Minimal hypersurfaces in manifolds of Ricci curvature bounded below

Qi Ding

doi:10.1515/crelle-2022-0050

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Minimal hypersurfaces in manifolds of Ricci curvature bounded below

Published/Copyright: August 27, 2022

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From the journal Journal für die reine und angewandte Mathematik (Crelles Journal) Volume 2022 Issue 791

Abstract

In this paper, we study the angle estimate of distance functions from minimal hypersurfaces in manifolds of Ricci curvature bounded from below using Colding’s method in [T. H. Colding, Ricci curvature and volume convergence, Ann. of Math. (2) 145 1997, 3, 477–501]. With Cheeger–Colding theory, we obtain the Laplacian comparison for limits of distance functions from minimal hypersurfaces in the version of Ricci limit space. As an application, if a sequence of minimal hypersurfaces converges to a metric cone C ⁢ Y × ℝ n - k ( 2 ≤ k ≤ n ) in a non-collapsing metric cone C ⁢ X × ℝ n - k obtained from ambient manifolds of almost nonnegative Ricci curvature, then we can prove a Frankel property for the cross section Y of CY. Namely, Y has only one connected component in X.

Funding source: National Natural Science Foundation of China

Award Identifier / Grant number: 11871156

Award Identifier / Grant number: 11922106

Funding statement: The author is partially supported by NSFC 11871156 and NSFC 11922106.

A Appendix I

For each integer i ≥ 1 , let ( X i , d i ) be a sequence of compact metric spaces. Let ( X ∞ , d ∞ ) be a compact metric space such that there is a sequence of ϵ i -Gromov–Hausdorff approximations Φ i : X i → X ∞ for some sequence ϵ i → 0 . For each i, let f i be a function on X i . For a function f on X ∞ , we say that f i → f if f i ⁢ ( x i ) → f ⁢ ( x ) for any x ∈ X ∞ and any sequence x i ∈ X i with x i → x . We further assume that all the functions f i are Lipschitz with lim sup i → ∞ ⁡ ( sup X i ⁡ | f i | + 𝐋𝐢𝐩 f i ) < ∞ .

Lemma A.1.

There are a subsequence i ′ → ∞ and a Lipschitz function f ∞ on X ∞ with Lip f ∞ ≤ lim sup i → ∞ ⁡ Lip f i such that f i ′ → f ∞ .

Proof.

Let δ k > 0 be a sequence of numbers with δ k → 0 as k → ∞ . For each k, let { y k , j } j = 1 m k be a finite δ k -net of X ∞ . For each j ∈ { 1 , … , m 1 } , let x i , j ∈ X i be a sequence converging to y 1 , j as i → ∞ . Then there is a subsequence { 1 i } of { i } such that f 1 i ⁢ ( x 1 i , j ) converges to a number t 1 , j ∈ ℝ . For each j ∈ { 1 , … , m 2 } and for any sequence x 1 i , j ∈ X 1 i , j converging as i → ∞ to y 2 , j , there is a subsequence { 2 i } of { 1 i } such that f 2 i ⁢ ( x 2 i , j ) converges to a number t 2 , j ∈ ℝ . Continue the procedure to get a sequence of functions { f k i } i , k ≥ 1 such that { ( k + 1 ) i } is a subsequence of { k i } , and for any y k , j and X k i ∋ x k i , j → y k , j , lim i → ∞ ⁡ f k i ⁢ ( x k i , j ) converges to t k , j ∈ ℝ as i → ∞ .

From lim sup i → ∞ ⁡ 𝐋𝐢𝐩 f i < ∞ , for y k , j = y k ′ , j ′ there holds t k , j = t k ′ , j ′ . We define a function f ∞ on ⋃ k ≥ 1 ⋃ 1 ≤ j ≤ m k { y k , j } by letting f ∞ ⁢ ( y k , j ) = t k , j . From

lim sup i → ∞ ⁡ 𝐋𝐢𝐩 f i < ∞ ,

for any sequence X k i ∋ x k i , j ′ → y k , j we have

lim i → ∞ ⁡ f k i ⁢ ( x k i , j ′ ) = lim i → ∞ ⁡ f k i ⁢ ( x k i , j ) = f ∞ ⁢ ( y k , j ) .

