Inverse mean curvature flow and Ricci-pinched three-manifolds

Gerhard Huisken; Thomas Koerber

doi:10.1515/crelle-2024-0040

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Inverse mean curvature flow and Ricci-pinched three-manifolds

Gerhard Huisken and Thomas Koerber

Published/Copyright: July 2, 2024

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From the journal Journal für die reine und angewandte Mathematik (Crelles Journal) Volume 2024 Issue 814

Abstract

Let ( M , g ) be a noncompact, connected, complete Riemannian three-manifold with nonnegative Ricci curvature satisfying Ric ≥ ε ⁢ tr ⁡ ( Ric ) ⁢ g for some ε > 0 . In this note, we give a new proof based on inverse mean curvature flow that ( M , g ) is either flat or has non-Euclidean volume growth. In conjunction with the work of J. Lott [On 3-manifolds with pointwise pinched nonnegative Ricci curvature, Math. Ann. 388 (2024), 3, 2787–2806] and of M.-C. Lee and P. Topping [Three-manifolds with non-negatively pinched Ricci curvature, preprint (2022), https://arxiv.org/abs/2204.00504], this gives an alternative proof of a conjecture of R. Hamilton recently proven by A. Deruelle, F. Schulze, and M. Simon [Initial stability estimates for Ricci flow and three dimensional Ricci-pinched manifolds, preprint (2022), https://arxiv.org/abs/2203.15313] using Ricci flow.

1 Introduction

Let ( M , g ) be a noncompact, connected, complete Riemannian three-manifold with nonnegative Ricci curvature. Recall that ( M , g ) is called Ricci-pinched if there is ε > 0 such that

(1.1) Ric ≥ ε ⁢ R ⁢ g .

Here, Ric and 𝑅 denote the Ricci curvature and the scalar curvature of ( M , g ) , respectively. The following theorem has been conjectured by R. Hamilton [4, Conjecture 3.39] and by J. Lott [13, Conjecture 1.1]. It has been proven by A. Deruelle, F. Schulze, and M. Simon [7, Theorem 1.3] under the additional assumption that ( M , g ) has bounded curvature. M.-C. Lee and P. Topping [11, Theorem 1.2] have subsequently shown that this additional assumption can be dispensed with.

Theorem 1

Theorem 1 ([11, Theorem 1.2])

Let ( M , g ) be a noncompact, connected, complete Riemannian three-manifold that is Ricci-pinched. Then ( M , g ) is flat.

Remark 2

Previous results in the direction of Theorem 1 have been obtained by B.-L. Chen and X.-P. Zhu [3, Main Theorem II] and by J. Lott [13, Theorem 1.4].

Remark 3

R. Hamilton has shown that every compact, connected, complete Riemannian three-manifold that is Ricci-pinched is either flat or smoothly isotopic to a spherical space form; see [8, Main Theorem] and the discussion on [7, p. 4].

To describe the contributions in [13, 7, 11], let p ∈ M and let

AVR = 3 4 ⁢ π ⁢ lim r → ∞ | B r ⁢ ( p ) | r 3

be the asymptotic volume ratio of ( M , g ) . Note that, by the Bishop–Gromov theorem, AVR is well-defined, independent of 𝑝, and satisfies AVR ∈ [ 0 , 1 ] . J. Lott [13] has shown that if ( M , g ) is Ricci-pinched and has bounded curvature, then there exists a smooth, Ricci-pinched solution of Ricci flow coming out of ( M , g ) . By performing a detailed asymptotic analysis of this flow, J. Lott has proven that if ( M , g ) is not flat, then ( M , g ) has positive asymptotic volume ratio; see [13, Corollary 1.7]. Subsequently, M.-C. Lee and P. Topping [11] have shown that the assumption that ( M , g ) has bounded curvature can be dispensed with. By contrast, if ( M , g ) has positive asymptotic volume ratio, A. Deruelle, F. Schulze, and M. Simon [7, Lemma 8.2] have observed that the asymptotic cone of ( M , g ) is a three-dimensional Alexandrov space with nonnegative curvature. Moreover, the previous work of A. Deruelle [6] respectively of F. Schulze and M. Simon [17] implies the existence of an expanding soliton solution with nonnegative curvature coming out of the asymptotic cone of ( M , g ) . Using their stability result [7, Theorem 1.2] to compare this solution with the Ricci-pinched solution constructed by J. Lott [13], they have concluded that ( M , g ) is in fact flat.

The goal of this paper is to provide a new proof based on inverse mean curvature flow of Theorem 1 under the additional assumption that ( M , g ) has positive asymptotic volume ratio.

Theorem 4

Let ( M , g ) be a noncompact, connected, complete Riemannian three-manifold that is Ricci-pinched. If ( M , g ) has positive asymptotic volume ratio, then ( M , g ) is isometric to flat R 3 .

