Weimin SHENG Lixia YUAN

1 Introduction

Let(Mn,g),n≥3 be a connected Riemannian manifold,andfa smooth function onM.The Ricci tensor onMis denoted by Ric(or Ricg).In order to study a log Sobolev inequality of the diffusion operator,Bakry and Emery[1]introduced the following Bakry-Emery Ricci tensor:

In fact,the Bakry-Emery Ricci tensor also occurs naturally in many different subjects(see[12–14]).It has been widely studied recently.Many important geometric results of this tensor have been obtained,such as the measured Gromov-Hausdorffconvergence theorem,volume comparison theorems,the splitting theorem,the rigidity theorem,etc.,see[2,12,17,20]and the references therein.Moreover,the Bakry-Emery Ricci tensor has a closed relation with Ricci flowThere are some other interesting results

In this paper,we consider the prescribing problems for this tensor.Letbe thek-th elementary symmetric function,namely,

is an open convex cone.Let Γ⊂nbe an open convex symmetric cone with the vertex at the origin satisfyingWe call a metricga Γ−-metric if it satisfies

whereλ(g−1Ricf)is ann-vector composed of the eigenvalues ofg−1Ricf.Let Γ−[g]denote the set of all Γ−-metrics that are conformal tog.

SupposeF:n→to be a general smooth symmetric homogeneous function of degree one withF=0 on∂Γ satisfying the following structure conditions in Γ:

(C1)Fis positive;

(C2)Fis concaveis negative semi-definite;

(C3)Fis monotoneis positive.

It follows from(C2)andF(0)=0 that there exists some uniform constant Θ>0 such that

SinceFis homogeneous and of degree one,by(C2),we have

whereeis the identity ofn(see[19]).

Letn≥3 be a smooth compact Riemannian manifold with the boundary∂M,andGiven a positive functionwe study the problem offinding a smooth complete metric∈Γ−[g]such that

whereand(resp.is the Ricci tensor(resp.Hessian)with respect to.Note that whenf=const.and(1.1)reduces to the following prescribedk-curvature equation:

In fact,the equation(1.2)has been extensively studied.Guan[4]and Gursky[5]proved that if Ricg<0,there exists a complete conformal metric of the negative Ricci curvature satisfying(1.2).By a theorem of Lohkamp in[11],there always exist compact smooth metrics onwith negative Ricci curvature.The results in[4–5]imply thatadmits a complete metricgsuch that thek-curvature de fined by the negative eigenvalues of the Ricci tensor equals any given positive function.Note that in the case ofk=1 andψ=const.,the equation(1.2)reduces to the Yamabe equation.Ifk=n,the equation(1.2)becomes the following Monge-Ampère type equation:

He and Sheng[7]solved the following equation:

provided that(M,g)has the semi-positive Ricci curvature with a totally geodesic boundary,and is not conformal equivalent to a hemisphere.In the case of∂M= ∅,Gursky and Viaclovsky in[6]found the solution metricsatisfying(1.2)with<0.Li and Sheng obtained the same result in[9]by using a parabolic argument.Trudinger and Wang[18]solved(1.4)by requiring that(M,g)is not conformally equivalent to the unit sphere and has positive Ricci curvature.

In[21],we solved the equation(1.1)on a closed manifold.In this paper we study the prescribing curvature problem(1.1)on a compact manifold with boundary.The method we used here is inspired by[4]and a recent work of Li and Sheng[10],in which they considered the prescribing problem on the modified Schouten tensor

Before stating our results,we first write out the corresponding partial differential equation of the equation(1.1).Let=e2ug,whereuis defined onM.Under this conformal change,we have

where the covariant derivative is taken with respect to the background metricg.

Set

For simplicity,we also use the notationF(A)to denoteF(λ(g−1A))for any smooth symmetric(0,2)-tensorA.Then the equation(1.1)becomes

In order to find a complete metric satisfying(1.1),we only need to solve the following Dirichlet problem with infinite boundary value,i.e.,

More generally,given a positive functionand a functionφ∈C∞(∂M),we consider the following equation:

where

forγ,s,t∈,γ>0,Tis a smooth symmetric(0,2)-tensor,anda(x),b(x)are two smooth functions defined onM. Clearly,(1.7)is fully nonlinear and elliptic for the solutionsuwithλ(g−1W[u])∈Γ (see[21]).Accordingly,we call a functionv∈C2(M)admissible ifλ(g−1W[v])∈Γ.

Our main results can be stated as follows.

Theorem 1.1Letn≥3be a smooth compact Riemannian manifold with theboundary∂M andIf T∈Γ,φ is a smooth function defined on a neighborhood of∂M,andsatisfies

then there exists a unique admissible solutionof the equation(1.7).

