A Generalised Monge-Amp`ere Equation

PINGALI Vamsi P.

Departmentof Mathematics,Johns HopkinsUniversity,Baltimore,MD 21218,USA.

Received 17 July 2014;Accepted 18 November 2014

A Generalised Monge-Amp`ere Equation

PINGALI Vamsi P.?

Departmentof Mathematics,Johns HopkinsUniversity,Baltimore,MD 21218,USA.

Received 17 July 2014;Accepted 18 November 2014

.We consider a generalised complex Monge-Amp`ere equation on a compact K¨ahler manifold and treat it using the method of continuity.For complex surfaces we prove an existence result.We also prove that(for three-folds and a related real PDE in a ball in R3)as long as the Hessian is bounded below by a pre-determined constant (whilst moving along the method of continuity path),a smooth solution exists.Finally, we prove existence for another real PDE in a 3-ball,which is a local real version of a conjecture of X.X.Chen.

Monge-Amp′ere equations;Hessian equations;Evans-Krylov theory.

1 Introduction

Let(X,ω)bean n-dimensionalcompact K¨ahler manifold.Hereweconsidera generalised complex Monge-Amp`ere PDE(to be solved for a smooth function φ)

When η＞0 and αi=0?i,Eq.(1.1)is the one introduced by Calabi and solved by Yau[1].Equations of this type are ubiquitous in geometry.A version of this generalised one appeared in[2]in the context of bounding the Mabuchi energy and was studied further in[3]using the J-flow.The geometric applications of this equation are explored elsewhere[4].Essentially,this equation arises out of the question-Given a form in the top Chern character class of a hermitian holomorphic vector bundle,can we conformally modify the metric so that the given form is the top Chern-Weil form of the corresponding Chern connection?

The aims of the paper are threefold-To introduce Eq.(1.1),to show that a local“toy model”ofit can be solved usingthe methodofcontinuity(thuspaving theway forstudying it on a manifold),and to apply the Evans-Krylov theory in a slightly unconventional way to obtain C2,αestimates in some examples under some assumptions.Indeed a similar technique was used in[5,6]to obtain C2,αestimates.The only difference is that in[5,6] a result of Caffarelli[7]was used instead of the Evans-Krylov theory.

2 Statements of results

We state a somewhat general theorem about uniqueness,openness and C0estimates.The proof is quite standard(adapted largely from[8]which is in turn based on[1]).Although the theorem is folklore,we have not found the precise statement(in this level of generality)in the literature on the subject.We need the notion of positivity of(p,p)forms,which is defined as follows.

Definition 2.1.A smooth(p,p)-form α is said to be strictly positive and denoted as α＞0 if there exists a positive integer N,a smooth function ?＞0,smooth functions fi≥0,?1≤i≤N,and smooth(1,0)-forms θikwhere 1≤k≤p such that

Let B be the product of Banach submanifolds of forms wherein an element ofBis of the form(α1,...,αn-1,φ)where αiare C1,β(i,i),closed forms and φ is a C3,βfunction satisfying

Theorem 2.1.If ωn+α1R∧ωn-1+...＞0,η＞0 and dαi=0,then,any smooth solution φ of(1.1)

One may formulate a version of the same problem locally as a Dirichlet problem on a pseudoconvex domain in Cn.In this context,we note that viscosity solutions to the Dirichlet problem exist by[9]in some cases.We also have the following(“short-time continuation”)result for a real version of the PDE:

Theorem 2.2.The following Dirichlet problem on the ball B?R3of radius 1 centred at the origin

has a unique smooth solution at t=T,if for all t∈[0,T),smooth solutions exist and satisfy D2ut＞3.

The number 36 does not play any role in Theorem 2.2. In particular, it is not sharp. Itis just chosen for convenience. A similar result holds for complex three-folds.

Theorem 2.3. If α＞0, ω＞0 are smooth K¨ahler forms on a 3-dimensional compact K¨ahler manifold(X,ω0), then, there exists a constant C＞0 depending only on α and ω0 such that the equation

Remark 2.1.We note that in the above equations,the set of“admissible”solutions(in the sense of[10])is much larger than the set of convex functions in the case of Theorem 2.2 or quasiplurisubharmonic(ω+ddcφ＞0)functions in the case of Theorem 2.3.However,the aim of the two theorems above is not so much to solve the equation(which is a challenging task given that it is not at all clear as to how to come up with an ellipticitypreserving continuity path or to prove a priori estimates)as to prove C2,αestimates using Evans-Krylov theory in a slightly novel way,and to show that these examples impose“incorrect”assumptions on the positivity(or the lack of thereof)of the αiin Eq.(1.1).

