Can One Perturb the Equatorial Zone on a Sphere with Larger Mean Curvature Baichuan HuXiang MaShengyang Wang

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Can One Perturb the Equatorial Zone on a

Sphere with Larger Mean Curvature?

Baichuan Hu∗Xiang Ma†Shengyang Wang‡

October 5, 2022

Abstract

We consider the mean curvature rigidity problem of an equatorial zone on a

sphere which is symmetric about the equator with width 2w. There are two diﬀerent

notions on rigidity, i.e. strong rigidity and local rigidity. We prove that for each

kind of these rigidity problems, there exists a critical value such that the rigidity

holds true if, and only if, the zone width is smaller than that value. For the rigidity

part, we used the tangency principle and a speciﬁc lemma (the trap-slice lemma we

established before). For the non-rigidity part, we construct the nontrivial pertur-

bations by a gluing procedure called the round-corner lemma using the Delaunay

surfaces.

Keywords: spheres, mean curvature, rigidity theorem, inﬁnitesimal deforma-

tion, Delaunay surfaces, tangency principle, gluing construction.

MSC(2010): 53C24, 53C42; see also 52C25.

1 Introduction

The central theme in this paper is about the so-called mean curvature rigidity phe-

nomenon. The ﬁrst result along this direction is by Gromov [4], who pointed out that

a hyperplane Min a Euclidean space Rn+1 cannot be perturbed on a compact set S

so that the perturbed hypersurface Σ has mean curvature HΣ≥0 unless HΣ≡0 and

Σ = Midentically. Very soon Souam [7] gave a simple proof of this fact using the

Tangency Principle and established rigidity results for horospheres, hyperspheres,

and hyperplanes in the hyperbolic space Hn+1. For other types of rigidity theorems

on spheres and hemispheres, we just mention the famous Min-Oo’s conjecture [6]

and a series of beautiful work [1, 5].

In a previous work [2], we found that similar mean curvature rigidity result holds

for compact CMC hypersurfaces like spheres, with the restriction that the perturbed

∗School of Mathematical Sciences, Peking University, Beijing 100871, People’s Republic of China.

1900010607@pku.edu.cn.

†LMAM, School of Mathematical Sciences, Peking University, Beijing 100871, P.R. China.

maxiang@math.pku.edu.cn, Fax:+86-010-62751801. Corresponding author. Supported by NSFC

grant 11831005.

‡School of Mathematical Sciences, Peking University, Beijing 100871, People’s Republic of China.

1900010752@pku.edu.cn. Supported by the Undergraduate Student Research Study program of

Peking University for the year 2021-2022.

arXiv:2210.00994v2 [math.DG] 4 Oct 2022

part is no more than a hemisphere. In other words, we considered perturbation of

a spherical cap whose boundary is ﬁxed up to C2. Here we turn to perturbations

on a doubly connected domain of a sphere, and our aim is to ﬁnd out when such

rigidity theorem still holds true. This is done by detailed analysis and comparison

with the Delaunay CMC surfaces and gluing constructions in the 3-dim space. Yet

this should be not diﬃcult to generalize to any n-dimensional space.

Generally, suppose Nn+1 is a Riemannian manifold, and Mnis an embedded

hypersurface in it. The second fundamental form of Mnand the mean curvature

H(x), x ∈Mnare deﬁned as usual with respect to a given normal unit vector ﬁeld

n∈Γ(T⊥M). S⊂Mnis a precompact open domain on Mn.

Deﬁnition 1.1. For k≥2, a Ck-perturbation of S⊂Mnrefers to another Ck

embedding Σ : Mn→Nn+1 with Σ = id in Nn+1\S. When k=∞, we say the

perturbation is smooth.

When there is no confusion, we will also use Σ to represent Σ(Mn). And we

will only talk about the smooth perturbation (which is easy to generalize to other

Ck-perturbations.

