Absence of bubbling phenomena for non-convex anisotropic nearly umbilical and quasi-Einstein hypersurfaces

Antonio De Rosa; Stefano Gioffrè

doi:10.1515/crelle-2021-0038

Article

Absence of bubbling phenomena for non-convex anisotropic nearly umbilical and quasi-Einstein hypersurfaces

Antonio De Rosa and Stefano Gioffrè

Published/Copyright: August 17, 2021

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From the journal Journal für die reine und angewandte Mathematik (Crelles Journal) Volume 2021 Issue 780

Abstract

We prove that, for every closed (not necessarily convex) hypersurface Σ in ℝn+1 and every p>n, the Lp-norm of the trace-free part of the anisotropic second fundamental form controls from above the W2,p-closeness of Σ to the Wulff shape. In the isotropic setting, we provide a simpler proof. This result is sharp since in the subcritical regime p≤n, the lack of convexity assumptions may lead in general to bubbling phenomena. Moreover, we obtain a stability theorem for quasi-Einstein (not necessarily convex) hypersurfaces and we improve the quantitative estimates in the convex setting.

Funding source: National Science Foundation

Award Identifier / Grant number: DMS-1906451

Award Identifier / Grant number: DMS-2112311

Funding statement: Antonio De Rosa has been partially supported by the NSF DMS Grant No. 1906451 and the NSF DMS Grant No. 2112311.

A Appendix

Proof of Proposition 2.5 and Lemma 5.9

We recall the equations we are going to study:

(A.1)∇⁡HF=div⁡SF,

(A.2)∇⁡R=2⁢nn-2⁢div⁡Ric̊.

These equations present clear similarities, since they are all variations of the equation

∇⁡𝔲=div⁡𝔣

in a closed manifold. In both cases, an immediate but naive covering argument may show the existence of a number λ such that

(A.3)∥𝔲-λ∥Lp⁢(M)≤C⁢(M)⁢∥𝔣∥Lp⁢(M).

The problem in such an argument is that we do not only have to obtain an estimate, but also to keep an eye on the constant C, which in our case has to depend only on general parameters. We are going to show an improved estimate which is basically (A.3), but gives a better control on the bounding constant. The technique we are going to use has been used and developed in [31], where the author deals with the isotropic version of equation (A.1). Considered the massive use we are making of this type of estimates and ideas throughout the paper, we have decided to report the proof. We split it into the following steps:

We show by direct computation in graph parametrization how the two equations can be written as particular cases of a more general lemma.
We obtain a local estimate of our desired inequality, with the bounding constant depending on determined parameters.
We show how to make the local estimate global without losing the information on the bounding constant.

Step 1: Unifying the equations

We recall that from Lemma 2.4, if M=Graph⁡(u,𝔹n) is a smooth graph, then the following formulas hold:

(A.4)gi⁢j=δi⁢j+∂i⁡u⁢∂j⁡u,

(A.5)gi⁢j=δi⁢j-∂i⁡u⁢∂j⁡u1+|∂⁡u|2,

(A.6)d⁢Vg=1+|∂⁡u|2⁢d⁢x,

(A.7)Γi⁢kkg=vk⁢hi⁢j,where ⁢v=∂⁡u1+|∂⁡u|2.

We compute the divergence term of equations (A.1), (A.2) in graph parametrization, and notice how this does not depend on Christoffel symbols.

Equation (A.1)

Firstly, we need to prove that equation (A.1) holds. This follows from the computation below. For notation simplicity, we drop the subscript F from (SF)ji=Sji and (AF)ji=Aji:

divgSk=∇iSki=∇i(Api|νhkp)=∇i(Api|ν)hkp+Api|ν∇ihpk

=DqApi|νhiqhkp+Api|ν∇ihkp=DpAqi|νhiqhkp+Api|ν∇phi⁢k

=∇k(Api|νhip)=∇kHF.

Now we notice how also the last divergence term can be written as a flat divergence. We find

divg⁡Sk=∇i⁡Ski=∂i⁡Ski+Γi⁢pi⁢Skp-Γi⁢kp⁢Spi=∂i⁡Ski+vi⁢hi⁢p⁢Skp-vp⁢hi⁢k⁢Spi

=∂i⁡Ski+vi⁢hi⁢p⁢Aqp⁢hkq-vp⁢hi⁢k⁢Apq⁢hqi=∂i⁡Ski+vi⁢Ap⁢q⁢(hi⁢p⁢hq⁢k-hp⁢k⁢hq⁢i)

=∂i⁡Ski+vi⁢hi⁢p⁢hq⁢k⁢(Ap⁢q-Aq⁢p)=∂i⁡Ski.

Equation (A.2)

We compute the divergence term in equation (A.2). Firstly we compute the divergence of the Ricci tensor:

∇i⁡Ricki=∂i⁡Ricki+Γi⁢pi⁢Rickp-Γi⁢kp⁢Ricpi=∂i⁡Ricki+vi⁢hi⁢p⁢Rickp-vp⁢hi⁢k⁢Ricpi

=∂i⁡Ricki+vi⁢hi⁢p⁢(H⁢hkp-hqp⁢hkq)-vp⁢hi⁢k⁢(H⁢hpi-hqi⁢hpq)

=∂i⁡Ricki+H⁢(vi⁢hi⁢p⁢hkp-vp⁢hi⁢k⁢hpi)⏟=vi⁢hi⁢p⁢hkp-vi⁢hp⁢k⁢hip⁣= 0+(vp⁢hi⁢k⁢hqi⁢hpq-vi⁢hi⁢p⁢hqp⁢hkq)⏟=vp⁢(hi⁢k⁢hqi⁢hpq-hp⁢q⁢hiq⁢hki)⁣= 0=∂i⁡Ricki.

