfies the hypotheses of Proposition 2.7 then
$$\begin{aligned} \int _a^b\int _{\Sigma _t^i} \phi Rc^i(\nu ,\nu )d\mu dt \rightarrow 0 \end{aligned}$$
(70)
If
$$\begin{aligned} m_H(\Sigma ^i_T)-m_H(\Sigma ^i_0)&\rightarrow 0, \end{aligned}$$
(71)
$$\begin{aligned} m_H(\Sigma _T)&\rightarrow m > 0, \end{aligned}$$
(72)
and \(\Sigma _t\) satisfies the hypotheses of Proposition 2.7 then
$$\begin{aligned} \int _a^b\int _{\Sigma _t}&\phi Rc^i(\nu ,\nu )d\mu d t \rightarrow \int _a^b\int _{\Sigma _t} \frac{-2}{r_0}me^{-t/2} \phi d\mu dt. \end{aligned}$$
(73)

Now a similar lemma in the hyperbolic setting.

Lemma 2.9

Let \(\Sigma ^i_0\subset M^3_i\) be a compact, connected surface with corresponding solution to IMCF \(\Sigma _t^i\). Then if \(\phi \in C^1(\Sigma \times (a,b))\), \(0\le a <b\le T\), and \(\sigma \) is the round metric on \(S^2\) with area element \(d \sigma \) we can compute the estimate,
$$\begin{aligned}&\int _a^b\int _{\Sigma _t^i} 2\phi Rc^i(\nu ,\nu )d\mu d t= \int _{\Sigma _a^i} \phi H_i^2 d\mu - \int _{\Sigma _b^i} \phi H_i^2 d\mu \end{aligned}$$
(74)
$$\begin{aligned}&+ \int _a^b\int _{\Sigma _t^i}2\phi \frac{|\nabla H_i|^2}{H_i^2}-2\frac{{\hat{g}}^j(\nabla \phi , \nabla H_i)}{H_i} +\phi (H_i^2-2|A|_i^2) d\mu . \end{aligned}$$
(75)
If
$$\begin{aligned} m_H^{{\mathbb {H}}}(\Sigma ^i_T) \rightarrow 0 \end{aligned}$$
(76)
and \(\Sigma _t\) satisfies the hypotheses of Proposition 2.7 then
$$\begin{aligned} \int _a^b\int _{\Sigma _t}&\phi Rc^i(\nu ,\nu )d\mu d t \rightarrow \int _a^b \int _{\Sigma } -2r_0^2 e^t \phi d\sigma dt. \end{aligned}$$
(77)
If
$$\begin{aligned}&m_H^{{\mathbb {H}}}(\Sigma ^i_T)-m_H^{{\mathbb {H}}}(\Sigma ^i_0) \rightarrow 0,&\end{aligned}$$
(78)
$$\begin{aligned}&m_H^{{\mathbb {H}}}(\Sigma _T) \rightarrow m > 0,&\end{aligned}$$
(79)
and \(\Sigma _t\) satisfies the hypotheses of Proposition 2.7 then
$$\begin{aligned} \int _a^b\int _{\Sigma _t} \phi Rc^i(\nu ,\nu )d\mu d t \rightarrow \int _a^b\int _{\Sigma } -2\left( \frac{1 }{r_0}me^{-t/2} + r_0^2 e^t\right) \phi d\sigma dt. \end{aligned}$$
(80)

We end this subsection with a lemma which allows us to control the metric on \(\Sigma _t^i\) in terms of the metric on \(\Sigma _T^i\).

Lemma 2.10

Assume that \(\Sigma _t^i\) is a solution to IMCF and let
$$\begin{aligned} \lambda _1^i(x,t)\le \lambda _2^i(x,t) \end{aligned}$$
(81)
be the eigenvalues of \(A^i(x,t)\) then
$$\begin{aligned} e^{\int _T^t\frac{2\lambda ^i_1(x,s)}{H^i(x,s)}ds} g^i(x,T) \le g^i(x,t)&\le e^{\int _T^t\frac{2\lambda ^i_2(x,s)}{H^i(x,s)}ds} g^i(x,T) \end{aligned}$$
(82)

2.2 New estimates for \(W^{1,2}\) convergence

In this section we prove new estimates which are in particular useful for proving \(W^{1,2}\) convergence. This \(W^{1,2}\) convergence will be defined with respect to \((\Sigma \times [0,T], \delta )\) and hence we are concerned with derivatives with respect to the polar coordinates defined in (5) with the coordinate vectors \(\{\partial _0=\partial _t,\partial _1,\partial _2\}\).

We begin by deriving an equation for the evolution of the average of H under IMCF.

Lemma 2.11

If we let \(\Sigma _t\) be a solution of IMCF and definethen

Now we can use Lemma 2.11 to derive a more specific equation.

Lemma 2.12

If we let \(\Sigma _t\) be a solution of IMCF and definethen

Proof

The evolution equation for H under IMCF is given by
$$\begin{aligned} \frac{\partial H}{\partial t} = -\Delta \left( \frac{1}{H} \right) -\frac{|A|^2}{H} - \frac{Rc(\nu ,\nu )}{H} \end{aligned}$$
(90)
and so if we take the average integral of both sides of this equation we findand so by the divergence theorem and Lemma 2.11 we find\(\square \)

Similarly, we can derive an equation for the average of \(H^2\).

Lemma 2.13

If we let \(\Sigma _t\) be a solution of IMCF and definethen

Proof

The evolution equation for H under IMCF is given by
$$\begin{aligned} \frac{\partial H^2}{\partial t} = -2H\Delta \left( \frac{1}{H} \right) -2\frac{|\nabla H|^2}{H^2} -2|A|^2 -2 Rc(\nu ,\nu ) \end{aligned}$$
(95)
and so if we take the average integral of both sides of this equation we findand so by integration by parts and Lemma 2.11 we find\(\square \)

We now use the previous lemmas to deduce what the evolution of the average mean curvature must converge to.

Corollary 2.14

Let \(\Sigma ^i\subset M^i\) be a compact, connected surface with corresponding solution to IMCF \(\Sigma _t^i\). If
$$\begin{aligned}&m_H(\Sigma _0)\ge 0\text { and }m_H(\Sigma ^i_T) \rightarrow 0 \end{aligned}$$
(98)
$$\begin{aligned}&\text { or }m_H^{{\mathbb {H}}}(\Sigma _0)\ge 0\text { and }m_H^{{\mathbb {H}}}(\Sigma ^i_T) \rightarrow 0 \end{aligned}$$
(99)
then for almost every \(t \in [0,T]\) and almost every \(x \in \Sigma _t\) we have that
$$\begin{aligned} \frac{\partial {\bar{H}}_i^2}{\partial t}&\rightarrow \frac{-4}{r_0^2} e^{-t} \frac{\partial {\bar{H}}_i}{\partial t} \rightarrow -\frac{e^{-t/2}}{r_0} \end{aligned}$$
(100)
If
$$\begin{aligned}&\left( m_H(\Sigma ^i_T)-m_H(\Sigma ^i_0) \right) \rightarrow 0\text { where }m_H(\Sigma _0) \rightarrow m > 0 \end{aligned}$$
(101)
$$\begin{aligned}&\text { or } \left( m_H^{{\mathbb {H}}}(\Sigma ^i_T)-m_H^{{\mathbb {H}}}(\Sigma ^i_0) \right) \rightarrow 0\text { where }m_H^{{\mathbb {H}}}(\Sigma _0) \rightarrow m > 0 \end{aligned}$$
(102)
then for almost every \(t \in [0,T]\) and almost every \(x\in \Sigma _t\) we have that
$$\begin{aligned} \frac{\partial {\bar{H}}_i^2}{\partial t}&\rightarrow \frac{-4}{r_0^2}\left( 1- \frac{2m}{r_0} e^{-t/2} \right) e^{-t} \frac{\partial {\bar{H}}_i}{\partial t} \rightarrow -\frac{1}{r_0}\sqrt{1- \frac{2m}{r_0} e^{-t/2} } e^{-t/2} \end{aligned}$$
(103)

Proof

This follows by combining Lemmas 2.12 and 2.13 with Corollary 2.5. \(\square \)

In the following lemma we obtain estimates on the derivatives in the \(\Sigma \) direction in the coordinate space \(\Sigma \times [0,T]\).