There also exists a unit normal vector ﬁeld n

n∈Γ(T⊥Σ) with n

n=e

nin Mn\S,

which gives the mean curvature of Σ, deﬁned as H(x),H(Σ(x)), x ∈M. Given a

constant α∈R, we say H(Σ) ≥αiﬀ ∀x∈S,H(x)≥α(similar for H(Σ) ≤α).

In convex geometry and isometric deformation problems, usually we talk about

two kinds of notions about deformations and rigidity. One is the so-called inﬁnites-

imal deformations which exist in an arbitrarily small neighborhood of the original

hypersurface; one can imagine that it comes from a one-parameter deformation pro-

cess. The other is large-scale perturbations which have to go far away. Here we need

also to distinguish between these two kinds of rigidity.

Deﬁnition 1.2. Given an open domain Θ⊂Nn+1 satisfying S⊂Θ. We say that

Shas H+(or H−) rigidity in Θif for any perturbation Σwith Σ(S)⊂Θ, the two

statements below are equivalent:

(1)H(x)≥e

H(x)(or H(x)≤e

H(x)), ∀x∈S

(2)Σ = id in S

We say Shas local H+(or H−) rigidity, if ∃Θ⊂Nn+1 satisfying S⊂Θ, and S

has H+(or H−) rigidity in Θ.

When Θ = Nn+1, we simply say Shas (strong) H+(or H−) rigidity.

We can simply say equivalently that Sis (local/strong) H+/H−rigid.

Remark 1.3.It is obvious that each of these four kinds of rigidity has monotonicity

property with respect to the domain S, i.e. for two precompact open domain S1, S2⊂

Mnwith S1⊃S2, we have:

(1) If S1has H+or H−rigidity, then this is also true for S2.

(2) If S1has local H+or H−rigidity, then S2also has this rigidity property.

We can ﬁnd that the strong rigidity considers a large-scale perturbation of S.

However, for the local Hrigidity, we only need to consider a local deformation of S,

since we only have to prove the existence of some Θ which can be arbitrarily small

enough. Notice that the strong rigidity implies the local rigidity.

As a demonstration of these rigidity notions, we review and summarize our pre-

vious results as below:

Theorem 1.4. [2] A spherical cap S⊂Snis H+rigid if and only if it is part of a

hemisphere. When Sis contained in a hemisphere, it is local H−rigid in a certain

dumb-bell shaped domain Θ.

In this follow-up work, we will mainly discuss those rigidity properties on doubly

connected domains symmetric about the equator on the unit sphere S2⊂R3.

Convention:

(1) S2⊂R3is the unit sphere with radius 1, deﬁned by

S2={(x1, x2, x3)∈R3|x2

1+x2

2+x2

3= 1}

S2divides R3into two connected components, and D3={(x1, x2, x3)∈R3|x2

2+x2

3≤1}is one of them. Also, we deﬁne Pithe coordinate hyperplane in R3,

with

Pi={(x1, x2, x3)∈R3|xi= 0}

In this passage we will usually consider some curves in P2, and we will use (x3, x1)

as the coordinate in P2.

(2) Suppose a∈(0,1) and Sais an annulus around the equator with width 2 arccos a,

i.e.

Sa={(x1, x2, x3)∈S2|x2

1+x2

2> a2}

And we use Σato refer to a perturbation of Sa.

(3) Deﬁne e

n(x1, x2, x3) = (−x1,−x2,−x3) as the unit inward normal vector ﬁeld on

S2. In this passage, if there is no other explanation, we will default that the unit

normal vector ﬁelds of Σawe talk about are all consistent with e

nat S2\Sa, which

is inward. And the mean curvature of Saand Σaalso come from it.

The main results in this paper are stated as below.

Theorem 1.5. For a∈(0,1), we have:

(1) Sais H+rigid iﬀ a≥√3/2.

(2) ∀a∈(0,1),Sais not H−rigid.

Theorem 1.6. There exists a constant a0≈0.5524 such that for a∈(0,1):

(1) Sahas local H+rigidity iﬀ a≥a0.

