A Third Derivative Estimate for Monge-Ampere Equations with Conic Singularities

Gang TIAN

(Dedicated to Haim Brezis on the occasion of his 70th birthday)

1 Introduction

In this note,we extend a third derivative estimate in[4]to Monge-Ampere equations in the conic case.For the purpose of our application,we will consider the complex Monge-Ampere equations in this note.The same result holds for real Monge-Ampere equations.

Let U be a neighborhood of(0,···,0),and u satisfy

and

whereβ∈(0,1)andωβis the standard conic flat metric on Cn:

Theorem 1.1Let V be an open neighborhood of(0,···,0),whose closure is contained in U.Then for any α 0 and Cα,which may depend on α,c0,V,inf?F and∥F∥C1,such that for any x∈ V and 0

where Br(x)denotes the ball with center x and radius r,? is the covariant derivative,? denotes the Laplacian and the norm is taken with respect to ωβ.

It follows from Theorem 1.1 and the standard arguments,e.g.,by using the Green function.

Corollary 1.1Let u and F be as in Theorem 1.1.Then for any α ∈ (0,β?1? 1)and α<1,is Cα-bounded with respect to ωβ.

This corollary was used by Jeffres-Mazzeo-Rubinstein in their proving the existence of K?hler-Einstein metrics with conic singularities.We refer the readers to Appendix B of[3]for details.

Theorem 1.1 has been known to me for some time.The proof is identical to that in[4].Its arguments were inspired by Giaquinta-Giusti’s work(see[2])on harmonic maps.For years,I had talked about this approach to the C2,α-estimate for complex Monge-Ampere equations in my courses on K?hler-Einstein metrics.

2 The Proof of Theorem 1.1

In this section,we prove Theorem 1.1 by following the arguments in Section 2 of[4].I will present the proof for complex Monge-Ampere equations in details.In[4],the proof was written for real Monge-Ampere equations,though it also applies to complex Monge-Ampere equations.

Without loss of generality,we may assume x∈V∩{z1=0}.In fact,one can apply known estimates(see,e.g.,[4])to u outside the singular set{z1=0}ofωβ.If F is smooth on U,we can also apply Calabi’s third derivative estimate to u outside{z1=0}(see[5]).

Define Bβ(r)to be the domain in C × Cn?1consisting of all(w1,w′),where w′=(w2,···,wn),satisfying

There is an r>0 such thatand w′=(z2,···,zn)defines a natural map frominto U.This map is isometric from the interior ofonto its image.For convenience,we will use w1,···,wnas coordinates.By scaling,we may assume thatr=1.

In terms of coordinates w1,···,wn,we have

and covariant derivatives of u become ordinary derivatives,e.g.,

We will denote det(ukl)uijby Uij,where(uij)denotes the inverse of

First we recall two elementary facts.

Lemma 2.1(see[4,Lemma 2.1])For each j,where all derivatives are covariant with respect to ωβ.

ProofRecall the identities

Differentiating them in wk-direction and summing over k,we get

Then the lemma follows.

Lemma 2.2(see[4,Lemma 2.2])For any positive λ1,···,λn,we have

where λ = λ1···λn,and C is a constant depending only on λiand

ProofIn[4],(2.2)is proved by the properties of determinants.Here we outline a simpler proof.First by using the homogeneity and positivity of,we only need to prove the case whenNext,if we denote the left-hand side of(2.2)by f(Λ),then by a direct computation,for i=1,···,n.Then(2.2)follows from the Taylor expansion of f at I.

In terms of w1,···,wn,(1.1)becomes

By a direct computation,we deduce from this

This system resembles the one for harmonic maps whose regularity theory were studied extensively in 70s and 80s.The idea of[4]is to apply the arguments,particularly in[2],from the regularity theory for harmonic maps.

The following lemma follows easily from the Sobolev embedding theorem.

Lemma 2.3There is a constant Cβ,which depends on β,such that for any smooth function h on Bβ=Bβ(1)with boundary condition

we have

Note that Cβblows up whenβtends to 1.

Lemma 2.4(see[4,Lemma 2.3])There is some q>2,which may depend on β,and,such that for any,we have

whereand C denotes a uniform constant.1Note that C,c always denote uniform constant though their actual values may vary in different places.

ProofFirst we assume y=x.Defineλijby

By using unitary transformations if necessary,wemay assumefor anyand i,j ≥ 2.It follows from(1.2)that

where I denotes the identity matrix.

Choose a cut-off function η:Br(x)7→ R satisfying

Using Lemma 2.1 and(2.4),we can deduce

whereand λ = λ1···λn.

Using Lemma 2.1,we have

Then by Lemma 2.2,we can deduce from the above that

By applying the Sobolev inequality to(i,j ≥ 2)and Lemma 2.3 toin the above,we get

This inequality still holds,if we replace Br(x)by any Br(y)which is disjoint from the singular set{z1=0}.This can be proved by using the same arguments,but Lemma 2.3 is not needed.One can easily deduce from this and a covering argument that for any ball B2r(y)?U,

Then(2.7)follows from Gehring’s inverse H?lder inequality(see[1]).

Lemma 2.5(see[4,Lemma 2.4])For any y∈V and B4r(y)?U andσ

Multiplying(2.10)bywe get

It follows that

Multiplying(2.4)by b w and integrating by parts,we have

Using the assumption that?F is bounded,we can easily deduce from this

By Lemma 2.4 and the Poincare inequality,we have

Without loss of generality,we may assume that q≥2(q?2).Sincevanishes on?Br(y),its L2-norm is controlled by the L2-norm ofand consequently,of|?ω|.Then we have

Next,we recall a simple lemma which can be proved by standard methods.

Lemma 2.6Let h be any harmonic function on Bβ=Bβ(1),such that

Then for any r<1,

In fact,we only need a weaker version of Lemma 2.6 in the subsequent arguments:In addition to the assumption(2.15),we may further assume

Remark 2.1If we replace(2.15)by

then we have a better estimate

We observe

Hence,by(2.11)and(2.18),we get

Clearly,(2.9)follows from(2.12)–(2.14)and(2.19).

In view of(2.9),we need the following lemma.

Lemma 2.7(see[4,Lemma 2.5])For any ?0>0,there is an ?depending only on ?0,and inf?F satisfying that for any e r>0 with Ber(y)? U,there issuch that

ProofIt follows from(2.4)that

where?′denotes the Laplacian ofω.

Letηbe a non-negative function on Br(y)satisfying thatη(z)=1 for anyη(z)=0 for any z near ?Br(y)andThen

where

SetIt follows from(2.21)that for a suitable constanti.e.,Z is super-harmonic with respect toω.Sinceω is equivalent toωβ,we can apply the standard Moser iteration to?′to get

It follows that

Hence,if(2.20)does not hold forthen

This is impossible if k is sufficiently large.So the lemma is proved.

Now we complete the proof of Theorem 1.1.Chooseξ=λ2αandλ∈(0,1),such that

Next we choose?0and r0sufficiently small,such that

Then we can deduce from(2.9)that forσ=λr and r≤r0satisfying(2.20),

whereν = β?1?1>α.Then(1.3)follows again from this and a standard iteration.

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