Lemma 2.15

We can find the following estimates on the coordinate derivatives of the metric g
$$\begin{aligned} \left| Dg(x,T)-Dg(x,t)\right|&\le \int _t^T \frac{|D A|_i}{H_i} + \frac{|A|_i|D H_i|}{H_i^2} ds \end{aligned}$$
(104)
$$\begin{aligned} \left| Dg(x,T)-e^{t-T}Dg(x,t)\right|&\le \int _t^Te^{s-T} \left( \frac{|D A|_i}{H_i} + \frac{|A|_i|D H_i|}{H_i^2}+ |Dg| \right) ds, \end{aligned}$$
(105)
where D is the covariant derivative with respect to \(\sigma \) and all norms are taken with respect to \(\sigma \).

Proof

We start by taking spatial derivatives of the equation,
$$\begin{aligned} \frac{\partial g_{lm}^i}{\partial t} = \frac{A_{lm}^i}{H_i}, \end{aligned}$$
(106)
in normal coordinates with respect to \(\sigma \) centered at x,
$$\begin{aligned} \frac{\partial }{\partial t} g_{lm,k}&= \frac{A_{lm,k}^i}{H_i} - \frac{A_{lm}^i}{H_i^2}H_{i,k} \end{aligned}$$
(107)
$$\begin{aligned} \frac{\partial }{\partial t} (e^{t-T} g_{lm,k})&= e^{t-T} \left( \frac{A_{lm,k}}{H_i} - \frac{A_{lm}}{H_i^2}H_{i,k}\right) + e^{t-T} g_{lm,k}. \end{aligned}$$
(108)
Now by taking norms with respect to \(\sigma \) of both sides yields the inequality,
$$\begin{aligned} \left| \frac{\partial }{\partial t}(e^{t-T} Dg)\right|&\le e^{t-T} \left( \frac{|D A|_i}{H_i} + \frac{|A|_i|D H_i|}{H_i^2} + |Dg|\right) , \end{aligned}$$
(109)
where D is the covariant derivative with respect to \(\sigma \).
Now to finish up we find,
$$\begin{aligned}&\left| Dg(x,t)-e^{t-T}Dg(x,0)\right| =\left| \int _0^t \frac{\partial }{\partial s}(e^{s-T}Dg)ds \right| \end{aligned}$$
(110)
$$\begin{aligned}&\le \int _0^t \left| \frac{\partial }{\partial s}(e^{s-T}Dg) \right| ds \end{aligned}$$
(111)
$$\begin{aligned}&\le \int _0^t e^{s-T} \left( \frac{|D A|_i}{H_i} + \frac{|A|_i|D H_i|}{H_i^2} + |Dg|\right) ds, \end{aligned}$$
(112)
where all norms are taken with respect to \(\sigma \), which yields the second estimate and the first estimate follows similarly. \(\square \)

In order for the previous lemma to be useful we will need to deduce integral estimates for \(|\nabla A|\). The following interpolation inequality will be key which can be found in section 12 of the work of Hamilton [26].

Lemma 2.16

If T is a tensor on \(\Sigma \) then there exists a constant C(nm), independent of the metric and the connection, so that the following estimate holds
$$\begin{aligned} \int _{\Sigma }|\nabla ^iT|^2 d \mu \le C \left( \int _{\Sigma }|\nabla ^mT|^2 d \mu \right) ^{i/m}\left( \int _{\Sigma } |T|^2 \right) ^{1-i/m} \end{aligned}$$
(113)
for \(0\le i \le m\).

We now use this interpolation inequality in combination with the assumptions of Definition 1.1 to show the desired integral convergence of |DA|.

Corollary 2.17

Let \(\Sigma ^i\subset M^i\) be a compact, connected surface with corresponding solution to IMCF \(\Sigma _t^i\) such that
$$\begin{aligned} \Vert A_i\Vert _{W^{2,2}(\Sigma \times [0,T])} \le C. \end{aligned}$$
(114)
If
$$\begin{aligned}&m_H(\Sigma _0)\ge 0\text { and } m_H(\Sigma ^i_T) \rightarrow 0 \end{aligned}$$
(115)
$$\begin{aligned}&\text { or }\left( m_H(\Sigma ^i_T)-m_H(\Sigma ^i_0) \right) \rightarrow 0\text { where }m_H(\Sigma _0) \rightarrow m > 0 \end{aligned}$$
(116)
then for almost every \(t \in [0,T]\)
$$\begin{aligned} \int _{\Sigma } |D A_i|^2 r_0^2e^td \sigma \rightarrow 0. \end{aligned}$$
(117)

Proof

Consider the tensor \(T=A-\frac{e^{-t/2}}{r_0}g\) and apply Lemma 2.16 with \(m=2\) and \(i=1\) to find
$$\begin{aligned} \int _{\Sigma _t^i}|D A|^2 d \mu&\le C \left( \int _{\Sigma _t^i}|D^2 A|^2 d \mu \right) ^{1/2}\left( \int _{\Sigma _t^i} |A-\frac{e^{-t/2}}{r_0}g|^2 \right) ^{1/2} \end{aligned}$$
(118)
$$\begin{aligned}&\le C \sqrt{A_2} \left( \int _{\Sigma _t^i} n \max \left\{ |\lambda _1-\frac{e^{-t/2}}{r_0}|^2,|\lambda _2-\frac{e^{-t/2}}{r_0}|^2 \right\} \right) ^{1/2} \rightarrow 0 \end{aligned}$$
(119)
where \(\lambda _1,\lambda _2\) are the eigenvalues of A. The last convergence result follows from the definition of \(|\nabla A|\) and |DA| as well as the assume \(L^2\) convergence. \(\square \)

Now we will prove estimates which allow us to use the evolution equation for H to gain \(W^{1,2}\) control on H at the price of assuming \(L^2\) or \(W^{1,2}\) control on the Ricci curvature.

Lemma 2.18

Let \(\Sigma _t\) be a solution of IMCF such that
$$\begin{aligned} 0< H_0 \le H(x,t) \le H_1 < \infty \end{aligned}$$
(120)
then
$$\begin{aligned}&\int _0^T\int _{\Sigma _t} 4|Rc(\nu ,\nu )|^2 +8|A|^2|Rc(\nu ,\nu )| + 4 |A|^4d\mu dt + \int _{\Sigma _0} \frac{|\nabla H|^2}{H^2}d \mu \end{aligned}$$
(121)
$$\begin{aligned}&\ge \int _0^T\int _{\Sigma _t} \left( \frac{\partial H^2}{\partial t}\right) ^2 + \frac{(\Delta H^2)^2}{H^4}d\mu dt + \sup _{t \in [0,T]} \int _{\Sigma _t}\frac{|\nabla H|^2}{H^2} d\mu . \end{aligned}$$
(122)
If the stronger estimate of the hessian of \(H^2\) is needed instead of just the laplacian then the above estimate can be improved to find
$$\begin{aligned}&\int _0^T\int _{\Sigma _t} 4|Rc(\nu ,\nu )|^2 +8|A|^2|Rc(\nu ,\nu )|+4|A|^4+ |Rc| \frac{|\nabla H^2|^2}{H_1^4}d\mu dt \end{aligned}$$
(123)
$$\begin{aligned}&+ \int _{\Sigma _0} \frac{|\nabla H|^2}{H^2}d \mu \ge \int _0^T\int _{\Sigma _t} \left( \frac{\partial H^2}{\partial t}\right) ^2 + \frac{|\nabla \nabla H^2|^2}{H_1^4}d\mu dt \end{aligned}$$
(124)
$$\begin{aligned}&+ \sup _{t \in [0,T]} \int _{\Sigma _t}\frac{|\nabla H|^2}{H^2} d\mu . \end{aligned}$$
(125)