Now we write Ric̊ji=Ricji-Rn⁢δji, and notice that δ is a symmetric tensor. The computation of it is identical to the previous one, and we are done.

Lastly we write in graph chart ∇⁡f=∂⁡f, since at the first order the Levi-Civita coincides with the classical derivations. These computations show how we have reduced the two problems to the following:

Lemma A.1.

Let M⊂Rn+1 be a closed hypersurface. Assume Σ has fixed volume V and satisfies the assumptions of Lemma 2.2, i.e. admits two numbers L and R such that around every q∈Σ we can find a chart defined on the ball BRn, which is the graph of a smooth, L-Lipschitz function uq. Assume there are u:M→R, f∈Γ⁢(T*⁢M⊗T*⁢M) that satisfy a differential relation which in every graph parametrization at every point admits the form

∂k⁡𝔲=∂i⁡𝔣ki in ⁢𝔹Rn.

Then there exists a λ∈R such that the following estimate holds:

∥𝔲-λ∥Lp⁢(M)≤C⁢(n,p,V,R,L)⁢∥𝔣∥Lp⁢(M).

Notice that in both cases we are studying, the manifold M satisfies the assumptions of Lemma 2.2, as explained in Remark 2.3. These will be crucial in the proof. In the next step, we prove Lemma A.1.

Step 2: Obtaining local estimates: Proof of Lemma A.1

We begin by working in the graph, and observe that 𝔲 has to satisfy the equation

Δδ⁢𝔲=∂k⁡∂i⁡𝔣ki,

where Δδ is the flat laplacian. The estimate for this equation follows by applying the classic Calderon–Zygmund theorem (see [31, Proposition 1.11] for a detailed proof in this particular case). We find a constant C0:=C⁢(n,p) and a number λ such that

(A.8)∥𝔲-λ∥Lp⁢(𝔹R/2n)≤C0⁢∥𝔣∥Lp⁢(𝔹Rn).

Estimate (A.8) is almost what we want. It is indeed a local estimate, but it concerns all Euclidean quantities. We show how to swap Euclidean measures with manifold metrics, and how to substitute Euclidean balls with geodesic balls.

The first follows easily from equation (A.6) and Remark 2.3. Since Lip⁡(u)≤L, we obtain indeed

d⁢x≤1+|∂⁡u|2⁢d⁢x=d⁢Vg≤1+L2⁢d⁢x.

Thus the measures are equivalent, and the control constants depend only on L. The same constant L controls the switch from the Euclidean metric δ to the metric g.

Now Lemma 2.2 allows us to pass from Euclidean to geodesic balls and grants our privileged covering of balls. In particular, we obtain the existence of a radius R such that

minλ∈ℝ∥𝔲-λ∥Lp⁢(Brg⁢(q))≤C(n,p,V,L,R)∥𝔣∥Lp⁢(M) for every 0<r≤R.

Step 3: Making the estimate global

To make the estimate global, we follow the technique used in [31, pp. 6–7] and prove the following lemma.

Lemma A.2.

Let M be a closed manifold, with fixed volume Voln⁡(M)=V. Suppose u∈C∞⁢(M) has the following property. There is a radius ρ such that for every x∈M the following local estimate is satisfied:

(A.9)∥𝔲-λ⁢(x)∥Lp⁢(Brg⁢(x))≤β,

where λ⁢(x) is a real number depending on x, r≤2⁢ρ and β does not depend on x. Then there exists λ∈R such that ∥u-λ∥Lp⁢(M)≤C⁢(n,ρ,V)⁢β.

Proof.

We choose a finite covering of balls {(Bjg,λj)}j=1N which satisfies the following properties. Every ball Bjg has radius 2⁢ρ, estimate (A.9) holds with λj, and for every j, k there exists a ball of radius ρ contained in Bjg∩Bkg. Therefore, given two balls Bjg and Bkg whose intersection is non-empty, we have

|λj-λk|=1Voln(Bjg∩Bkg)1p⁢∥λj-λk∥Lp⁢(Bjg∩Bkg)

=1Voln(Bjg∩Bkg)1p⁢∥λj-𝔲+𝔲-λk∥Lp⁢(Bjg∩Bkg)

≤1Voln(Bjg∩Bkg)1p⁢(∥𝔲-λk∥Lp⁢(Bjg∩Bkg)+∥𝔲-λk∥Lp⁢(Bjg∩Bkg))

≤2⁢βVoln(Bjg∩Bkg)1p.

Using the properties of the covering, we obtain

|λj-λk|≤2Voln(Bρg)-1pβ.

Define λmin:=min1≤j≤n⁡λj and λmax:=max1≤j≤n⁡λj. Consider a path joining the ball in the cover with λmin to the one with λmax. Since this path can cross at most N different balls, we obtain

|λmax-λmin|≤2NVoln(Bρg)-1pβ=C(n,p,ρ)β.

For every λmin≤λ≤λmax we have

∥𝔲-λ∥Lσp⁢(𝕊n)≤∑j=1N∥𝔲-λ∥Lσp⁢(Bjg)≤∑j=1N∥𝔲-λj+λj-λ∥Lσp⁢(Bjg)

≤∑j=1N∥𝔲-λj∥Lσp⁢(Bjg)+|λj-λ|Voln(Bjg)-1p

≤∑j=1N∥𝔲-λj∥Lσp⁢(Bjg)+|λmax-λmin|Voln(Bjg)-1p

≤C2⁢(n,p,ρ)⁢β

and the proof of Lemma A.2 is completed. ∎

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Received: 2018-04-13

Revised: 2021-05-30

Published Online: 2021-08-17

Published in Print: 2021-11-01

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https://doi.org/10.1515/crelle-2021-0038