Remark 5

In conjunction with the results of J. Lott [13] and M.-C. Lee and P. Topping [11], Theorem 4 gives a new proof of Theorem 1.

Remark 6

Our technique extends to the case where ( M , g ) has superquadratic volume growth and the volume of geodesic unit balls in ( M , g ) is noncollapsed; see Theorem 12.

We now outline the proof of Theorem 4. Suppose, for a contradiction, that ( M , g ) is not flat. Since ( M , g ) has nonnegative Ricci curvature and is Ricci-pinched, the scalar curvature of ( M , g ) must be strictly positive at one point. It follows that there is a closed, connected, outward-minimizing surface Σ ⊂ M such that

(1.2) ∫ Σ H 2 ⁢ d μ < 16 ⁢ π .

Here, d ⁢ μ and 𝐻 denote the area element and the mean curvature of Σ, respectively. Recall that a nested family { E t } t = 0 ∞ of precompact, open sets E t ⊂ M with smooth and strictly mean-convex boundary ∂ E t flows by inverse mean curvature flow if

d ⁢ x d ⁢ t = H − 1 ⁢ ν .

Here, 𝑥 and 𝜈 are the position and the outward normal of ∂ E t , respectively. Using that ( M , g ) has positive asymptotic volume ratio, the work of K. Xu [19] shows that there exists a weak solution { E t } t = 0 ∞ of inverse mean curvature flow with ∂ ∗ E 0 = Σ in the sense of the work of T. Ilmanen and the first-named author [10]. Here, ∂ ∗ denotes the reduced boundary. Using (1.2) and that ( M , g ) is Ricci-pinched, we show that

lim t → ∞ ∫ ∂ ∗ E t H 2 ⁢ d μ = 0 .

By contrast, the work of V. Agostiniani, M. Fogagnolo, and L. Mazzieri [1] implies that, for every t ≥ 0 ,

(1.3) ∫ ∂ ∗ E t H 2 ⁢ d μ ≥ 16 ⁢ π ⁢ AVR ,

a contradiction; see also the work of X. Wang [18] for an alternative proof of (1.3).

2 Proof of Theorem 4

In this section, we assume that ( M , g ) is a noncompact, connected, complete Riemannian three-manifold with nonnegative Ricci curvature satisfying (1.1) for some ε > 0 .

The goal of this section is to prove Theorem 4. We recall the following result of S.-H. Zhu [20], which extends previous work of R. Schoen and S.-T. Yau [16, Theorem 3].

Proposition 7

If ( M , g ) is not flat, then ( M , g ) is diffeomorphic to R 3 .

Proof

Using (1.1), we see that there is p ∈ M with Ric ⁢ ( p ) > 0 . The assertion follows from [20]. ∎

Let Σ ⊂ M be a closed, connected surface with area measure d ⁢ μ , designated normal 𝜈, and mean curvature 𝐻 with respect to 𝜈. In Lemma 8 below, h ̊ denotes the traceless second fundamental form of Σ.

Lemma 8

If genus ⁡ ( Σ ) ≥ 1 , there holds

(2.1) 2 ⁢ ∫ Σ Ric ⁢ ( ν , ν ) + | h ̊ | 2 ⁢ d ⁢ μ ≥ ∫ Σ H 2 ⁢ d μ

and, if genus ⁡ ( Σ ) = 0 , there holds

(2.2) 2 ⁢ ∫ Σ Ric ⁢ ( ν , ν ) ⁢ d μ ≥ ε ⁢ ( 16 ⁢ π − ∫ Σ H 2 ⁢ d μ ) .

Proof

Integrating the contracted Gauss equation and using the Gauss–Bonnet theorem, we have

∫ Σ H 2 ⁢ d μ = 16 ⁢ π ⁢ ( 1 − genus ⁡ ( Σ ) ) + 2 ⁢ ∫ Σ | h ̊ | 2 ⁢ d μ + ∫ Σ 4 ⁢ R ⁢ i ⁢ c ⁢ ( ν , ν ) − 2 ⁢ R ⁢ d ⁢ μ .

Using that Ric ≥ 0 , we have R ≥ Ric ⁢ ( ν , ν ) . This implies (2.1).

In the case where genus ⁡ ( Σ ) = 0 , we have, using that Ric ≥ 0 ,

2 ⁢ ∫ Σ R ⁢ d μ ≥ 16 ⁢ π − ∫ Σ H 2 .

In conjunction with (1.1), we obtain (2.2). ∎

Lemma 9

Suppose that ( M , g ) is not flat. There exists a sequence { Σ i } i = 1 ∞ of closed, connected surfaces Σ i ⊂ M with

lim i → ∞ ∫ Σ i H 2 ⁢ d μ = 0 .

Proof

In the case where AVR = 0 , the assertion follows from [1, Theorem 1.1].