Proof

We start by integrating the square of the following linear PDE for \(H^2\)
$$\begin{aligned} \left( \partial _t - \frac{\Delta }{H^2} \right) H^2 = -2|A|^2-2Rc(\nu ,\nu ) \end{aligned}$$
(126)
in order to find
$$\begin{aligned} \int _{\Sigma _t}(-2|A|^2-2Rc(\nu ,\nu ))^2 d \mu&= \int _{\Sigma _t}\left[ \left( \partial _t - \frac{\Delta }{H^2} \right) H^2 \right] ^2 d \mu \end{aligned}$$
(127)
from which we obtain
$$\begin{aligned}&\int _{\Sigma _t}4|A|^4+8|A|^2|Rc(\nu ,\nu )|+ 4|Rc(\nu ,\nu ))|^2 d \mu \end{aligned}$$
(128)
$$\begin{aligned}&\ge \int _{\Sigma _t}\left( \frac{\partial H^2}{\partial t} \right) ^2 - 2 \frac{\partial H^2}{\partial t}\frac{\Delta H^2}{H^2} + \frac{(\Delta H^2)^2}{H^4} d \mu \end{aligned}$$
(129)
$$\begin{aligned}&\ge \int _{\Sigma _t}\left( \frac{\partial H^2}{\partial t} \right) ^2 +2\frac{\partial }{\partial t}\nabla _k H^2 \frac{\nabla ^kH^2}{H^2}- 2 \frac{\partial H^2}{\partial t}\frac{|\nabla H^2|^2}{H^3} +\frac{(\Delta H^2)^2}{H_1^4} d \mu \end{aligned}$$
(130)
$$\begin{aligned}&= \int _{\Sigma _t}\left( \frac{\partial H^2}{\partial t} \right) ^2 + \frac{\partial }{\partial t} \left( \frac{|\nabla H^2|}{H^2} \right) -\frac{\nabla ^kH^2 \nabla _k\nabla ^i\nabla _iH^2}{H_1^4} d \mu \end{aligned}$$
(131)
$$\begin{aligned}&= \int _{\Sigma _t}\left( \frac{\partial H^2}{\partial t} \right) ^2 + \frac{\partial }{\partial t} \left( \frac{|\nabla H^2|}{H^2} \right) -\frac{\nabla ^kH^2 \nabla _i\nabla ^i\nabla _kH^2}{H_1^4} -\frac{R_{kl} \nabla ^kH^2\nabla ^lH^2}{H_1^4} d \mu \end{aligned}$$
(132)
$$\begin{aligned}&= \int _{\Sigma _t}\left( \frac{\partial H^2}{\partial t} \right) ^2 + \frac{\partial }{\partial t} \left( \frac{|\nabla H^2|}{H^2} \right) +\frac{|\nabla \nabla H^2|^2}{H_1^4} -\frac{Rc(\nabla H^2,\nabla H^2)}{H_1^4} d \mu \end{aligned}$$
(133)
So now we integrate from 0 to \(t'\) with respect to t, \(t' \in [0,T]\), to find
$$\begin{aligned}&\int _0^{t'}\int _{\Sigma _t} 4|Rc(\nu ,\nu )|^2+8|A|^2|Rc(\nu ,\nu )|+4|A|^4 + \frac{|Rc||\nabla H^2|^2}{H_1^4}d\mu dt \end{aligned}$$
(134)
$$\begin{aligned}&\ge \int _0^{t'}\int _{\Sigma _t} 4|Rc(\nu ,\nu )|^2 +8|A|^2|Rc(\nu ,\nu )|+4|A|^4+ \frac{Rc(\nabla H^2, \nabla H^2)}{H_1^4}d\mu dt \end{aligned}$$
(135)
$$\begin{aligned}&\ge \int _0^{t'}\int _{\Sigma _t} \left( \frac{\partial H^2}{\partial t}\right) ^2 + \frac{|\nabla \nabla H^2|^2}{H_0^4}d\mu dt + \int _{\Sigma _{t'}}\frac{|\nabla H|^2}{H^2} d\mu - \int _{\Sigma _0}\frac{|\nabla H|^2}{H^2} d\mu \end{aligned}$$
(136)
and then taking the \(\sup _{t' \in [0,T]}\) of both sides we find the desired estimate. \(\square \)

We now show how to use the previous Lemma to show new higher order convergence results.

Lemma 2.19

If \(\Sigma _t^i\) is a sequence of IMCF solutions where
$$\begin{aligned}&m_H(\Sigma _t) \rightarrow 0\text { as }i \rightarrow \infty , \end{aligned}$$
(137)
$$\begin{aligned}&m_H^{{\mathbb {H}}}(\Sigma _t) \rightarrow 0\text { as }i \rightarrow \infty , \end{aligned}$$
(138)
$$\begin{aligned}&m_H(\Sigma _{T}^i)- m_H(\Sigma _{0}^i) \rightarrow 0\text { and }m_H(\Sigma _t^i)\rightarrow m > 0, \end{aligned}$$
(139)
$$\begin{aligned}&\text { or }m_H^{{\mathbb {H}}}(\Sigma _{T}^i)- m_H^{{\mathbb {H}}}(\Sigma _{0}^i) \rightarrow 0\text { and }m_H^{{\mathbb {H}}}(\Sigma _t^i)\rightarrow m > 0, \end{aligned}$$
(140)
and
$$\begin{aligned}&0< H_0 \le H(x,t) \le H_1 < \infty , \end{aligned}$$
(141)
$$\begin{aligned}&|A|(x,t) \le A_0 < \infty , \end{aligned}$$
(142)
$$\begin{aligned}&\Vert Rc^i(\nu ,\nu )\Vert _{W^{1,2}(\Sigma \times [0,T])} \le C \end{aligned}$$
(143)
then
$$\begin{aligned}&\int _0^T\int _{\Sigma _t} \left( \frac{\partial H_i^2}{\partial t}\right) ^2-4|A|_i^4 d \mu dt \rightarrow 0, \end{aligned}$$
(144)
$$\begin{aligned}&\int _0^T\int _{\Sigma _t} \frac{(\Delta H_i^2)^2}{H_i^4} d \mu dt \rightarrow 0, \end{aligned}$$
(145)
$$\begin{aligned}&\sup _{t \in [0,T]} \int _{\Sigma _t}\frac{|\nabla H_i|^2}{H_i^2} d\mu \rightarrow 0. \end{aligned}$$
(146)

Proof

Notice that by combining (143) with Lemma 2.8 we find that
$$\begin{aligned} \Vert Rc^i(\nu ,\nu )\Vert _{L^2(\Sigma \times [0,T])} \rightarrow 0. \end{aligned}$$
(147)
Then we note that Lemma 2.18 implies
$$\begin{aligned}&\int _0^T\int _{\Sigma _t} 4|Rc^i(\nu ,\nu )|^2 +8|A|_i^2|Rc^i(\nu ,\nu )| d\mu dt + \int _{\Sigma _0} \frac{|\nabla H_i|^2}{H_i^2}d \mu \end{aligned}$$
(148)
$$\begin{aligned}&\ge \int _0^T\int _{\Sigma _t} \left( \frac{\partial H_i^2}{\partial t}\right) ^2- 4 |A|_i^4 + \frac{(\Delta H_i^2)^2}{H_i^4}d\mu dt + \sup _{t \in [0,T]} \int _{\Sigma _t}\frac{|\nabla H_i|^2}{H_i^2} d\mu \end{aligned}$$
(149)
and (147) implies
$$\begin{aligned}&\int _0^T\int _{\Sigma _t}|A|_i^2|Rc^i(\nu ,\nu )| d \mu dt \end{aligned}$$
(150)
$$\begin{aligned}&\le \left( \int _0^T\int _{\Sigma _t}|A|_i^4 d \mu dt\right) ^{1/2} \left( \int _0^T\int _{\Sigma _t}|Rc^i(\nu ,\nu )|^2 d \mu dt\right) ^{1/2} \rightarrow 0. \end{aligned}$$
(151)
Now since Open image in new window we notice that Corollary 2.14 implies
$$\begin{aligned} \lim _{i \rightarrow \infty } \left[ \left( \frac{\partial H_i^2}{\partial t}\right) ^2-4|A|_i^4 \right] \ge 0, \end{aligned}$$
(152)
for a.e. \(t \in [0,T]\) and \(x \in \Sigma \). Combining (152) with Fatou’s Lemma and the assumption \(|A|_i \le A_0\) we find,
$$\begin{aligned}&\int _0^T\int _{\Sigma _t} \left( \frac{\partial H_i^2}{\partial t}\right) ^2-4|A|_i^4 d \mu dt \rightarrow 0, \end{aligned}$$
(153)
$$\begin{aligned}&\int _0^T\int _{\Sigma _t} \frac{(\Delta H_i^2)^2}{H_i^4} d \mu dt \rightarrow 0, \end{aligned}$$
(154)
$$\begin{aligned}&\sup _{t \in [0,T]} \int _{\Sigma _t}\frac{|\nabla H_i|^2}{H_i^2} d\mu \rightarrow 0. \end{aligned}$$
(155)
\(\square \)

2.3 Consequences of rigidity results

In this subsection we use rigidity results for Riemannian manifolds to deduce important consequences for the metric on \(\Sigma _t\), \(g^i(x,t)\).