Remark 1.1Different from the results of[10,15],in this theorem,we need not add any restriction ona(x),b(x)and the coefficientsγ,s,t∈R,and just requireγ>0.

Applying Theorem 1.1 to the quotient of the elementary symmetric functions,i.e.,F=0≤l

Corollary 1.1Letn≥3be a smooth compact Riemannian manifold with theboundary∂M andIfthen for each function φ∈C∞(∂M)and a positive function,there exists a unique smooth metricsatisfying

By solving the Dirichlet problem(1.7)with infinite boundary data,we can obtain a complete metric satisfying(1.6)(compare with[4,10]).

Theorem 1.2Letn≥3be a smooth compact Riemannian manifold with theboundary∂M andGiven any smooth positive functionif T∈Γ,then there exists a unique admissible solutionof the equation

Moreover,there exist positive constants C and0<θ≤1,depending only on g,γ,s,t,such that

where ρ(x)is the distance function from x to∂M with respect to the background metric g.

Then we have the following corollary.

Corollary 1.2Letn≥3be a smooth compact Riemannian manifold with theboundary∂M andthen there exists a unique smoothcomplete metric satisfyingand

In particular,for k=n and l=0,we have

Remark 1.2Let(M,g)be a Riemannian manifold andfbe a smooth function onM.By[2,13],theN-Bakry-Emery Ricci tensorand theN-Ricci tensor RicNare defined respectively by

Note that df⊗dfis invariant under conformal changes.IfMhas a boundary,all the conclusions above are also valid for itsN-Bakry-Emery Ricci tensor and theN-Ricci tensor.

This paper is organized as follows.In Sections 2–3,we establish a priori boundary and interior estimates of admissible solutions of(1.7).Then we prove Theorem 1.1 in Section 4 by using the a priori estimates and the standard continuity method.In Section 5,we solve the Dirichlet equation(1.10)by constructing two suitable barrier functions,and then prove Theorem 1.2.

2 Boundary Estimates

In this section,we establish a priori boundary estimates for the first and second derivatives of admissible solutions of(1.7)with a smooth Dirichlet dataφ.We always assume thatφ∈throughout this paper.

2.1 Boundary C1estimates

For convenience,set

and

Then

A functionwis said to be a subsolution of(1.7)if it satisfies the following equation:

Changing the direction of the inequalities,one gets the definition of the supsolution of(1.7).

To estimate the gradient on the boundary,we need the following maximum principle for(1.7).

Proposition 2.1(Maximum Principle)Suppose that w and v are smooth sub-and supsolutions of the equation(1.7)with w|∂M≤v|∂M,respectively.If∂zΨ >0in M×R,then

One may prove this proposition by a contradictory argument.We omit its proof,and see[10,21]for details.

By the maximum principle and the boundary distance functionρ(x):=distg(x,∂M),we can construct two barrier functions later to control the gradient derivatives.Given any small positive constantδ,we set

Since∂Mis smooth and|∇ρ|=1 on∂M,we may assume thatρ(x)is smooth andinMδforδsufficiently small.

For any fixed pointx0∈∂M,we choose a local orthonormal coordinate systeminMδ,such that∂Mis the planexn=0.Let{e1,···,en−1,en}be the corresponding coordinate vector fields,whereenis the interior normal vector andeαis the tangential direction vector,α=1,2,···,n−1.

Lemma 2.1Letbe an admissible solution of(1.7).Iffor someconstantµ>0,andthen we have

where the constant C depends on

µ,g,γ,s,t,and

ProofIf there exists a local subsolutionof the equation(1.7),i.e.,

then we complete the lemma.In fact,the maximum principle implies thatu−(x)≤u(x)onConsequently,for anyx0∈∂M,we have

which implies that

Now,we construct a local subsolutionu−of(1.7)by using the method whichis similar to that of[4,10].Set

whereθis a positive constant to be fixed.Thenand

Choosingwe getu−(x)≤u(x)on∂Mδ∂M.Thus,u−(x)≤u(x)on∂Mδ.

It remains to verifyF(W[u−])≥Ψ(x,u−)inMδ.Since

we have

where the constantsC′andC′′depend onand

Chooseθ−1≥maxThenγ−tθ>0 and 1+sθ>0.ByinMδand(2.4),we have

Note thatρ+δ2≤2δand(ρ+δ2)2≤4δ2<4δ.By choosingδ

which implies that

Hence,W[u−]∈Γ andu−is admissible.Denote We know that{Fij}is positive definite(see[3]),andNote thatFis homogeneous and of degree one.ThenThus,by(2.5),we have

Since∂zΨ >0 forδsufficiently small,we obtain

Similarly,we can get the upper bound of∂nuon∂M.