Moreover, for any y , y ′ ∈ ⋃ k ≥ 1 ⋃ 1 ≤ j ≤ N k { y k , j } it follows that

| f ∞ ⁢ ( y ) - f ∞ ⁢ ( y ′ ) | ≤ ( lim sup i → ∞ ⁡ 𝐋𝐢𝐩 f i ) ⁢ d ∞ ⁢ ( y , y ′ ) .

For any y ∈ X ∞ , there is a sequence { y i } ⊂ ⋃ k ≥ 1 ⋃ 1 ≤ j ≤ N k { y k , j } with y i → y , and we define f ∞ ⁢ ( y ) = lim i → ∞ ⁡ f ∞ ⁢ ( y i ) . From the above inequality, the definition of the function f ∞ ⁢ ( y ) is independent of the particular choice of y i . Then 𝐋𝐢𝐩 f ∞ ≤ lim sup i → ∞ ⁡ 𝐋𝐢𝐩 f i and f ∞ ⁢ ( y ) = lim i → ∞ ⁡ f i i ⁢ ( y i ′ ) for any y i ′ ∈ X i i with y i ′ → y . ∎

For each integer i ≥ 1 , let E i be a closed set in X i . Suppose that Φ i ⁢ ( E i ) converges in the Hausdorff sense to a closed set E ∞ in X ∞ . From the triangle inequality, one has

| ρ E ∞ ⁢ ( x ) - ρ E ∞ ⁢ ( y ) | ≤ d ∞ ⁢ ( x , y ) for any ⁢ x , y ∈ X ∞ .

In other words, ρ E ∞ has the Lipschitz constant 𝐋𝐢𝐩 ρ E ∞ ≤ 1 on X ∞ .

Lemma A.2.

For any point x ∈ X ∞ and any sequence x i ∈ X i with x i → x as i → ∞ , we have ρ E i ⁢ ( x i ) → ρ E ∞ ⁢ ( x ) .

Proof.

The proof is routine. For each x ∈ X ∞ , there is a point y ∈ E ∞ so that

d ∞ ⁢ ( x , y ) = ρ E ∞ ⁢ ( x ) .

Let y i ∈ E i with y i → y . Then for any sequence x i ∈ X i with x i → x , there holds

(A.1) ρ E ∞ ⁢ ( x ) = d ∞ ⁢ ( x , y ) = lim i → ∞ ⁡ d i ⁢ ( x i , y i ) ≥ lim sup i → ∞ ⁡ ρ E i ⁢ ( x i ) .

On the other hand, there is a point z i ∈ E i so that ρ E i ⁢ ( x i ) = d i ⁢ ( x i , z i ) . Suppose there is a sequence i ′ → ∞ such that lim inf i → ∞ ⁡ ρ E i ⁢ ( x i ) = lim i ′ → ∞ ⁡ ρ E i ′ ⁢ ( x i ′ ) . Then there is a subsequence i ′′ of i ′ such that z i ′′ → z ∈ E ∞ . Hence

(A.2) lim inf i → ∞ ⁡ ρ E i ⁢ ( x i ) = lim i ′ → ∞ ⁡ d i ′ ⁢ ( x i ′ , z i ′ )

= lim i ′′ → ∞ ⁡ d i ′′ ⁢ ( x i ′′ , z i ′′ )

= d ∞ ⁢ ( x , z ) ≥ ρ E ∞ ⁢ ( x ) .

Combining (A.1) and (A.2), we complete the proof. ∎

B Appendix II

Let R i ≥ 0 be a sequence with the property that R i → ∞ as i → ∞ . Let B R i ⁢ ( p i ) be a sequence of ( n + 1 ) -dimensional smooth geodesic balls with Ricci curvature ≥ - n ⁢ R i - 2 such that ( B R i ⁢ ( p i ) , p i ) converges to a metric cone ( 𝐂 , 𝐨 ) in the pointed Gromov–Hausdorff sense. Suppose lim inf i → ∞ ⁡ ℋ n + 1 ⁢ ( B 1 ⁢ ( p i ) ) > 0 , and for some integer 1 ≤ k ≤ n there is a k-dimensional compact metric space X such that 𝐂 = C ⁢ X × ℝ n - k , where CX is a metric cone with the vertex o of the cross section X.