Theorem 2.20

(Theorem 6.4 of Petersen [36]) Let \(n \ge 2\) \(\Lambda , v, D > 0\) and \(c \in {\mathbb {R}}\) be given. There exists \(\epsilon =\epsilon (n,\Lambda ,v, D)\) such that any \((\Sigma ,g)\) satisfying
$$\begin{aligned}&|Rc| \le \Lambda \end{aligned}$$
(156)
$$\begin{aligned}&vol(\Sigma )\ge v \end{aligned}$$
(157)
$$\begin{aligned}&diam(\Sigma ) \le D \end{aligned}$$
(158)
$$\begin{aligned}&\left( \int _{\Sigma } \Vert Rm - \lambda g \circ g\Vert ^{n/2} d \mu \right) ^{2/n}\le \epsilon \end{aligned}$$
(159)
is \(C^{1+\alpha }\) close to a constant curvature metric for any \(\alpha < 1\).

Note that when \(n=2\) the Riemann curvature tensor is \(Rm = K g \circ g\), where \(g \circ g\) represents the Kulkarni-Nomizu product, and so \(\Vert Rm - \lambda g \circ g\Vert ^2 = \Vert g \circ g\Vert ^2 |K - \lambda |^2 = 2^4|K - \lambda |^2\).

Lemma 2.21

Let \(\Sigma _t^i\) be a sequence of solutions to IMCF such that \(\Sigma _T^i\) is a sequence of hypersurfaces such that
$$\begin{aligned}&m_H(\Sigma _t) \rightarrow 0\text { as }i \rightarrow \infty , \end{aligned}$$
(160)
$$\begin{aligned}&m_H^{{\mathbb {H}}}(\Sigma _t) \rightarrow 0\text { as }i \rightarrow \infty , \end{aligned}$$
(161)
$$\begin{aligned}&m_H(\Sigma _{T}^i)- m_H(\Sigma _{0}^i) \rightarrow 0\text { and }m_H(\Sigma _t^i)\rightarrow m > 0, \end{aligned}$$
(162)
$$\begin{aligned}&\text { or }m_H^{{\mathbb {H}}}(\Sigma _{T}^i)- m_H^{{\mathbb {H}}}(\Sigma _{0}^i) \rightarrow 0\text { and }m_H^{{\mathbb {H}}}(\Sigma _t^i)\rightarrow m > 0, \end{aligned}$$
(163)
If we assume,
$$\begin{aligned}&\Vert Rc^i(\nu ,\nu )\Vert _{W^{1,2}(\Sigma \times \{T\})} \le C, \end{aligned}$$
(164)
$$\begin{aligned}&\Vert R^i\Vert _{L^2(\Sigma \times \{T\})} \le C, \end{aligned}$$
(165)
$$\begin{aligned}&diam(\Sigma _T^i) \le D \text { } \forall \text { } i, \end{aligned}$$
(166)
$$\begin{aligned}&|K^i| \le C \text { on } \Sigma _T, \end{aligned}$$
(167)
then we find that
$$\begin{aligned} g^i(x,T) \rightarrow r_0^2 e^T\sigma (x) \end{aligned}$$
(168)
in \(C^{1,\alpha }\), \(0<\alpha < 1\).

Proof

By the assumption that \(\Vert Rc^i(\nu ,\nu )\Vert _{W^{1,2}(\Sigma \times \{T\})} \le C\) we know by Sobolev embedding that a subsequence converges strongly in \(L^2(\Sigma \times \{T\})\) with the measure \(r_0^2d\sigma \), i.e.
$$\begin{aligned} \int _{\Sigma }|Rc^j(\nu ,\nu )|^2 r_0^2d\sigma \rightarrow k(x,t) \in L^2(\Sigma \times \{T\}). \end{aligned}$$
(169)
By combining (169) with Lemma 2.8 we find that
$$\begin{aligned} \int _{\Sigma }|Rc^j(\nu ,\nu )|^2 r_0^2d\sigma \rightarrow 0. \end{aligned}$$
(170)
Then since \(R^i\) converges to 0 in \(L^1\) by Corollary 2.5 we can combine with the assumed \(L^2\) bound and an interpolation inequality to conclude that \(R^i\) converges to 0 in \(L^2\). Then we have that \(\int _{\Sigma }|K_{12}^i| d\mu ^{\infty }_0 \rightarrow 0\) and hence
$$\begin{aligned} \int _{\Sigma }|K^i-\frac{1}{r_0^2}|r_0^2d\sigma&=\int _{\Sigma }|K_{12}^i+ \lambda _1^i\lambda _2^i - \frac{1}{r_0^2}|r_0^2d\sigma \end{aligned}$$
(171)
$$\begin{aligned}&\le 2\int _{\Sigma }|K_{12}^i|+|\lambda _1^i\lambda _2^i - \frac{1}{r_0^2}| r_0^2d\sigma \rightarrow 0 \end{aligned}$$
(172)
where we use the pointwise a.e. convergence of \(\lambda _j^i\), \(j=1,2\), that is implied by Corollary 2.5, on a subsequence, and the bound \(|A|_i\le C\).

This shows that \(\int _{\Sigma } |K^i - \frac{1}{r_0^2}| d \mu _0^{\infty } \rightarrow 0\) and hence by combining with the diameter bound diam\((\Sigma _0^i)\le D\) and the pointwise Gauss curvature bound \(K^i \le C\) on \(\Sigma _T\) then we can apply the rigidity result of Petersen [36], Corollary 2.20, which implies that \(|g^i(x,0) - r_0^2\sigma (x)|_{C^{1+\alpha }} \rightarrow 0\) as \(i \rightarrow \infty \) where \(\alpha < 1\). \(\square \)

Remark 2.22

By combining Lemma 2.21 with Lemma 2.10 and the bounds assumed in Definition 1.1 we now have shown that \({\hat{g}}^i\) has a uniform upper bound. Note that this implies that the metrics \(g^i\) and \(\sigma \) are uniformly equivalent on \(\Sigma _t\) for \(t \in [0,T]\) and hence quantities which are converging to zero in coordinates will converge to zero in norm with respect to either metric.

Theorem 2.23

(Corollary 1.5 of Petersen and Wei [37]) Given any integer \(n \ge 2\), and numbers \(p > n/2\), \(\lambda \in {\mathbb {R}}\), \(v >0\), \(D < \infty \), one can find \(\epsilon = \epsilon (n,p,\lambda , D) > 0\) such that a closed Riemannian \(n-\)manifold \((\Sigma ,g)\) with,
$$\begin{aligned}&\text {vol}(\Sigma )\ge v, \end{aligned}$$
(173)
$$\begin{aligned}&\text {diam}(\Sigma ) \le D, \end{aligned}$$
(174)
$$\begin{aligned}&\frac{1}{|\Sigma |} \int _{\Sigma } \Vert Rm - \lambda g \circ g\Vert ^p d \mu \le \epsilon (n,p,\lambda ,D), \end{aligned}$$
(175)
is \(C^{\alpha }\), \(\alpha < 2 - \frac{n}{p}\), close to a constant curvature metric on \(\Sigma \).

Lemma 2.24

Let \(\Sigma _t^i\) be a sequence of solutions to IMCF such that
$$\begin{aligned}&m_H(\Sigma _t) \rightarrow 0\text { as }i \rightarrow \infty , \end{aligned}$$
(176)
$$\begin{aligned}&m_H^{{\mathbb {H}}}(\Sigma _t) \rightarrow 0\text { as }i \rightarrow \infty , \end{aligned}$$
(177)
$$\begin{aligned}&m_H(\Sigma _{T}^i)- m_H(\Sigma _{0}^i) \rightarrow 0\text { and }m_H(\Sigma _t^i)\rightarrow m > 0, \end{aligned}$$
(178)
$$\begin{aligned}&\text { or }m_H^{{\mathbb {H}}}(\Sigma _{T}^i)- m_H^{{\mathbb {H}}}(\Sigma _{0}^i) \rightarrow 0\text { and }m_H^{{\mathbb {H}}}(\Sigma _t^i)\rightarrow m > 0, \end{aligned}$$
(179)
If we assume,
$$\begin{aligned} \Vert Rc^i(\nu ,\nu )\Vert _{W^{1,2}(\Sigma \times [0,T])}&\le C, \end{aligned}$$
(180)
$$\begin{aligned} \Vert R^i\Vert _{L^2(\Sigma \times [0,T])}&\le C, \end{aligned}$$
(181)
$$\begin{aligned} Diam(\Sigma _t)&\le C \text { for } t \in [0,T], \end{aligned}$$
(182)
then on a subsequence,
$$\begin{aligned} g^k(x,t) \rightarrow r_0^2 e^t \sigma (x), \end{aligned}$$
(183)
in \(C^{\alpha }\) for a.e. \(t \in [0,T]\).