Lemma 2.2Letbe an admissible solution of(1.7).Iffor someconstantµ>0,and∂zΨ >0on M×,then we have

where the constant C depends onµ,g,γ,s,t,and

ProofSinceuis admissible andwe have

Now,we construct a local supsolutionof(2.6),that is,the functionu+satisfies

Letτbe a small positive constant to be decided.Define

A direct calculation shows that

Chooseτ,such thatwe have

where the constantsC′andC′′depend on theand other known data.Then forδsufficiently small,we have

Note thatandFor a smallδ,we have

Hence,u+satisfies(2.7).By the maximum principle,we haveTherefore,for anyx0∈∂M,

which implies that

Combining the above two lemmas,we obtain the following proposition.

Proposition 2.2Suppose thatis an admissible solution of(1.7).Iffor some constantµ>0,and∂zΨ >0,then we have

where the constant C depends on g,γ,s,t,µ,and

2.2 Boundary C2estimates

The method we use here to derive the second derivative estimates is similar to that of[4–5,10].For any fixed pointx0∈∂M,define a half ballcentered atx0of radiusδby

We may assume thatr(x)is smooth infor smallδ.Then|∇r|=1 inChoose a local orthonormal framee1,···,enatx0,whereenis the inward unit normal vector.Since∇ijr2(x0)=2δij,we also assume thatin

Letbe an admissible solution of(1.7).Define a linearized operatorLby

whereand

In order to get the estimates of the normal and tangential derivatives of mixed type,we need the following two lemmas.

Lemma 2.3For any constant β>0,there exist positive constants δ sufficiently small and N sufficiently large such that the barrier function

satisfies

where δ and N depend on β,and other known data.

ProofIt is easy to check that

where the constantC∗depends onγ,|s|,|t|,n,g,andSince{Fij}is positive definite andwe have

By choosingandN>max,we obtain

Finally,for the fixedN,we can chooseto ensure thatw≥0 on

Lemma 2.4LetIf h≤0on∂M,h(x0)=0and

for some positive constant D,then we have

where the constant C depends on D,β,and other known data,and βisthe same constant as in Lemma2.3.

ProofBy Lemma 2.3,we can chooseA≫β≫1 such thatAw(x)+βr2(x)−h(x)≥0 onIt is clear thatwhereC′depends onγandδ.Note thatF≥F(e).By choosing,we have

It follows from the maximum principle thatAw+βr2−h≥0 inB+δ.Since

which implies that∇nh(x0)≤C.

Now,we can get the following boundary estimates for the second derivatives.

Proposition 2.3Suppose thatis an admissible solution of(1.7).Then

where the constant C depends onthe geometric quan-tities ofand other known data.

ProofSinceu−φ=0 on∂M,for any pointx0∈∂M,we have

where 1≤α,β≤n−1 and Π denotes the second fundamental form of∂M.Then one can get

whereCdepends onand the geometric quantities of

To get the estimates of the normal and tangential derivatives of mixed type∇αnu,we differentiate the equation(1.7)with respect toek,that is,

By the Ricci identities(2.9)and(2.12),we have

Then,for eachk=1,···,n,we obtain that

wheredepends onn,g,γ,|s|,|t|,

Applying Lemma 2.4 toh=±∇α(u−φ),we immediately get the estimates

It remains to estimate the bound of∇nnu.Sinceuis admissible,by(2.6),we have△u(x0)≥−C.Thus,∇nnu(x0)has a lower bound by(2.11).Without loss of generality,we can assumeunn(x0)≥0(otherwise we have done).Orthogonally decomposing the matrixW[u]atx0∈∂Min terms ofeαanden,and using the known bounds in(2.11)and(2.13),we have

where the constantCalso depends only onand

Then,we have

SinceF≥F(e),we then obtain the upper bound of∇nnu(x0).

3 Global Estimates

In this section,we first calculate the local interior estimates for admissible solutions of(1.7).By combining the local interior estimates and the boundary estimates in Section 2,we will derive a priori globalC2estimates.Now,we divide the procedure into three steps.

3.1 Global C0estimate

Since the manifold is compact,we can get a globalC0estimate easily.

Proposition 3.1Let T∈Γ,φ∈C∞(∂M)andsatisfies(1.9),then for any admissible solutionof(1.7),we have

where the constant C0only depends on

ProofLetxminbe the minimum point of the functionuonthen the lower bound ofucan be obtained by

Ifxminis an interior point ofM,then atxmin,we have∇u(xmin)=0,∇2u(xmin)≥0.Note thatγ>0 andT∈Γ.Then

Hence,we can get the lower bound ofuby the condition

Similarly,we can get the upper bound ofuby considering its maximum and using the fact that

3.2 Global C1estimate

LetBr⊂Mbe a geodesic ball of radiusr>0.There exists a cutofffunctionsuch thatand

for some constantb0>0.