For any point x ∈ X and r , t > 0 , let B r ⁢ ( t ⁢ x ) be the ball of radius r and centered at t ⁢ x ∈ C ⁢ X in CX, let ℬ r ⁢ ( t ⁢ x ) denote the metric ball in t ⁢ X = ∂ ⁡ B t ⁢ ( o ) with the radius r and centered at tx, and set

𝒞 r ⁢ ( x ) ≜ { t ⁢ ξ ∈ C ⁢ X : ξ ∈ ℬ r ⁢ ( x ) , | t - 1 | < r 2 } .

For | t - 1 | < r 2 and 0 < r ≤ 1 , with Cauchy inequality we get

t 2 + 1 - 2 ⁢ t ⁢ cos ⁡ ( t ⁢ r ) = ( t - 1 ) 2 + 4 ⁢ t ⁢ sin 2 ⁡ ( t ⁢ r 2 )

≥ ( t - 1 ) 2 + 4 π 2 ⁢ t 3 ⁢ r 2 ⁢ r 2 ⁢ ( ( t - 1 ) 2 + 2 ⁢ t 2 π 2 )

≥ r 2 9 ⁢ ( ( t - 1 ) 2 + t 2 + 8 ⁢ ( t - 1 ) 2 + t 2 8 )

≥ r 2 9 ⁢ ( ( t - 1 ) 2 + t 2 + 2 ⁢ t ⁢ ( t - 1 ) )

= r 2 9 .

Hence from (6.3) we have B r / 3 ⁢ ( x ) ⊂ 𝒞 r ⁢ ( x ) for all r ∈ ( 0 , 1 ] . Moreover, for | t - 1 | < r 2 and 0 < r ≤ 1

( t - 1 ) 2 + 4 ⁢ t ⁢ sin 2 ⁡ ( t ⁢ r 2 ) ≤ ( t - 1 ) 2 + t 3 ⁢ r 2 ≤ r 2 4 + ( 1 + r 2 ) 3 ⁢ r 2 ≤ 4 ⁢ r 2 ,

which implies 𝒞 r ⁢ ( x ) ⊂ B 2 ⁢ r ⁢ ( x ) for all r ∈ ( 0 , 1 ] . In all, we have

(B.1) B r / 3 ⁢ ( x ) ⊂ 𝒞 r ⁢ ( x ) ⊂ B 2 ⁢ r ⁢ ( x ) for all ⁢ r ∈ ( 0 , 1 ] .

For each r ∈ ( 0 , 1 ) , from the co-area formula (6.2)

(B.2) ℋ k + 1 ⁢ ( 𝒞 r ⁢ ( x ) ) = ∫ 0 ∞ ℋ k ⁢ ( ∂ ⁡ B t ⁢ ( o ) ∩ 𝒞 r ⁢ ( x ) ) ⁢ 𝑑 t

= ∫ 1 - r 2 1 + r 2 ℋ k ⁢ ( t ⁢ ℬ r ⁢ ( x ) ) ⁢ 𝑑 t

= ∫ 1 - r 2 1 + r 2 t k ⁢ ℋ k ⁢ ( ℬ r ⁢ ( x ) ) ⁢ 𝑑 t

= 1 k + 1 ⁢ ( ( 1 + r 2 ) k + 1 - ( 1 - r 2 ) k + 1 ) ⁢ ℋ k ⁢ ( ℬ r ⁢ ( x ) ) .

Combining (B.1), (B.2) and the Bishop–Gromov volume comparison, there is a constant c k ≥ 1 depending only on k such that

(B.3) ℋ k ⁢ ( ℬ R ⁢ ( x ) ) ≤ c k ⁢ R k r k ⁢ ℋ k ⁢ ( ℬ r ⁢ ( x ) ) for each ⁢ 0 < r ≤ R ≤ 1 .