Proof

By (180) combined with Lemma 2.8 we find that
$$\begin{aligned} \int _0^T\int _{\Sigma }|Rc^i(\nu ,\nu )|^2 d \mu dt \rightarrow 0. \end{aligned}$$
(184)
By combining (181) with Corollary 2.5 and an interpolation inequality we find that
$$\begin{aligned} \int _0^T\int _{\Sigma }|R^i|^p d \mu dt \rightarrow 0 \end{aligned}$$
(185)
for \(1<p<2\) and hence
$$\begin{aligned} \int _0^T\int _{\Sigma }|K_{12}^i|^p d \mu dt \le \int _0^T\int _{\Sigma }|Rc^i(\nu ,\nu )|^p+|R^i|^p d \mu dt \rightarrow 0 \end{aligned}$$
(186)
for \(1< p < 2\). Now we notice that by combining with Corollary 2.5 we find
$$\begin{aligned} \int _0^T\int _{\Sigma }|K^i- \frac{1}{r_0^2}|^p d \mu dt \le \int _0^T\int _{\Sigma }|K_{12}^i|^p+|\lambda _1^i\lambda _2^i - \frac{1}{r_0^2}|^p d \mu dt \rightarrow 0 \end{aligned}$$
(187)
for \(1<p<2\). Now by combining with the rigidity result Theorem 2.23 and the diameter bound (182) we find that on a subsequence
$$\begin{aligned} g^k(x,t) \rightarrow r_0^2 e^t \sigma (x) \end{aligned}$$
(188)
in \(C^{\alpha }\) for a.e. \(t \in [0,T]\). \(\square \)

3 Convergence to prototype spaces

In this section we successively show the pairwise convergence of interpolating metrics in \(W^{1,2}\) from \({\hat{g}}^i(x,t)\) to \(\delta \), \(g_S\), \(g_{{\mathbb {H}}}\), or \(g_{ADSS}\). By combining all the pairwise convergence results using the triangle inequality we will be able to prove the main theorems of this work. In this section derivatives with respect to \(\{\partial _0 = \partial _t,\partial _1,\partial _2\}\) denote Euclidean covariant derivatives with respect to polar coordinates on \(\Sigma \times [0,T]\).

Theorem 3.1

Let \(U_{T}^i \subset M_i^3\) be a sequence such that \(U_{T}^i\subset {\mathcal {M}}_{r_0,H_0,I_0}^{T,H_1,A_1}\) and assume
$$\begin{aligned}&m_H(\Sigma _t) \rightarrow 0\text { as }i \rightarrow \infty , \end{aligned}$$
(189)
$$\begin{aligned}&m_H^{{\mathbb {H}}}(\Sigma _t) \rightarrow 0\text { as }i \rightarrow \infty , \end{aligned}$$
(190)
$$\begin{aligned}&m_H(\Sigma _{T}^i)- m_H(\Sigma _{0}^i) \rightarrow 0\text { and }m_H(\Sigma _t^i)\rightarrow m > 0, \end{aligned}$$
(191)
$$\begin{aligned}&\text { or }m_H^{{\mathbb {H}}}(\Sigma _{T}^i)- m_H^{{\mathbb {H}}}(\Sigma _{0}^i) \rightarrow 0\text { and }m_H^{{\mathbb {H}}}(\Sigma _t^i)\rightarrow m > 0. \end{aligned}$$
(192)
If we consider (4) and define the metric,
$$\begin{aligned} g^i_1(x,t)&= \frac{1}{{\overline{H}}_i(t)^2}dt^2 + g^i(x,t), \end{aligned}$$
(193)
on \(U_T^i\) then,
$$\begin{aligned} \int _{U_T}|{\hat{g}}^i -g^i_1|^2 dV+ \sum _{k=0}^2\int _{U_T}|\partial _k {\hat{g}}^i -\partial _k g^i_1|^2 dV \rightarrow 0, \end{aligned}$$
(194)
as \(i \rightarrow \infty \) where dV is the volume form on \(U_T=\Sigma \times [0,T]\) with respect to \(\delta \). Where \(\{\partial _0 = \partial _t,\partial _1,\partial _2\}\) denotes the derivatives with respect to the coordinates on \(\Sigma \times [0,T]\).

Proof

We have previously shown that,
$$\begin{aligned} \int _{U_T}|{\hat{g}}^i -g^i_1|^2 dV \rightarrow 0, \end{aligned}$$
(195)
so now we concentrate on the derivative terms. We denote \(\partial _t = \partial _0\) and then \(\partial _1,\partial _2\) are the two spacial derivatives with respect to \(\Sigma \). We can compute,
$$\begin{aligned} \partial _k {\hat{g}}^i&=\frac{-2}{H_i(x,t)^3} \partial _k H_i(x,t) dt^2 + \partial _kg^i(x,t), \end{aligned}$$
(196)
$$\begin{aligned} \partial _k {\hat{g}}_1^i&=\frac{-2}{{\overline{H}}_i(t)^3} \partial _k {\bar{H}}_i(t) dt^2 + \partial _kg^i(x,t), \end{aligned}$$
(197)
where we notice that we already know that,
$$\begin{aligned} \sum _{k=1}^2 \int _{U_T}|\partial _k{\hat{g}}^i -\partial _kg^i_1|^2 dV =\int _{U_T}\frac{|\nabla H_i|^2}{H^3_i} dV \rightarrow 0. \end{aligned}$$
(198)
$$\begin{aligned} \int _{U_T}|\partial _0{\hat{g}}^i -\partial _0g^i_1|^2 dV&= \int _{U_T^i} \left| \frac{2}{H_i(x,t)^3} \partial _0 H_i(x,t) -\frac{2}{{\overline{H}}_i(t)^3} \partial _0 {\bar{H}}_i(t) \right| ^2 dV \end{aligned}$$
(199)
$$\begin{aligned}&= \int _{U_T} \frac{2}{H_i^6{\overline{H}}_i^6} \left| {\bar{H}}^3_i \frac{\partial H_i}{\partial t} - H_i^3\frac{\partial {\bar{H}}_i}{\partial t} \right| ^2dV \end{aligned}$$
(200)
$$\begin{aligned}&\le \frac{2}{H_0^{12}}\int _{U_T} {\bar{H}}^3_i \left| \frac{\partial H_i}{\partial t} + \frac{e^{-t/2}}{r_0} \right| ^2dV \end{aligned}$$
(201)
$$\begin{aligned}&+\frac{2}{H_0^{12}} \int _{U_T}\frac{e^{-t}}{r_0^2}\left| {\bar{H}}^3_i - H_i^3 \right| ^2dV \end{aligned}$$
(202)
$$\begin{aligned}&+\frac{2}{H_0^{12}}\int _{U_T} H_i^3\left| \frac{e^{-t/2}}{r_0} + \frac{\partial {\bar{H}}_i}{\partial t} \right| ^2 dV \end{aligned}$$
(203)
Notice that (201) goes to zero since by the evolution equation for H we find
$$\begin{aligned}&\int _{U_T} {\bar{H}}^3_i \left| \frac{\partial H_i}{\partial t} + \frac{e^{-t/2}}{r_0} \right| ^2dV \end{aligned}$$
(204)
$$\begin{aligned}&=\int _{U_T} {\bar{H}}^3_i \left| \frac{\Delta H_i}{H_i^2} - \frac{|A|_i^2}{H_i} - \frac{Rc^i(\nu ,\nu )}{H_i} + \frac{e^{-t/2}}{r_0} \right| ^2dV \end{aligned}$$
(205)
$$\begin{aligned}&\le H_1^3 \int _{U_T}\frac{|\Delta H_i|^2}{H_0^2} +\frac{|Rc(\nu ,\nu )|^2}{H_0} + \left| \frac{e^{-t/2}}{r_0}- \frac{|A|_i^2}{H_i}\right| ^2 dV \end{aligned}$$
(206)
and the last equation goes to zero on a subsequence by Corollary 2.5 and Lemma 2.19.