In the case where AVR > 0 , using that ( M , g ) is not flat and that Ric ≥ 0 , we see that there is p ∈ M with R ⁢ ( p ) > 0 . Consequently,

(2.3) ∫ B r ⁢ ( p ) H 2 ⁢ d μ < 16 ⁢ π

provided that r > 0 is sufficiently small; see, e.g., [15, Proposition 3.1]. Let Σ ′ ⊂ M be the outward-minimizing hull of B r ⁢ ( p ) ; see [10, p. 371]. Using [10, (1.15)], we see that

∫ Σ ′ H 2 ⁢ d μ ≤ ∫ B r ⁢ ( p ) H 2 ⁢ d μ .

By the Bishop–Gromov theorem, we have

| B 1 ⁢ ( q ) | ≥ 4 ⁢ π 3 ⁢ AVR

for every q ∈ M . Using this and that AVR > 0 , the work of T. Coulhon and L. Saloff-Coste [5] implies that there is γ > 0 such that | ∂ Ω | ≥ γ ⁢ | Ω | 2 / 3 for every precompact, open set Ω ⊂ M with smooth boundary ∂ Ω ; see the remarks below [5, Theorème 3] and also [1, 2]. In conjunction with [19, Theorem 1.2], it follows that there exists a proper weak solution { E t } t = 0 ∞ of inverse mean curvature flow in the sense of [10, p. 368] such that Σ ′ = ∂ ∗ E 0 . Let Σ t = ∂ ∗ E t . By Proposition 7 and [10, Lemma 4.2], Σ t is connected for every t ≥ 0 . According to the results in [10, §5] and [9, Korollar 5.6], Σ t is of class W 2 , 2 ∩ C 1 , 1 and there holds

(2.4) ∫ Σ 0 H 2 ⁢ d μ ≥ ∫ Σ t H 2 ⁢ d μ + 2 ⁢ ∫ 0 t ∫ Σ s Ric ⁢ ( ν , ν ) + | h ̊ | 2 ⁢ d ⁢ μ ⁢ d ⁢ s

for every t ≥ 0 . Clearly, the function

(2.5) [ 0 , ∞ ) → R , t ↦ ∫ Σ t H 2 ⁢ d μ

is nonincreasing. We claim that

lim t → ∞ ∫ Σ t H 2 ⁢ d μ = 0 .

Indeed, suppose, for a contradiction, that there is δ > 0 such that

∫ Σ t H 2 ⁢ d μ ≥ δ

for every t ≥ 0 . Shrinking δ > 0 , if necessary, and using (2.3), we may assume that

∫ Σ t H 2 ⁢ d μ ≤ 16 ⁢ π − δ

for every t ≥ 0 . Using Lemma 8, we have

2 ⁢ ∫ Σ s Ric ⁢ ( ν , ν ) + | h ̊ | 2 ⁢ d ⁢ μ ≥ min ⁡ { 1 , ε } ⁢ δ

for every t ≥ 0 . In conjunction with (2.4), we see that

∫ Σ t H 2 ⁢ d μ < 0

for every t ≥ 16 π min { 1 , ε } − 1 δ − 1 . The assertion follows from this contradiction. ∎

Proof of Theorem 4

Let ( M , g ) be a noncompact, connected, complete Riemannian three-manifold that is Ricci-pinched and has positive asymptotic volume ratio AVR . Suppose, for a contradiction, that ( M , g ) is not flat R 3 . By Lemma 9, there exists a closed, connected surface Σ ⊂ M with

∫ Σ H 2 ⁢ d μ < 16 ⁢ π ⁢ AVR .

As this is incompatible with [1, Theorem 1.1], the assertion follows. ∎

Funding source: Austrian Science Fund

Award Identifier / Grant number: M3184

Funding statement: The second-named author acknowledges the support of the Lise-Meitner-Project M3184 of the Austrian Science Fund. For open access purposes, the authors have applied a CC BY public copyright license to any author accepted manuscript version arising from this submission.

A Superquadratic volume growth

In this section, we assume that ( M , g ) is a noncompact, connected, complete Riemannian three-manifold with nonnegative Ricci curvature satisfying (1.1) for some ε > 0 . Moreover, we assume that there is p ∈ M and α > 0 with

(A.1) lim inf r → ∞ | B r ⁢ ( p ) | r 1 + α > 0

and that

(A.2) inf q ∈ M | B 1 ⁢ ( q ) | > 0 .

Note that, by the Bishop–Gromov theorem, α ≤ 2 . Moreover, note that if α ≤ 1 , then ( M , g ) is parabolic; see [12, Corollary 2.3].

The goal of this section is to give an alternative proof, based on inverse mean curvature flow, of the fact that ( M , g ) is either flat or has subquadratic volume growth, that is, α ≤ 1 .

