Regularity of Weak Solutions to a Class of Complex Hessian Equations on K¨ahler Manifolds

Weimin Shengand Jiaxiang Wang

School of Mathematical Sciences,Zhejiang University,Hangzhou 310027,China.

Abstract.We prove the smoothness of weak solutions to a class of complex Hessian equations on closed K¨ahler manifolds,by use of the smoothing property of the corresponding gradient f low.

Key words:Complex Hessian equation,regularity of weak solutions,pluripotential theory,parabolic f lows.

1 Introduction

wheref(x,z)∈C2,α(M×R).Whenk=1,it is a quasilinear equation;whenk=n,it is the complex Monge-Amp′ereequation,which has been studied extensively in the literature.

The pluri-potential theory for complex Hessian equations have also attracted many attentions,and it was developed in[2,8,16,17]etc.In[16],Lu has considered the weak solution to(1.1)and obtained its existence under some integral conditions.He also studied the approximation properties and the stability of this equation.Furthermore,in[18],Nguyen has studied the H¨older continuity of the solution whenf(x,u)=f(x)∈Lp(M)with varying choice ofp.

In this paper,we are going to study the regularity of equation(1.1)for smooth functionf(x,z)under theassumption that theequation(1.1)hasaweak solution.Our method is based on the arguments in[19].As an application,we show the regularity of the solutions of the equation which has been studied by Lu in[16].We also improved our previous result in[20].

Our main result is

Similar to[19],we will consider the corresponding parabolic equation

where we supposeu0∈PSHω,k(M)∩C2,α(M).Hereu0∈PSHω,k(M)means thatu0is an(ω,k)-plurisubharmonic function.The regularity of the weak solution follows from an approximation argument based on the regularity of the f low.The def inition of(ω,k)-plurisubharmonic function can be found in Def inition 2.1.We will prove the short time existenceand regularity in Section 3,and then prove the main result in Section 4.

2 Preliminaries

In this section,we introduce some notions and some recent results of(ω,k)?plurisubharmonic functions,following[2](see also[8]).

First,we recall the elementary symmetric functions(see[2,5,10]).Letσk(λ1,···,λn)be thek-th elementary symmetric function,i.e.

where(λ1,···,λn)∈Rn,and 1≤k≤n.Letλω{aiˉj}denote the eigenvalues of Hermitian symmetric matrix{aiˉj}with respect to the K¨ahler formω.We def ine

By a simple calculation,it can be shown that

Whenk=1,it is quasilinear operator;whenk=n,then-Hessian operator correspondsto the complex Monge-Amp′ereoperator,which plays a central role in K¨ahler geometry.We require the functionusatisfying a natural admissible condition

In this case,the operator logσkis concave with respect tou.

In[2],B?ockideveloped the corresponding pluri-potential theory of complex Hessian equation.Here is the def inition ofk-plurisubharmonic function.

Def inition 2.1.We call u an(ω,k)?plurisubharmonic function with respect to a K¨ahler form ω,if u∈L1(M)is upper-semi-continuous,and satisf ies

(ω+ddcu)j∧ωn?j≥0j=1,...,k

in current sense,and denotePSHω,k(M)theset of all(ω,k)-plurisubharmonic functionson M.

B?ockialso proved the following weak continuity theorem and comparison theorem,which show the ellipticity of the complex Hessian equations.

in thesenseof measure.

Theorem 2.2(see[2],or preliminary in[8]).Let u,v∈PSHω,k(ω)∩L∞,if{u>v}is a non-empty subset of themanifold,then

Then it is natural to ask whether there is a solution in PSHω,k(M)to the equation(1.1)with measurable right-hand side.Thanks to Lu[16],where he gave some existence results of weak solutions under some conditions,and proved a weak stability result.

Theorem 2.3([16]).Let(M,ω)bean n?dimensional compact K¨ahler manifold and f:M×R→[0,+∞)beafunction satisfying thefollowing conditions:

(1)f(x,t)isnon-decreasing and continuouswith respect to t for almost all x∈M;

(2)f(x,t)∈Lp(M)for any f ixed t∈R;

Then thereexists asolution?∈PSHω,k(M)∩C(M)to(1.1).

In Section 3 of[17],Lu proved the following approximation theorem.

Theorem 2.4.For any?∈PSHω,k(M),there exists a sequence?j∈PSHω,k(M)∩C∞(M)decreasing to?on M.

Using this theorem,Lu also proved the general stability for the complex Hessian equations.

Based on the stability result,Nguyen[18]proved the following H¨older continuity of the solution in Theorem 2.3.

(a)If?u belongs to L2(?),then u∈Cγ(?)for any0≤γ

(b)If thetotal massof△u isf inite,then u∈Cγ(?)for any0≤γ≤min{v,2γ2},whereγ1andγ2aredef ined as,

In this paper,we are going further to study the regularity in the casef(x,z)being smooth.

3 Short time existence and estimate on|u|and|ut|

We f irst show the solution to the f low(1.2)exists in a short time interval.Consider the following map F def ined fromC4,α(M×[0,T))toC2,α(M×[0,T))

For any initial functionu0∈PSHω,k(M)∩C∞(M),weextendu0to afunctionΦinC2,α(M×[0,T))so that

The linearization of the map F atΦis

Proposition 3.1.Let(M,ω)beaclosed K¨ahler manifold of dimension n≥2and u0∈PSHω,k(M).Then thef low(1.2)has auniquesolution on interval[0,t0)for somet0>0.

when t∈[0,T].