Then we see that (202) goes to zero by Proposition 2.7, and (203) goes to zero since Lemma 2.13 implies that \(\frac{d {\bar{H}}_i}{dt}\) is bounded so we can apply the dominated convergence theorem to Corollary 2.14. The proof is almost exactly the same in all three other cases where the quantity \(\frac{e^{-t/2}}{r_0}\) is replaced with the corresponding quantity for the case being considered. \(\square \)

Theorem 3.2

Let \(U_{T}^i \subset M_i^3\) be a sequence such that \(U_{T}^i\subset {\mathcal {M}}_{r_0,H_0,I_0}^{T,H_1,A_1}\) and assume
$$\begin{aligned}&m_H(\Sigma _t) \rightarrow 0\text { as }i \rightarrow \infty , \end{aligned}$$
(207)
$$\begin{aligned}&m_H^{{\mathbb {H}}}(\Sigma _t) \rightarrow 0\text { as }i \rightarrow \infty , \end{aligned}$$
(208)
$$\begin{aligned}&m_H(\Sigma _{T}^i)- m_H(\Sigma _{0}^i) \rightarrow 0\text { and }m_H(\Sigma _t^i)\rightarrow m > 0, \end{aligned}$$
(209)
$$\begin{aligned}&\text { or }m_H^{{\mathbb {H}}}(\Sigma _{T}^i)- m_H^{{\mathbb {H}}}(\Sigma _{0}^i) \rightarrow 0\text { and }m_H^{{\mathbb {H}}}(\Sigma _t^i)\rightarrow m > 0. \end{aligned}$$
(210)
If we define the metrics,
$$\begin{aligned} g^i_1(x,t)&= \frac{1}{{\overline{H}}_i(t)^2}dt^2 + g^i(x,t), \end{aligned}$$
(211)
$$\begin{aligned} g^i_2(x,t)&= \frac{1}{{\overline{H}}_i(t)^2}dt^2 + e^{t-T}g^i(x,T), \end{aligned}$$
(212)
on \(U_T^i\) then we have that,
$$\begin{aligned} \int _{U_T}|g^i_1 -g^i_2|_{g_3^i}^2 dV+ \sum _{k=0}^2\int _{U_T}|\partial _k g_1^i -\partial _k g^i_2|^2 dV \rightarrow 0, \end{aligned}$$
(213)
as \(i \rightarrow \infty \) where dV is the volume form on \(U_T=\Sigma \times [0,T]\) with respect to \(\delta \).

Proof

We already have that
$$\begin{aligned} \int _{U_T^i}&|g^i_1 -g^i_2|^2 dV \rightarrow 0, \end{aligned}$$
(214)
and we can compute that,
$$\begin{aligned}&\int _{U_T}|\partial _0g^i_1 -\partial _0 g^i_2|^2 dV = \int _{U_T}|\frac{\partial g^i}{\partial t}(x,t) -e^{t-T}g^i(x,T)|^2 dV \end{aligned}$$
(215)
$$\begin{aligned}&\quad =\int _{U_T}|\frac{2A^i}{H_i} -e^{t-T}g^i(x,T)|^2 dV \end{aligned}$$
(216)
$$\begin{aligned}&\quad \le \int _{U_T} \max _{j=1,2} \left\{ \left| \frac{2\lambda _j^i(x,t)g^i(x,t)}{H_i} -e^{t-T}g^i(x,T)\right| ^2 \right\} dV \end{aligned}$$
(217)
$$\begin{aligned}&\quad \le \int _{U_T}|g^i(x,T)|^2 \max _{j=1,2} \left\{ \left| \frac{2\lambda _j^i(x,t)}{H_i}e^{\int _T^t\frac{2\lambda ^i_j(x,s)}{H^i(x,s)}ds} -e^{t-T}\right| ^2 \right\} dV, \end{aligned}$$
(218)
where \(\lambda _1^i(x,t), \lambda _2^i(x,t)\) are the smallest and largest eigenvalue of \(A^i(x,t)\), respectively. Note by combining Lemma 2.10 with Lemma 2.5 we find that (218) goes to zero on a subsequence.
Now we calculate for \(k=1,2\)
$$\begin{aligned} \int _{U_T^i}|\partial _k g^i_1 -\partial _k g^i_2|^2 dV&= \int _{U_T^i}|\partial _k g^i(x,t) -e^{t-T}\partial _k g^i(x,T)|^2 dV. \end{aligned}$$
(219)
Note that by Lemma 2.21 we have that
$$\begin{aligned} |\partial _kg^i(x,T)| \rightarrow 0, \end{aligned}$$
(220)
and hence by combining Lemmas 2.10, 2.15, and 2.17 we find that
$$\begin{aligned}&\int _0^T\int _{\Sigma }|\partial _kg^i(x,t)|r_0^2 e^td \mu dt \end{aligned}$$
(221)
$$\begin{aligned}&\quad \le \int _0^T \int _{\Sigma }\int _0^t \frac{|DA|_i}{H_i} + \frac{|A|_i|DH|_i}{H_i^2} ds r_0^2 e^td \mu dt \end{aligned}$$
(222)
$$\begin{aligned}&\quad = \int _0^T r_0^2 e^t \int _0^t \int _{\Sigma } \frac{|DA|_i}{H_i} + \frac{|A|_i|DH|_i}{H_i^2}d \mu ds dt\rightarrow 0. \end{aligned}$$
(223)
Then by applying the result of Lemma 2.15 again in a similar way we find that (219) goes to zero, as desired.

We can get rid of the need for a subsequence by assuming to the contrary that for \(\epsilon > 0\) there exists a subsequence so that \(\int _{U_T^k}|g_1^k -g^k_2|^2 dV \ge \epsilon \), but this subsequence satisfies the hypotheses of Theorem 3.2 and hence by what we have just shown we know a subsequence must converge which is a contradiction. \(\square \)

Now we want to use the fact that we know that the average of the mean curvature is converging to that of a sphere in euclidean space in order to complete the convergence to the warped product \(g_3^i\).