Finally,we present a local real version of a conjecture of X.X.Chen(conjecture 4 in[2] made in thecompact complexmanifold case).Some progresshas beenmade ina fewspecial cases[11].However,in all these cases the problem was reduced to an inverse Hessian equation.We prove existence in a special case here using the method of continuity.Actually,a far more general result was proven in[10],but results on the Bellman equation were used(as opposed to a direct method of continuity).Such results may not carry over in an obvious way to the manifold case and hence our proof of this“toy model.”Another important point to note is that our proof of the C2a priori estimate strongly requires the C1estimate and we expect this behaviour to carry over to the manifold case as well.Even for the usual Monge-Amp`ere equation on a K¨ahler manifold,the gradient estimate was proven only recently by B?ocki[12].Indeed,these estimates are used in[4]to prove an existence theorem for Eq.(1.1)on a complex torus.

Theorem 2.4.If f＞0 is a smooth function onˉB(0,1)(the closed unit ball),then the following Dirichlet problem has a unique smooth convex solution.

3 Standard results used

For the convenience of the reader,we have included statements of some standard results in the form that we use in the proofs.

Our principal tool to study fully nonlinear PDE like Eq.(1.1)is the method of continuity(It is like a flow technique.In fact this analogy was exploited more seriously,to great advantage,in[13]).To solve Lu=f where L is a nonlinear operator,one considers the family of equations Lut=γ(t)where,γ(1)=f and γ(0)=g such that at t=0 one has a solution Lu0=g0.Then,one proves that the set of t∈[0,1]for which the equation has a solution is both,open and closed(and clearly non-empty).In order to prove openness, one considers L to be a map between appropriate Banach spaces.Then the implicit function theorem of Banach spaces proves openness.However,while dealing with equations like Monge-Amp`ere equations one has to verify that certain conditions like ellipticity are preserved along the“continuity path”.This is crucial because,in order to solve the linearised equation and to prove that indeed one has a solution in an appropriate Banach space,one needs ellipticity in these cases.In fact,in a few of the cases we will consider, ellipticity is not preserved and hence the best we can do is a“short-time”existence result.In order to prove closedness one needs to prove uniform(i.e.independent of t)a priori estimates for u.In our case,we will need these estimates in C2,αin order to use the Arzela-Ascoli theorem to conclude closedness.These estimates are usually proven by improving on lower order estimates.Once one produces a C2,αsolution,one“bootstraps”the regularity(at each t∈[0,1])using the Schauder estimates[14].

In order to derive a priori estimates we will use standard techniques as in[1],and[8] for the manifold case,and[15]for the Euclidean case.The main blackbox is the Evans-Krylov-Safonov theory for proving C2,αestimates from C2ones.This requires(apart from uniform ellipticity)concavity of the equation.There is a similar version for the complex case.The real version is:

Theorem 3.1.[14,16]Let u be a smooth function on the unit ball satisfying,

on the unit ball in Rncentred at the origin B(0,1)with u=0 on the boundary of the ball.Here, F is a smooth function defined on a convex open set of symmetric n×n matrices×R×Rnwhich satisfies,

a)Uniform ellipticity on solutions:There exist positive constants λ and Λ so that 0＜λ|ξ|2≤Fij(D2u,x,Du)ξiξj≤Λ|ξ|2for all vectors ξ and all u satisfying the equation.

b)Concavity on a convex open set:F is a concave function on a convex open set of symmetric matrices(containing D2u for all solutions u).

The complex,interior version(that we need)is:

Theorem 3.2.[17,18]Let u be a C4function on the unit ball in Cnsatisfying

for a C2,βfunction F,satisfying,

a)Uniform ellipticity on solutions:There exist positive constants λ and Λ so that 0＜for all vectors ξ and all u satisfying the equation.

b)Concavity on a convex open set:F is a concave function on a convex open set of hermitian matrices(containing uiˉjfor all solutions u).

Then,kukC2,α(B(0,12))≤C,where C and α depend on λ,Λ,n,andand uniform bounds on the first and second derivatives of F evaluated at u.

To conclude,we add a few words about uniqueness.The usual technique for demonstratinguniqueness(due toCalabi)of Lu=f,where L is a nonlinearoperator,is toassume two solutions u1and u2,and to write

If the integrand is an elliptic operator,then by the maximum principle u1=u2.