(2) Sahas local H−rigidity iﬀ a>a0.

This paper is organized as follows. In Section 2, we review the trap-slice lemma

in [2] and the Tangency Principle (see also [3] and [7]). Together with suitably

chosen trap and comparison surface we establish the strong H+rigidity in 1.5.

Then in Section 3 we establish local H+(H−) rigidity by detailed analysis of the

related ODE. The round-corner lemma is established in Section 4, which is applied

to a gluing construction using Delaunay surfaces to ﬁnd non-trivial deformations

increasing or decreasing the mean curvature, hence establish the only if part of the

above two theorems. This ﬁnishes the proof to the main theorems. Some technical

details involving elliptical integrals are left to the appendix.

Acknowledgement.

We would like to thank Yichen Cheng, who participated in our seminar and

suggested the construction of trap (comparison surfaces) using two spheres in Step 3

of the proof of Theorem 2.6. We also thank other participants of our seminar: Zheng

Yang, Yi Sha, Tianming Zhu, Shunkai Zhang, for their interests and many helpful

discussions. This research project is partially supported by NSFC grant 11831005

and the Undergraduate Student Research Study program of Peking University for

the year 2021-2022.

2 The trap-slice lemma and the strong rigidity

The Tangency Principle [3, 7] is an important instrument for mean curvature rigidity

problems.

Theorem 2.1. [The Tangency Principle] Let Mn

1and Mn

2be hypersurfaces of Nn+1

that are tangent at pand let η0be a unit normal vector of M1at p. Denote by Hi

r(x)

the r-mean curvature at x∈Wof Mi, i = 1,2,respectively. Suppose that with respect

to this given η0, we have:

1. Locally M1≥M2, i.e., M1remains above M2in a neighborhood of p;

2. H2

r(x)≥H1

r(x)in a neighborhood of zero for some r, 1≤r≤n; if r≥2,

assume also that M2is r-convex at p.

Then M1and M2coincide in a neighborhood of p.

Corollary 2.2. For a∈(0,1), suppose Σais a perturbation of Sa, then:

(1)If H(Σa)≥1and Σa6=id, then Σa∩˚

D36=∅.

(2)If H(Σa)≤1and Σa6=id, then Σa∩(D3)c6=∅.

Proof. We only prove (1), and the proof of (2) is similar.

Consider the collection

ST={x∈S2: Σ(x)∈S2}

It is apparent that STis closed in S2. If Σa∩˚

D3=∅, then for all x∈ST, Σ will be

tangent with S2at x. Hence, from Tangency Principle, Σawill coincide with S2in a

neighborhood of x, showing that STis also open in S2. Therefore, we have Σa=id,

which is a contradiction.

The trap-slice lemma is an encapsulated version of the Tangency Principle, which

was ﬁrst established in our previous work [2].

Theorem 2.3. [The trap-slice lemma]

Let the trap Ω ⊂Rnbe a domain enclosed by two connected hypersurfaces B0, B1

sharing a boundary A=B0∩B1and ∂Ω = B0∪B1.

The slice is a foliation of Ωby a one-parameter family of hypersurfaces {Ft} ⊂ Ω

(with or without boundaries). When ∂Ft6=∅, we assume ∂Ft⊂B1. Each Ftdivides

Ωinto two sub-domains, one having B0on its boundary, and Ωtis the other one

away from B0.

Fix a real constant α∈R. With respect to the outward normal of ∂Ωt⊃Ft,

suppose that the mean curvature function of Ftalways satisﬁes H(Ft)≥α.

Given the trap and the slice as above, there does NOT exist any hypersurface Σ∗

with boundary ∂Σ∗satisfying all of the following conditions:

1. Σ∗, the interior of the compact hypersurface Σ∗= Σ∗∪∂Σ∗, is embedded in

Ωwith boundary ∂Σ∗⊂B0⊂∂Ω. In particular, Σ∗divides Ωinto two sub-

domains; sub-domain Ω∗is the one of them that having B1on its boundary.