Theorem 3.3

Let \(U_{T}^i \subset M_i^3\) be a sequence such that \(U_{T}^i\subset {\mathcal {M}}_{r_0,H_0,I_0}^{T,H_1,A_1}\) and \(m_H(\Sigma _{T}^i) \rightarrow 0\) as \(i \rightarrow \infty \). If we define the metrics,
$$\begin{aligned} g^i_2(x,t)&= \frac{1}{{\bar{H}}^i(t)^2}dt^2 + e^{t-T}g^i(x,T), \end{aligned}$$
(224)
$$\begin{aligned} g^i_3(x,t)&= \frac{r_0^2}{4}e^tdt^2 + e^tg^i(x,0), \end{aligned}$$
(225)
on \(U_T^i\) then we have that,
$$\begin{aligned} \int _{U_T}|g^i_2 -g^i_3|^2 dV + \sum _{k=0}^2\int _{U_T}|\partial _k g^i_2 -\partial _k g^i_3|^2 dV\rightarrow 0, \end{aligned}$$
(226)
as \(i \rightarrow \infty \) where dV is the volume form on \(U_T=\Sigma \times [0,T]\) with respect to \(\delta \).
Instead, if \(m_H(\Sigma _{T}^i)- m_H(\Sigma _{0}^i) \rightarrow 0\) and \(m_H(\Sigma _t^i)\rightarrow m > 0\) and we define,
$$\begin{aligned} g^i_3(x,t)= \frac{r_0^2}{4}\left( 1 - \frac{2}{r_0} m e^{-t/2} \right) ^{-1} e^t dt^2 + e^{t-T}g^i(x,T), \end{aligned}$$
(227)
on \(U_T^i\) then we have that,
$$\begin{aligned} \int _{U_T}|g^i_2 -g^i_3|^2 dV + \sum _{k=0}^2\int _{U_T}|\partial _k g_2^i -\partial _k g^i_3|^2 dV\rightarrow 0, \end{aligned}$$
(228)
as \(i \rightarrow \infty \) where dV is the volume form on \(U_T=\Sigma \times [0,T]\) with respect to \(\delta \).
If \(m_H^{{\mathbb {H}}}(\Sigma _{T}^i) \rightarrow 0\) as \(i \rightarrow \infty \). If we define the metrics,
$$\begin{aligned} g^i_3(x,t)&= \frac{1}{4}\left( 1+ \frac{e^{-t}}{r_0^2} \right) ^{-1}dt^2 + e^{t-T}g^i(x,T), \end{aligned}$$
(229)
on \(U_T^i\) then we have that,
$$\begin{aligned} \int _{U_T}|g^i_2 -g^i_3|^2 dV + \sum _{k=0}^2\int _{U_T}|\partial _k g^i_2 -\partial _k g^i_3|^2 dV\rightarrow 0 , \end{aligned}$$
(230)
as \(i \rightarrow \infty \) where dV is the volume form on \(U_T=\Sigma \times [0,T]\) with respect to \(\delta \).
Instead, if \(m_H^{{\mathbb {H}}}(\Sigma _{T}^i)- m_H^{{\mathbb {H}}}(\Sigma _{0}^i) \rightarrow 0\) and \(m_H^{{\mathbb {H}}}(\Sigma _t^i)\rightarrow m > 0\) and we define,
$$\begin{aligned} g^i_3(x,t)= \frac{1}{4}\left( \frac{1}{r_0^2}e^{-t} - \frac{2}{r_0^3} m e^{-3t/2}+1 \right) ^{-1} dt^2 + e^{t-T}g^i(x,T), \end{aligned}$$
(231)
on \(U_T^i\) then we have that,
$$\begin{aligned} \int _{U_T}|g^i_2 -g^i_3|^2 dV + \sum _{k=0}^2\int _{U_T}|\partial _k g^i_2 -\partial _k g^i_3|^2 dV\rightarrow 0 , \end{aligned}$$
(232)
as \(i \rightarrow \infty \) where dV is the volume form on \(U_T=\Sigma \times [0,T]\) with respect to \(\delta \).

Proof

We have previously shown,
$$\begin{aligned} \int _{U_T}|{\hat{g}}_2^i -g^i_3|^2 dV \rightarrow 0, \end{aligned}$$
(233)
and we note that the only coordinate derivatives we need to consider in this case are with respect to t,
$$\begin{aligned} \partial _0 g_2^i&=\frac{-2}{{\bar{H}}_i(t)^3} \frac{\partial {\bar{H}}_i}{\partial t}(t) dt^2 + e^{t-T}g^i(x,T), \end{aligned}$$
(234)
$$\begin{aligned} \partial _0 g_3^i&=\frac{r_0^2}{4}e^t dt^2 + e^{t-T}g^i(x,T), \end{aligned}$$
(235)
and so we compute
$$\begin{aligned} \int _{U_T}|\partial _0{\hat{g}}_2^i -\partial _0g^i_3|^2 dV&=\int _{U_T}\left| \frac{r_0^2}{4}e^t +\frac{2}{{\bar{H}}_i(t)^3} \frac{\partial \bar{H_i}}{\partial t}(x,t) \right| ^2\rightarrow 0, \end{aligned}$$
(236)
which follows from applying Corollary 2.14 and Lemma 2.1 which give pointwise almost everywhere convergence of the integrand in (236) on a subsequence. Notice that Lemma 28 also implies that \(\frac{d{\bar{H}}^2}{dt}\) is bounded and hence we can use the dominated convergence theorem to conclude that the integral in (236) is \(\rightarrow 0\). The proof is almost exactly the same in all three other cases where the quantity \(\frac{r_0^2}{4}e^t\) is replaced with the corresponding quantity for the case being considered.

We can get rid of the need for a subsequence by assuming to the contrary that for \(\epsilon > 0\) there exists a subsequence so that \(\int _{U_T}|g_2^k -g^k_3|^2 dV \ge \epsilon \), but this subsequence satisfies the hypotheses of Theorem 3.3 and hence by what we have just shown we know a subsequence must converge which is a contradiction. \(\square \)

We now finish by completing the \(W^{1,2}\) convergence to the appropriate prototype spaces.

Theorem 3.4

Let \(U_{T}^i \subset M_i^3\) be a sequence such that \(U_{T}^i\subset {\mathcal {M}}_{r_0,H_0,I_0}^{T,H_1,A_1}\) and \(m_H(\Sigma _{T}^i) \rightarrow 0\) as \(i \rightarrow \infty \). Considering the metric (225) on \(U_T^i\) then,
$$\begin{aligned} \int _{U_T}|g^i_3 -\delta |^2 dV + \sum _{k=0}^2\int _{U_T}|\partial _k g^i_3 -\partial _k \delta |^2 dV\rightarrow 0, \end{aligned}$$
(237)
as \(i \rightarrow \infty \) where dV is the volume form on \(U_T=\Sigma \times [0,T]\) with respect to \(\delta \).
Instead, if \(m_H(\Sigma _{T}^i)- m_H(\Sigma _{0}^i) \rightarrow 0\) and \(m_H(\Sigma _t^i)\rightarrow m > 0\) and we consider (227) on \(U_T^i\) then,
$$\begin{aligned} \int _{U_T}|g^i_3 -g_S|^2 dV + \sum _{k=0}^2\int _{U_T}|\partial _k g_3^i -\partial _k g_S|^2 dV\rightarrow 0 , \end{aligned}$$
(238)
as \(i \rightarrow \infty \) where dV is the volume form on \(U_T=\Sigma \times [0,T]\) with respect to \(\delta \).
If \(m_H^{{\mathbb {H}}}(\Sigma _{T}^i) \rightarrow 0\) as \(i \rightarrow \infty \) and we consider (229) on \(U_T^i\) then,
$$\begin{aligned} \int _{U_T}|g^i_3 -g_{{\mathbb {H}}}|^2 dV + \sum _{k=0}^2\int _{U_T}|\partial _k g^i_3 -\partial _k g_{{\mathbb {H}}}|^2 dV\rightarrow 0, \end{aligned}$$
(239)
as \(i \rightarrow \infty \) where dV is the volume form on \(U_T=\Sigma \times [0,T]\) with respect to \(\delta \).
Instead, if \(m_H^{{\mathbb {H}}}(\Sigma _{T}^i)- m_H^{{\mathbb {H}}}(\Sigma _{0}^i) \rightarrow 0\) and \(m_H^{{\mathbb {H}}}(\Sigma _t^i)\rightarrow m > 0\) and we consider (231) on \(U_T^i\) then,
$$\begin{aligned} \int _{U_T}|g^i_3 -g_{ADSS}|^2 dV + \sum _{k=0}^2\int _{U_T}|\partial _k g^i_3 -\partial _k g_{ADSS}|^2 dV\rightarrow 0, \end{aligned}$$
(240)
as \(i \rightarrow \infty \) where dV is the volume form on \(U_T=\Sigma \times [0,T]\) with respect to \(\delta \).

Proof

This follows by Lemma 2.21. \(\square \)

Proof of Main Theorems:

Proof

The proof of the main theorems now follows by combining Theorems 3.1, 3.2, 3.3 and 3.4 via the triangle inequality. Note that \(W^{1,2}\) convergence implies \(L^6\) convergence in dimension 3 which is how we can conclude that the volume of regions are converging. \(\square \)