4 Proofs of the Theorems

4.1 Proof of Theorem 2.1

Proof.This proof is similar to the one for the usual Monge-Amp`ere equation[8].

The C0estimate:As usual,withoutlossofgeneralitywe may change the normalisation to supφ=-1,i.e.,we may add-1-supφ to φ.Indeed,if the new φ has a C0estimate, thenR

The Moser-iteration procedure gives sup|φ|≤CkφkL2.If we prove that the right hand side is controlled by the L1norm of φ we will be done.Indeed,

where we have used the Poincar′e inequality.Hence proved.

Uniqueness:If φ1and φ2are two solutions,upon subtraction we have,

Thus,by the maximum principle φ2-φ1is a constant.

The mixed derivatives estimate:When αi＞0,

where C＞0.Since 0＜ω+ddcφ,the eigenvalues of ddcφ are bounded above.Thus,the mixed second derivatives of φ are bounded.Note that by the Schauder estimate[20]the first derivatives are bounded as well.

Openness:The map T is smooth.Its G?ateaux derivative is DT(0,0,...,0,χ)=(nα0(ω+ ddcφ)n-1+(n-1)α1∧(ω+ddcφ)n-2+...+αn-1)∧ddcχ.It is clearly a bounded surjection (by the Schauder theory)onto its image if nα0(ω+ddcφ)n-1+(n-1)α1∧(ω+ddcφ)n-2+ ...+αn-1＞0.If DT isrestrictedtovectorsoftheform(0,0,...,0,χ),thenitis aBanach space isomorphism.Hence,by the implicit function theorem of Banach manifolds,openness is guaranteed.In fact,it also guarantees that on a level set φ can be solved for(locally),in terms of αi.

The n=2 case:The equation we have is equivalent to

This is just the usual Monge-Amp`ere equation and hence we are done.

4.2 Proof of Theorem 2.2

Proof.Uniqueness is proven as before.We will only prove existence.Let Lu=det(D2u)+ Δu.To this end,we use the method of continuity.Consider the equation

Closedness:Suppose there is a sequence ti→t such that there are smooth solutions utisatisfying D2u＞3.Then we wish to prove that a subsequence of the uticonverges to a smooth solution utin the C2,βtopology.This requires apriori estimates(the convergence following from the Arzela-Ascoli theorem).We will prove the same for the Eq.(2.2). We just have to prove the C2,αestimate in order to ensure smoothness(by the Schauder theory).

Just as before,by adding or subtracting a large multiple of

HencekukC1≤C.

C2estimate:Since Δu≤Lu≤f and Δu＞0,kuijkC0≤C.Hence,kukC2≤C.

C2,αestimate:Sofar,we have not usedanything about thesequenceexceptthat D2uti＞0.This will change presently.For any function F:R→R,F(det(D2u)+Δu)=F(f). If we choose the function appropriately then the resulting equation will be a concave, uniformly elliptic Monge Amp`ere PDE to which we may apply the Evans-Krylov theory to extract a C2,αestimate.We also note the standard fact that since F is purely a function of symmetric polynomials of the eigenvalues of D2u,in order to check concavity on the space of symmetric matrices it is enough to check it for diagonal ones[19].

We will prove the aforementioned fact:Let x=∑λi+λ1λ2λ3.We see thatand is less than 1+3f where we have evaluated the derivative at the eigenvalues of the Hessian of a solution of Eq.(2.2).Hence,it is uniformly elliptic.

If(v1,v2,v3)∈R3,then

This is in turn equal to which is positive?β2-4αγ≤0 and

Continuing further,

Hence proved.

Remark 4.1.Writing Eq.(1.1)for n=3 we have

A local real version of the above is Eq.(2.2).

4.3 Proof of Theorem 2.3

Proof.Once again we apply the method of continuity.We will impose several conditions on C as we go along.It should be large enough so that whenever β＞Cω0,β3＞3α2β (Indeed,if K＞0 and B＞0 are given,det(A)＞Ktr(BA)for sufficiently large A＞0). Obviously at t=0,u=0 solves the equation.Openness and uniqueness follow from Theorem 2.1.As before,if ti→t is a sequence such that there exist smooth solutions uisatisfying ω+ddcui＞Cω0,then we will prove that a subsequence converges to a smooth solution u in the C2,βtopology.As usual we need apriori estimates for this.

The C0and the mixed derivative estimates follow directly from Theorem2.1.We have to prove the C2,αestimate(thus proving existence and smoothness as before).It suffices to prove a local(interior)estimate.We will accomplish this via the complex version of the(interior)Evans-Krylov theory done in[17]and[18].