We orient Σ∗by the outward normal of ∂Ω∗.

2. The boundary ∂Σ∗has a neighborhood Utin Σ∗not contained in Ωtfor any t.

3. Given the orientation of Σ∗, the mean curvature function H(Σ∗)≤α.

Corollary 2.4. [2] Assumptions on the trap Ω⊂Rn, ∂Ω = B0∪B1and the slice

{Ft}are as in the trap-slice lemma (Theorem 2.3). Moreover, we suppose that:

1. B0is also one leave of the foliation {Ft}(we may suppose B0=F0is an open

subset of ∂Ω);

2. For any other t6= 0, either ∂B0∩∂Ft=∅, or B0intersects with Ftat their

boundaries transversally.

Then B0admits no non-trivial perturbation Σ0(with ﬁxed boundary up to C2and

the same orientation on ∂Σ = ∂B0) such that H(Σ0)≤α, unless two hypersurfaces

Σ0and B1intersect at their interior points.

Remark 2.5.The trap-slice lemma and Corollary 2.4 above are still true when the

assumptions are changed as below: Σ∗and Ftare oriented by the inward normal

vectors with respect to Ω∗and Ωt, respectively, and the inequality on His reversed

H(Ft)≤α≤H(Σ∗).

Now we consider the H+rigidity of Sa:

Theorem 2.6. Suppose a∈[√3/2,1), then Sahas H+rigidity.

Proof. From Remark 1.3, we only need to consider a=√3/2. Assuming there is a

perturbation Σa6=id of Sasuch that H(Σa)≥1, we will try to ﬁnd contradiction.

Step 1: Denote B1=S2∩{x3≤ −1/2}, and B0the symmetrical surface of B1with

respect to x3=−1/2. They enclose an open domain Ω ⊂D3, which is our ”trap”.

Then we translate B0by the vector v

vt= (0,0,−t),0≤t < 1, denoted the translated

surface as Bt. Denote Ft=Bt∩Ω, which is our slice. The normal of B0and Ftare

all inward about Ω.

We assert that Σa∩Ω = ∅, because if not, we can choose a connected component

of Σa∩Ω and denote it Σ∗. Might as well, assume there exist p1∈Σ∗and p2∈B1

such that the open line segment p1p2∩Σa=∅(this is reasonable for Σais an

embedded map of S2). Then the normal on Σ∗will suit the condition 1 in trap-slice

lemma and Remark 2.5.

Also, it is apparent that the boundary ∂Σ∗⊂B0suits condition 2 in trap-slice

lemma, since ∂Ft⊂B1and Σais an embedded map. Hence, we get the contradiction

by Remark 2.5.

Similarly, denote ˜

B1=S2∩ {x3≥1/2}, and ˜

B0the symmetrical surface of ˜

with respect to x3= 1/2. They enclose an open domain ˜

Ω⊂D3, which is symmet-

rical with Ω with respect to P3. We can also get Σa∩˜

Ω = ∅.

Step 2: Since we have had

Σa∩(Ω ∪˜

Ω) = ∅,(1)

we will then further consider where Σais.

From Corollary 2.2, we know Σa∩˚

D36=∅. Hence, we can select a connected

component of Σa∩D3, denoted as Σ∗.

We can prove that

Σ∗∩ {x2

1+x2

2<1

4} 6=∅.

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CanOnePerturbtheEquatorialZoneonaSpherewithLargerMeanCurvature?BaichuanHu*XiangMaShengyangWangOctober5,2022AbstractWeconsiderthemeancurvaturerigidityproblemofanequatorialzoneonaspherewhichissymmetricabouttheequatorwithwidth2w.Therearetwodierentnotionsonrigidity,i.e.strongrigidityandlocalrigidity....

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Can One Perturb the Equatorial Zone on a Sphere with Larger Mean Curvature Baichuan HuXiang MaShengyang Wang

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