Entropy bounds, compactness and finiteness theorems for embedded self-shrinkers with rotational symmetry

John Man Shun Ma; Ali Muhammad; Niels Martin Møller

doi:10.1515/crelle-2022-0073

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Entropy bounds, compactness and finiteness theorems for embedded self-shrinkers with rotational symmetry

John Man Shun Ma , Ali Muhammad und Niels Martin Møller

Veröffentlicht/Copyright: 11. November 2022

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Aus der Zeitschrift Journal für die reine und angewandte Mathematik (Crelles Journal) Band 2022 Heft 793

Abstract

In this work, we study the space of complete embedded rotationally symmetric self-shrinking hypersurfaces in ℝ n + 1 . First, using comparison geometry in the context of metric geometry, we derive explicit upper bounds for the entropy of all such self-shrinkers. Second, as an application we prove a smooth compactness theorem on the space of all such shrinkers. We also prove that there are only finitely many such self-shrinkers with an extra reflection symmetry.

Funding source: Independent Research Fund Denmark

Award Identifier / Grant number: DFF Sapere Aude 7027-00110B

Funding source: Danish National Research Foundation

Award Identifier / Grant number: CPH-GEOTOP-DNRF151

Funding source: Carlsberg Foundation

Award Identifier / Grant number: CF21-0680

Funding statement: The authors were partially supported by DFF Sapere Aude 7027-00110B, by CPH-GEOTOP-DNRF151 and by CF21-0680 from respectively the Independent Research Fund Denmark, the Danish National Research Foundation and the Carlsberg Foundation.

A The sequence ( E n )

In this appendix, we show the following lemma.

Lemma A.1.

The sequence ( E n ) defined in (3.1) satisfies 2 < E n ≤ E 2 and

(A.1) lim n → ∞ ⁡ E n = 4 ⁢ π 3 .

Proof.

It is proved in [38, A.4 Lemma] that the entropy of the n-sphere λ ⁢ ( 𝕊 n ) satisfies

(A.2) λ ⁢ ( 𝕊 n ) = ( n 2 ⁢ π ⁢ e ) n 2 ⁢ ω n

and the sequence ( λ ⁢ ( 𝕊 n ) ) is strictly decreasing. Also,

(A.3) lim n → ∞ ⁡ λ ⁢ ( 𝕊 n ) = 2 .

From (3.1) and (A.2) we obtain

(A.4) E n = 2 ⁢ π 3 ⁢ 1 + x n 1 + 2 ⁢ x n 3 ⁢ ( 1 e ⁢ ( 1 + x n ) 1 x n ) a n 4 ⁢ λ ⁢ ( 𝕊 n - 1 ) ,

where

a n = y n - 2 ⁢ ( n - 1 ) , x n = a n 2 ⁢ ( n - 1 ) .

Direct calculations give

(A.5) 1 2 < a n < 2 3 , 0 < x n < 1

and

(A.6) lim n → ∞ ⁡ a n = 2 3 , lim n → ∞ ⁡ x n = 0 .

Using (A.4), (A.3) and (A.6), one obtains (A.1).

Next we show 2 < E n ≤ E 2 . By the Taylor expansion of ln ⁡ ( 1 + x ) , we have

x + x 3 3 > ln ⁡ ( 1 + x ) > x - x 2 2 for all x ∈ ( 0 , 1 ) .

Thus

e x 2 3 > 1 e ⁢ ( 1 + x ) 1 x > e - x 2 for all ⁢ x ∈ ( 0 , 1 ) .

Together with λ ⁢ ( 𝕊 n - 1 ) > 2 , (A.5) and (A.4),

(A.7) 2 ⁢ π ⁢ ( 3 + ( n - 1 ) - 1 ) 3 ⁢ e 1 162 ⁢ ( n - 1 ) 2 ⁢ λ ⁢ ( 𝕊 n - 1 ) > E n > 4 ⁢ π 3 ⁢ e - 1 36 ⁢ ( n - 1 ) .

Since λ ⁢ ( 𝕊 n - 1 ) is decreasing, the upper bound in (A.7) is strictly decreasing in n. Also, the lower bound in (A.7) is strictly increasing in n. Plugging in n = 4 in the upper and lower bound of (A.7) gives

2.21823 ∼ 10 ⁢ π e 1093 / 729 > E n > 4 ⁢ π 3 ⁢ e - 1 108 ∼ 2.02780 for all ⁢ n ≥ 4 .

The inequality implies 2 < E n < E 2 for all n ≥ 4 . The cases n = 2 and n = 3 can be checked directly. ∎

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Received: 2022-05-25

Revised: 2022-09-02

Published Online: 2022-11-11

Published in Print: 2022-12-01

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