Notes

References

  1. 1.
    Abbott, L.F., Deser, S.: Stability of gravity with a cosmological constant. Nucl. Phys. B 195(1), 76–96 (1982)ADSCrossRefGoogle Scholar
  2. 2.
    Allen, B.: Inverse mean curvature flow and the stability of the PMT and RPI under \(L^2\) convergence. Ann. Henri Poincaré 19(4), 1283–1306 (2018)ADSMathSciNetCrossRefGoogle Scholar
  3. 3.
    Allen, B.: Stability of the PMT and RPI for asymptotically hyperbolic manifolds foliated by IMCF. J. Math. Phys. 59, 082501 (2018)ADSMathSciNetCrossRefGoogle Scholar
  4. 4.
    Allen, B.: Long Time Existence of Inverse Mean Curvature Flow in Metrics Conformal to Warped Product Manifolds. arXiv:1708.02535 [math.DG] (2017)
  5. 5.
    Allen, B.: Inverse Mean Curvature Flow and the Stability of the Positive Mass Theorem. arXiv:1807.08822 [math.DG] (2018)
  6. 6.
    Allen, B., Sormani, C.: Contrasting Various Notions of Convergence in Geometric Analysis. arXiv:1803.06582 [math.MG] (2018)
  7. 7.
    Andersson, L., Cai, M., Galloway, G.J.: Rigidity and positivity of mass for asymptotically hyperbolic manifolds. Ann. Henri Poincaré 9(1), 1–33 (2008)ADSMathSciNetCrossRefGoogle Scholar
  8. 8.
    Ashtekar, A., Megnon, A.: Asymptotically anti-de Sitter space–times. Class. Quantum Gravity 1(4), L39–L44 (1984)ADSMathSciNetCrossRefGoogle Scholar
  9. 9.
    Bray, H.: Proof of the Riemannian Penrose inequality using the positive mass theorem. J. Differ. Geom. 59(2), 177–267 (2001)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Bray, H., Finster, F.: Curvature estimates and the positive mass theorem. Commun. Anal. Geom. 2, 291–306 (2002)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Brendle, S., Hung, P.-K., Wang, M.-T.: A Minkowski inequality for hypersurfaces in the anti-de Sitter–Schwarzschild manifold. Commun. Pure Appl. Math. 69(1), 124–144 (2016)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Bryden, E.: Stability of the Positive Mass Theorem for Axisymmetric Manifolds. arXiv:1806.02447 [math.DG] (2018)
  13. 13.
    Cabrera Pacheco, A.J.: On the Stability of the Positive Mass Theorem for Asymptotically Hyperbolic Graphs. arXiv:1803.01899 [math.DG] (2018)
  14. 14.
    Chruściel, P., Herzlich, M.: The mass of asymptotically hyperbolic Riemannian manifolds. Pac. J. Math. 2, 393–443 (2003)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Corvino, J.: A note on asymptotically flat metrics on \(\mathbb{R}^{3}\) which are scalar-flat and admit minimal spheres. Proc. Am. Math. Soc. 12, 3669–3678 (2005). (electronic)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Dahl, M., Gicquaud, R., Sakovich, A.: Penrose type inequalities for asymptotically hyperbolic graphs. Ann. Henri Poincaré 14(5), 1135–1168 (2013)ADSMathSciNetCrossRefGoogle Scholar
  17. 17.
    Dahl, M., Gicquaud, R., Sakovich, A.: Asymptotically hyperbolic manifolds with small mass. Commun. Math. Phys. 325, 757–801 (2014)ADSMathSciNetCrossRefGoogle Scholar
  18. 18.
    de Lima, L.L., Girão, F.: An Alexandrov–Fenchel-type inequality in hyperbolic space with an application to a Penrose inequality. Ann. Henri Poincaré 17(4), 979–1002 (2016)ADSMathSciNetCrossRefGoogle Scholar
  19. 19.
    Ding, Q.: The inverse mean curvature flow in rotationally symmetric spaces. Chin. Ann. Math. Ser. B 32(1), 27–44 (2011)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Finster, F.: A level set analysis of the Witten spinor with applications to curvature estimates. Math. Res. Lett. 1, 41–55 (2009)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Finster, F., Kath, I.: Curvature estimates in asymptotically flat manifolds of positive scalar curvature. Commun. Anal. Geom. 5, 1017–1031 (2002)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Gerhardt, C.: Flow of nonconvex hypersurfaces into spheres. J. Differ. Geom. 32, 299–314 (1990)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Gerhardt, C.: Inverse curvature flows in hyperbolic space. J. Differ. Geom. 89, 487–527 (2011)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Geroch, R.: Energy extraction. Ann. N. Y. Acad. Sci. 224, 108–117 (1973)ADSCrossRefGoogle Scholar
  25. 25.
    Gibbons, G.W., Hawking, S.W., Horowitz, G.T., Perry, M.J.: Positive mass theorems for black holes. Commun. Math. Phys. 88(3), 295–308 (1983)ADSMathSciNetCrossRefGoogle Scholar
  26. 26.
    Hamilton, R.: Three-manifolds with positive Ricci curvature. J. Differ. Geom. 17, 255–306 (1982)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Huang, L.-H., Lee, D., Sormani, C.: Intrinsic flat stability of the positive mass theorem for graphical hypersurfaces of Euclidean space. J. fur die Riene und Ang. Math. (Crelle’s Journal) 727, 1–299 (2015)Google Scholar
  28. 28.
    Hung, P.-K., Wang, M.-T.: Inverse mean curvature flows in the hyperbolic 3-space revisited. Calc. Var. PDE 54, 119–126 (2015)MathSciNetCrossRefGoogle Scholar
  29. 29.
    Huisken, G., Ilmanen, T.: The inverse mean curvature flow and the Riemannian Penrose inequality. J. Differ. Geom. 59, 353–437 (2001)MathSciNetCrossRefGoogle Scholar
  30. 30.
    Lee, D.: On the near-equality case of the positive mass theorem. Duke Math. J. 1, 63–80 (2009)ADSMathSciNetCrossRefGoogle Scholar
  31. 31.
    Lee, D., Sormani, C.: Near-equality in the Penrose inequality for rotationally symmetric Riemannian manifolds. Ann. Henri Poinc. 13, 1537–1556 (2012)ADSCrossRefGoogle Scholar
  32. 32.
    Lee, D., Sormani, C.: Stability of the positive mass theorem for rotationally symmetric Riemannian manifolds. J. fur die Riene und Ang. Math. (Crelle’s Journal) 686, 187–220 (2014)MathSciNetzbMATHGoogle Scholar
  33. 33.
    LeFloch, P., Mardare, C.: Definition and stability of Lorentzian manifolds with distributional curvature. Portugaliae Mathematica 64, 535–574 (2007)MathSciNetCrossRefGoogle Scholar
  34. 34.
    LeFloch, P., Sormani, C.: The nonlinear stability of spaces with low regularity. J. Funct. Anal. 268(7), 2005–2065 (2015)MathSciNetCrossRefGoogle Scholar
  35. 35.
    Neves, A.: Insufficient convergence of inverse mean curvature flow on asymptotically hyperbolic manifolds. J. Differ. Geom. 84, 191–229 (2010)MathSciNetCrossRefGoogle Scholar
  36. 36.
    Petersen, P.: Convergence theorems in riemannian geometry. In: Grove, K., Petersen, P. (eds.) Comparison Geometry, vol. 30, pp. 167–202. MSRI Publications, Cambridge University Press, Cambridge (1997)zbMATHGoogle Scholar
  37. 37.
    Petersen, P., Wei, G.: Relative volume comparison with integral curvature bounds. Geom. Funct. Anal. 7, 1031–1045 (1997)MathSciNetCrossRefGoogle Scholar
  38. 38.
    Sakovich, A., Sormani, C.: Almost rigidity of the positive mass theorem for asymptotically hyperbolic manifolds with spherical symmetry. Gen. Relativ. Gravit. 49, 125 (2017)ADSMathSciNetCrossRefGoogle Scholar
  39. 39.
    Scheuer, J.: The inverse mean curvature flow in warped cylinders of non-positive radial curvature. Adv. Math. 306, 1130–1163 (2017)MathSciNetCrossRefGoogle Scholar
  40. 40.
    Scheuer, J.: Inverse curvature flows in Riemannian warped products. J. Funct. Anal. 276(4), 1097–1144 (2019)MathSciNetCrossRefGoogle Scholar
  41. 41.
    Schoen, R., Yau, S.-T.: On the proof of the positive mass conjecture in general relativity. Commun. Math. Phys. 65, 45–46 (1979)ADSMathSciNetCrossRefGoogle Scholar
  42. 42.
    Urbas, J.: On the expansion of starshaped hypersurfaces by symmetric functions of their principal curvatures. Math. A. 205, 355–372 (1990)MathSciNetzbMATHGoogle Scholar
  43. 43.
    Wang, X.: The mass of asymptotically hyperbolic manifolds. J. Differ. Geom. 2, 273–299 (2001)MathSciNetCrossRefGoogle Scholar
  44. 44.
    Zhou, H.: Inverse mean curvature flows in warped product manifolds. J. Geom. Anal. 28(2), 1749–1772 (2018)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.US Military AcademyWest PointUSA

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