The local(in a ball)version of the equation is

We wish to prove that g′′(V,V)＜0 for every hermitian V.Let us diagonalise the positive-definiteform B-1,i.e.,PB-1P?=I forsomematrix P.Define?A=(P?)-1AP-1and ?V=(P?)-1VP-1.Now using a unitary matrix U,we may diagonalise?A i.e.?A=U?AU?= diag(a1,a2,a3)where a1≤a2≤a3and?V=U?VU?.This implies that det(?A)det(B)=det(A) and tr(?A)=tr(B-1A).Let?Vii=vi.Hence,

where

As before we want Q2-4PR＜0.Assume(without loss of generality)that v3=1.We see that

where

where the first inequality follows from the assumption that det(A)=det(B)a1a2a3＞3tr(B-1A)＞3∑ai.Hence,

Thus,K2-4JL＜0 implies that g is concave.The C2,αestimate follows from Theorem 3.2.

4.4 Proof of Theorem 2.4

Proof.We use the method of continuity again.As before,openness follows easily using theImplicit functiontheoremonBanachspaces.Hereweproveonlytheapriori estimates. Smoothness follows by bootstrapping as indicated earlier.Lastly,we will also prove the uniqueness of convex solutions.

which is in turn equal to

Thus det(D2φ)-det(D2u)-Δ(φ-u)=L(φ-u),where L is a linear second order differential operator depending on φ and u.We may write the expression above in terms of the eigenvalues λiof D2u:

We see that sinceμ2＞3,L is an elliptic operator acting on φ-u with L(φ-u)＞0.It is in fact uniformly elliptic on the ball because f＞0 onˉB.By the maximum principle φ＜u. This gives us a C0estimate on u.

C1estimate:As before we will use ellipticity and the maximum principle.We differentiate the equation and obtain

Using the equation itself we see that λ1λ2-1=(f+λ2+λ1)/λ3.Hence the right hand side of(4.3)is bounded below by 6μ-fiwhich is positive for large enoughμ.It is also clear that H(φ-ui)is an elliptic operator,and hence,by the maximum principle,φ-uiis bounded above by its value on the boundary.Applying the same argument to H(ui)+ H(φ),we see that uiis controlled by its boundary values.On the boundary the same argument of[15]as in Theorem 2.2 proves the result.

C2estimate:For future use notice that atleast two of the eigenvalues of D2u are largerthan 1.Taking derivatives of the equation we have(let u0be the minimum of u),

Let A=det(D2u)(D2u)-1-I.Consider g=Δu+μ(u-u0)＞0(we will choose the constantμ＞0 later.It can depend onkukC1and other constants).Notice that if g is bounded,then so is Δu and thus D2u is bounded.At the maximum of g(if it occurs in the interior),(Δu)i=-μuiand tr(AD2g)≤0.This implies that

Hence,Δu is boundedat that point.Thus,g is boundedat that point.This implies that Δu is bounded everywhere.If the maximum of g occurs on the boundary(call the max g0),we will have to analyse it separately.Let?g=g+g0(1-2r2).Clearly,the maximum of ?g has to occur in the interior.There,D?g=0 and tr(AD2?g)≤0.Hence(here,we assume that tr(A)=∑(λiλj-1)and that g0are sufficiently large compared to constants;if not, we are done.),

Choosingμ＞E,we see that g0is bounded.Notice that this also implies a lower bound on D2u.This is because Mλi＞λ1λ2λ3＞f.

C2,αestimate:Notice that the setY of positive,symmetric matrices satisfying det(A)-tr(A)＞0 is a convex open set([10,Lemma 4.16]).Also,our equation maybe written as

which is certainly concave on Y by the same lemma in[10].It is uniformly elliptic on solutions as long as the eigenvalues of the Hessian are bounded below and above(which they are by the C2estimates).Theorem 3.1 yields the desired estimates.

Uniqueness:If u1and u2aretwoconvexsolutionsoftheequation F(u)=-1(as above), then upon subtraction,

where L is elliptic.By the maximum principle u1=u2+C for some constant C.

Acknowledgement

The author thanks Leon A.Takhtajan and Dror Varolin for suggesting this direction of study and for sparing time generously to discuss the same.

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?Corresponding author.Email addresses:vpingali@math.jhu.edu(V.Pingali)

10.4208/jpde.v27.n4.4 December 2014

AMS Subject Classifications:35J96,53C07

Chinese Library Classifications:O175.25