Let ϕ be a Lipschitz function on X, and let λ be a Lipschitz function on [ 0 , ∞ ) . Put

f ⁢ ( t ⁢ ξ ) = λ ⁢ ( t ) ⁢ ϕ ⁢ ( ξ ) for each ⁢ t ⁢ ξ ∈ C ⁢ X ⁢ with ⁢ ξ ∈ X .

From (2.2) and (6.3), we have

Lip ⁡ ϕ ⁢ ( ξ ) = lim sup η → ξ , η ≠ ξ ⁡ | ϕ ⁢ ( ξ ) - ϕ ⁢ ( η ) | d X ⁢ ( ξ , η ) .

and

(B.4) ( Lip ⁡ f ) 2 ⁢ ( t ⁢ ξ ) = lim sup τ ⁢ η → t ⁢ ξ , τ ⁢ η ≠ t ⁢ ξ ⁡ | f ⁢ ( t ⁢ ξ ) - f ⁢ ( τ ⁢ η ) | 2 d C ⁢ X ⁢ ( t ⁢ ξ , τ ⁢ η ) 2

= lim sup τ → t , η → ξ , τ ⁢ η ≠ t ⁢ ξ ⁡ | λ ⁢ ( t ) ⁢ ϕ ⁢ ( ξ ) - λ ⁢ ( τ ) ⁢ ϕ ⁢ ( η ) | 2 t 2 + τ 2 - 2 ⁢ t ⁢ τ ⁢ cos ⁡ d X ⁢ ( ξ , η ) .

Note that

| λ ⁢ ( t ) ⁢ ϕ ⁢ ( ξ ) - λ ⁢ ( τ ) ⁢ ϕ ⁢ ( η ) | 2 = | λ ⁢ ( t ) ⁢ ϕ ⁢ ( ξ ) - λ ⁢ ( τ ) ⁢ ϕ ⁢ ( ξ ) | 2 + | λ ⁢ ( τ ) ⁢ ϕ ⁢ ( ξ ) - λ ⁢ ( τ ) ⁢ ϕ ⁢ ( η ) | 2

+ 2 ⁢ ( λ ⁢ ( t ) ⁢ ϕ ⁢ ( ξ ) - λ ⁢ ( τ ) ⁢ ϕ ⁢ ( ξ ) ) ⁢ ( λ ⁢ ( τ ) ⁢ ϕ ⁢ ( ξ ) - λ ⁢ ( τ ) ⁢ ϕ ⁢ ( η ) ) .

Let

τ - t = sin ⁡ θ ⁢ ( τ - t ) 2 + t ⁢ τ ⁢ d X ⁢ ( ξ , η ) 2 ,

and

t ⁢ τ ⁢ d X ⁢ ( ξ , η ) = cos ⁡ θ ⁢ ( τ - t ) 2 + t ⁢ τ ⁢ d X ⁢ ( ξ , η ) 2 for θ ∈ [ 0 , 2 ⁢ π ) .

Note that

t 2 + τ 2 - 2 ⁢ t ⁢ τ ⁢ cos ⁡ d X ⁢ ( ξ , η ) = ( t - τ ) 2 + t ⁢ τ ⁢ d X ⁢ ( ξ , η ) 2 + t ⁢ τ ⁢ O ⁢ ( d X ⁢ ( ξ , η ) 3 ) .

For the fixed θ, we have

(B.5) | λ ⁢ ( t ) - λ ⁢ ( τ ) | 2 ⁢ | ϕ | 2 ⁢ ( ξ ) + | ϕ ⁢ ( ξ ) - ϕ ⁢ ( η ) | 2 ⁢ λ 2 ⁢ ( τ ) t 2 + τ 2 - 2 ⁢ t ⁢ τ ⁢ cos ⁡ d X ⁢ ( ξ , η )

+ 2 ⁢ ( λ ⁢ ( t ) - λ ⁢ ( τ ) ) ⁢ ( ϕ ⁢ ( ξ ) - ϕ ⁢ ( η ) ) ⁢ ϕ ⁢ ( ξ ) ⁢ λ ⁢ ( τ ) t 2 + τ 2 - 2 ⁢ t ⁢ τ ⁢ cos ⁡ d X ⁢ ( ξ , η )