By the work of T. Coulhon and L. Saloff-Coste, using (A.1) and (A.2), the isoperimetric inequality

(A.3) | ∂ Ω | ≥ γ ⁢ min ⁡ { | Ω | 2 / 3 , | Ω | α / ( 1 + α ) }

holds for some γ > 0 and every precompact open set Ω ⊂ M with smooth boundary ∂ Ω ; see the remarks below [5, Theorème 3].

Lemma 10

Suppose that ( M , g ) is not flat. Then there exists a proper weak solution { E t } t = 0 ∞ of inverse mean curvature flow such that, for every t ≥ 0 ,

∫ ∂ ∗ E t H 2 ⁢ d μ < 16 ⁢ π ⁢ e − t .

Proof

By Lemma 9, there is a closed, connected surface Σ ⊂ M satisfying

ε ⁢ ( 16 ⁢ π − ∫ Σ H 2 ⁢ d μ ) ≥ ∫ Σ H 2 ⁢ d μ .

Replacing Σ by its outward-minimizing hull and using [10, (1.15)], we may assume that Σ is outward-minimizing. By [19, Theorem 1.2], using (A.3), there is a proper weak solution { E t } t = 0 ∞ of inverse mean curvature flow with ∂ ∗ E 0 = Σ . Let Σ t = ∂ ∗ E t . As in the proof of Lemma 9, we see that the function (2.5) is nonincreasing so that, for every t ≥ 0 ,

(A.4) ε ⁢ ( 16 ⁢ π − ∫ Σ t H 2 ⁢ d μ ) ≥ ∫ Σ t H 2 ⁢ d μ .

Again, as in the proof of Lemma 9, using (A.4) and Lemma 8, we see that

∫ Σ H 2 ⁢ d μ ≥ ∫ Σ t H 2 ⁢ d μ + ∫ 0 t ∫ Σ s H 2 ⁢ d μ ⁢ d s .

Using that

∫ Σ H 2 ⁢ d μ < 16 ⁢ π ,

the assertion follows. ∎

Remark 11

The existence of a proper weak solution of inverse mean curvature flow as asserted in Lemma 10 would also follow from [14, Remark 1.6 and Theorem 1.7], even without assuming (A.2), but replacing (A.1) by

0 < lim r → ∞ | B r ⁢ ( p ) | r 1 + α < ∞ .

However, we were not able to verify all arguments in the proof of [14, Theorem 3.6].

Theorem 12

Let ( M , g ) be a noncompact, connected, complete Riemannian three-manifold that is Ricci-pinched and satisfies (A.1) for some α > 0 and (A.2). If ( M , g ) is not flat, then α ≤ 1 .

Proof

Let { E t } t = 0 ∞ be a proper weak solution of inverse mean curvature flow as in Lemma 10. Let Σ t = ∂ ∗ E t . By [10, Lemma 5.1], there holds H > 0 d ⁢ μ -almost everywhere on Σ t and H − 1 ∈ L 1 ⁢ ( Σ t ) for almost every t ≥ 0 . Using Hölder’s inequality, we have, for almost every t ≥ 0 ,

(A.5) | Σ t | 3 / 2 ≤ ∫ Σ t H − 1 ⁢ d μ ⁢ ( ∫ Σ t H 2 ⁢ d μ ) 1 / 2 .

Recall from [10, Exponential Growth Lemma 1.6] that

(A.6) | Σ t | = | Σ 0 | ⁢ e t .

In conjunction with Lemma 10 and (A.5), we obtain that, for almost every t ≥ 0 ,

(A.7) ∫ Σ t H − 1 ⁢ d μ ≥ | Σ 0 | 3 16 ⁢ π ⁢ e 2 ⁢ t .

Arguing as in [10, §5], we see that

| E t | ≥ | E 0 | + ∫ 0 t ∫ Σ s H − 1 ⁢ d μ ⁢ d s

for every t ≥ 0 . In conjunction with (A.6) and (A.7), it follows that

(A.8) lim inf t → ∞ | Σ t | − 2 ⁢ | E t | > 0 .

By contrast, using (A.3), we have

| Σ t | − 2 ⁢ | E t | ≤ γ − 2 ⁢ | E t | ( 1 − α ) / ( 1 + α ) .

As this is incompatible with (A.8) unless α ≤ 1 , the assertion follows. ∎

Acknowledgements

This work originated during the authors’ visit to the Hebrew University during the first-named author’s Mark Gordon Distinguished Visiting Professorship. The authors thank the Hebrew University for its hospitality. The authors thank Or Hershkovits, Marco Pozetta, and Miles Simon for helpful discussions. The authors thank the referees for the valuable comments and corrections.

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Received: 2023-07-25

Revised: 2024-04-08

Published Online: 2024-07-02

Published in Print: 2024-09-01

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