Proof.We derive the estimates by the maximal principle.Due to the positivity off(x,z),we set

At f ixedt∈[0,T),we denotexmaxthe maximal point ofu(x,t).Then we have

By the comparison principle of O.D.E.,there existsLandT>0,such thatu(xmax,t)≤L1.In a similar argument we can showu(xmin,t)≥?L1.

Supposexmaxbe the maximal point ofut(x,t)at f ixedt,we obtain

Now,using the estimates of Theorems 1.3,1.4 in[20],we have the existence and regularity of solutions to(1.2)onM×[0,T],where the derivatives of the solutions rely on the derivatives of the initial values.In the next section,we obtain that for anyε>0,the second order spatial derivatives estimatesof thesolution onM×[ε,T]areindependent of sup|?u|(x,0)and sup|ddcu|(x,0).

4 Estimates up to order 2

In this section,we are going to prove a regularity result.

Theorem 4.1.There is a uniform T>0,such that the solution to(1.2)exists for any u0∈C2,α∩PSHω,k(M)and t∈[0,T].Moreover,for any0<ε

By the standard Evans-Krylov aruguments,it suff ices to derive the a prioriestimates up to the second order derivatives onM×[ε,T]for f ixed 0<ε

First,we derive the second order estimate,which will be controlled by the square of the gradient estimates linearly.

Lemma 4.1.For anyε>0,thereexists C>0which relies only on L,H,T and U,such that

Proof.Denoteh(x,z):=logf(x,z).Our arguments are parabolic version of those in[12],but we need to avoid the case that the maximum attains att=0.Consider

whereξis an unit tangent vector at the corresponding point,and

ChooseAas

Let(x0,t0,ξ)be the maximum point ofW(x,t,ξ)onM×[0,T]×S2n?1.Note that ift0=0,W(x0,t0,ξ)=1 is always bounded.Hence we can assumet0>0.Now we are going to deriveW(x0,t0,ξ0)≤C,whereC>0 is independent ofKandε.Once the estimate established,the result will follow by

as well as 0

Choose the coordinates such that the matrix{giˉj+uiˉj}is diagonal at the point,and satisf iesλ1≥λ2≥···,where

λi=1+uiˉi.

After a straight calculation,wesee

Then the rest of the proof is essentially the same as we did in[20].

Take thederivative of Eq.(1.2)on the both sides in the?/?zpdirection,we have

By commuting the covariant derivatives

Similarly

Hence we have

Take the second derivative of Eq.(4.8)on both sides,we have

Since

Hence we have

We may supposeλ1≥K,andK≥1,otherwise we have done.Thus

Then(4.11)becomes

The third term of(4.5)satisf ies

The seventh term of(4.7)satisf ies

The forth term of(4.7)satisf ies

Substituting(4.9)–(4.15)into(4.7),we obtain

Now we set

We separate the rest calculations into two cases.

Case1.λn≤?δλ1.

The f irst term of(4.16)is

Next,we use

Since 2φ′2=φ′′,the f irst term of(4.18)cancels out the third term of(4.16).The second term of(4.18)can be estimated as

Substituting(4.17)-(4.19)into(4.16),we obtain

which implies

then wecan estimate it as

Moreover,we can assume

otherwise,we would derive

w hich implies,at(x0,t0,ξ0),

and the estimate established.Therefore,(4.20)becomes

The choice ofAimples that the coeff icient of F is negative.So we have

Case2.λn>?δλ1.

It is easy to verify 1/∈I.Then we separate the second term of(4.16)into two parts,one part is

The last term in(4.21)can be estimated by

We claim the another part of the second term of(4.16)satisf ies

Then,by(4.21)-(4.23),(4.16)becomes

otherwise,

and the proof has been f inished.

otherwisewe have,

Therefore

λ1≤C(K+1).

Now in order to f inish theproof of Theorem 1.3,weneed to provetheclaim(4.23).In fact it was proved in[12],so we omit it here.

Therefore,wehaveobtained that onM×[ε,T],|?ˉ?u|≤C(K+1).Then wecan provethe f irst-order estimateby blow up analysis and a Liouville type theorem[8].It is essentially the same as[20].

Lemma 4.2.Suppose T,εaredef ined in Lemmas3.1and4.1,thederivatives of solution u(x,t)to Eq.(1.2)isuniformly bounded for t∈[ε,T],and thebound isindependent ofsupM|Du(x,0)|2andsupM|ddcu(x,0)|.

Proof.By Theorem 4.1,we have obtained|?ˉ?u|≤C(K+1)whent∈[ε,T],andCis independent of the choice of supM|Du(x,0)|2and supM|ddcu(x,0)|.Then we can employ a blow-up method on this time interval analogous to that in[7]and reduce the problem to a Liouville Type theorem which was proved in[7].Thus we obtain a contradiction.We omit the proof.See[20]for details.

5 Proof of Theorem 1.1

Proof of Theorem 1.1.Suppose Eq.(1.1)admits a continuous PSHω,k?solution?0,and wechoose?jto bea sequenceof smooth functions approximate decreasingly to?0.Then by the continuity of?0,?j→?0inL∞.Letuj∈PSHω,k(M)∩C2,α(M)solves

(ω+ddcuj)k∧ωn?k=f(x,?j)ωn.

Next,we consider the corresponding parabolic equation(1.2)with initial valuesuj,

We denoteuj(x,t)the solution,whereuj(x,t)|t=0=uj(x).

Therefore,

u(x,t)≡?0.

Consequently,?0must be aC2,α-solution.

Acknowledgments

The authors are supported by by the National Natural Science Foundation of China(Grant Nos 11971424 and 12031017).