→ ( λ ′ ⁢ ( t ) ) 2 ⁢ ϕ 2 ⁢ ( ξ ) ⁢ sin 2 ⁡ θ + λ 2 ⁢ ( t ) t 2 ⁢ cos 2 ⁡ θ ⁢ ( Lip ⁡ ϕ ) 2 ⁢ ( ξ )

+ 2 ⁢ λ ⁢ ( t ) ⁢ λ ′ ⁢ ( t ) t ⁢ ϕ ⁢ ( ξ ) ⁢ sin ⁡ θ ⁢ cos ⁡ θ ⁢ Lip ⁡ ϕ ⁢ ( ξ )

= ( λ ′ ⁢ ( t ) ⁢ ϕ ⁢ ( ξ ) ⁢ sin ⁡ θ + λ ⁢ ( t ) t ⁢ cos ⁡ θ ⁢ Lip ⁡ ϕ ⁢ ( ξ ) ) 2

as ( τ - t ) 2 + t ⁢ τ ⁢ d X ⁢ ( ξ , η ) 2 → 0 . From (B.4)–(B.5), for every t ⁢ ξ ∈ V we have

(B.6) ( Lip ⁡ f ) 2 ⁢ ( t ⁢ ξ ) = sup θ ∈ [ 0 , 2 ⁢ π ) ⁡ ( λ ′ ⁢ ( t ) ⁢ ϕ ⁢ ( ξ ) ⁢ sin ⁡ θ + λ ⁢ ( t ) t ⁢ cos ⁡ θ ⁢ Lip ⁡ ϕ ⁢ ( ξ ) ) 2

= λ 2 ⁢ ( t ) t 2 ⁢ ( Lip ⁡ ϕ ) 2 ⁢ ( ξ ) + ( λ ′ ⁢ ( t ) ) 2 ⁢ ϕ 2 ⁢ ( ξ )

= λ 2 ⁢ ( t ) t 2 ⁢ | d ⁢ ϕ | 2 ⁢ ( ξ ) + ( λ ′ ⁢ ( t ) ) 2 ⁢ ϕ 2 ⁢ ( ξ ) ,

where d ⁢ ϕ is the differential of ϕ on X defined below (6.5).

From the Poincaré inequality (2.5) and the proof of [21, Theorem 1], it follows that

(B.7) ∫ 𝒞 r ⁢ ( x ) | f - ⨍ 𝒞 r ⁢ ( x ) f | q ≤ c k ⁢ r q ⁢ ∫ 𝒞 r ⁢ ( x ) | d ⁢ f | q

for any x ∈ X , r ∈ ( 0 , 1 ] , q ≥ 1 , where c k is a general constant depending only on k. From the co-area formula (6.2), if f ⁢ ( t ⁢ ξ ) = ϕ ⁢ ( ξ ) for each t ⁢ ξ ∈ t ⁢ X with ξ ∈ X , then we have

(B.8) ⨍ 𝒞 r ⁢ ( x ) f = 1 ℋ k + 1 ⁢ ( 𝒞 r ⁢ ( x ) ) ⁢ ∫ 1 - r 2 1 + r 2 ( ∫ t ⁢ ℬ r ⁢ ( x ) f ) ⁢ 𝑑 t

= 1 ℋ k + 1 ⁢ ( 𝒞 r ⁢ ( x ) ) ⁢ ∫ 1 - r 2 1 + r 2 ( t k ⁢ ∫ ℬ r ⁢ ( x ) ϕ ) ⁢ 𝑑 t

= 1 ℋ k ⁢ ( ℬ r ⁢ ( x ) ) ⁢ ∫ ℬ r ⁢ ( x ) ϕ

= ⨍ ℬ r ⁢ ( x ) ϕ .

With (B.6), substituting (B.8) into (B.7) gives

(B.9) ∫ ℬ r ⁢ ( x ) | ϕ - ⨍ ℬ r ⁢ ( x ) ϕ | q ≤ c k ⁢ r q ⁢ ∫ ℬ r ⁢ ( x ) | d ⁢ ϕ | q

for any q ≥ 1 .

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Received: 2021-09-07

Revised: 2022-06-20

Published Online: 2022-08-27

Published in Print: 2022-10-01

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