Lemma 3.1Letandbe an admissible solu-tion of(1.7).Then there exists a constant C depending only onand C0such that

ProofConsider the following auxiliary function:

whereandηis a function to be chosen later.Suppose thatGattains its maximum at an interior pointx0∈Br.Choose a local normal coordinate frameei,i=1,···,natx0with respect togsuch thatW[u](x0)is diagonal.Then atx0we have

that is

and

By(3.5)and the Schwarz inequality,we have

whereαis any positive constant and we will choose a suitable one later.Note that

Substituting(3.7)–(3.8)into(3.6),we have

Note that the first term of the inequality above is non-negative,then by(3.2)and the Ricci identities,we have

where the constantCdepends onnandb0.Similarly,we can get

By(3.9)–(3.10),we have

It follows from(3.5),(3.8)and(2.12)that

By(3.2),the equality above implies

where the constantCdepends only on

SinceFis homogeneous and of degree one,then

Note that∇kΨ = Ψk+Ψuuk.It follows from(3.11)–(3.13)that

whereCalso depends onγ,g,and

Letη(u)=v−N,whereNis a large positive constant to be determined later and

Since<0 andthen

Then for any fixedα∈(0,1),we have

Note that 1≤v<2C0.We may chooseandNlarge enough such that

Therefore,by(3.14)–(3.15),

We can assume|∇u(x0)|≥1(otherwise we have done).Then

which implies that

where the constantCalso depends onNandC0,from which it is easy to derive(3.3).

Choosingζ≡1,we obtain the following global gradient estimates by(2.8)and(3.3).

Proposition 3.2Let T∈ΓandThen for any admissible solutionof(1.7),we have

where the constant C1depends onand

3.3 Global C2estimate

As in Section 3.2,we first establish the interior second derivative estimates.

Lemma 3.2Let T∈Γ,Ψ(x,z)∈C∞(M×R),and u∈C4(Br)be an admissible solu-tion of(1.7).Then there exists a constant C depending only on g,b0,γ,r−1,|s|,|t|,such that

ProofConsider the following auxiliary function:

whereξ∈TxMis a unit vector,satisfiesand(3.2),and the functionηis chosen later.Suppose thatHattains its maximum at a pointx0∈Brandξ∈Tx0M.Choose a local orthnormal frame{ei,i=1,···,n}atx0with respect togsuch thate1(x0)=ξand{W[u]ij}(x0)is diagonal.DenoteWithout loss of generality,we can assumeK(x0)>1.Then atx0,we have

that is

and

By(3.20)–(3.21),we have

Note that

By(3.22)–(3.23)and the Ricci identities,we have

where the constantCdepends only ong,b0andSimilarly,we have

By(3.24)–(3.25),we have

By(2.12),we have

where the constantCdepends onand

Differentiating the equation(1.7)twice,by the concavity ofF,we obtain

where the constantCdepends only ong,γ,|s|,|t|,and

Substituting(3.27)–(3.28)into(3.26),we have

By(3.20),(3.23)and the Ricci identities,we have

Note that

where the constantCdepends only on

Substituting(3.30)–(3.31)into(3.29),we have

Since{W[u]ij(x0)}is diagonal,|uij|=|−suiuj−2buifj−Tij|≤Cfori≠j.Hence,

Note that 1

Since

we have

Let

whereandαis a small positive constant to be chosen later.Then we have

If we choosethen

Then by(2.12)and(3.35),we have

Combining(3.34)and(3.36),we have

Multiplyingζon both sides of(3.37),we obtain

Hence,This implies that

Since,△uhas a lower bound by Lemma 3.2.Then by(3.39),we get

Thus,(3.18)follows from(3.39)–(3.40).

Letζ≡1.By(3.18)and(2.10),we derive the following global estimate for the second derivatives.

Proposition 3.3Let T∈ΓandThen for any admissible solutionof(1.7),we have

where the constant C2depends only onand

4 Proof of Theorem 1.1

For any functionhonM,define

Then any solutionuof(1.7)satisfiesP[u]=0.Letup=u+pvforp∈R.The linearized operator of equation(1.7)is

Lemma 4.1be an admissible solution of equation(1.7).If∂zΨis positiveonis invertible.

ProofSince∂zΨ is positive onthe coefficient of the zero order term in(4.1)is strictly negative.Hence,is invertible in the H¨older space

Proof of Theorem 1.1Note that the maximum principle in Proposition 2.1 ensures the uniqueness of solutions of(1.7).Now,we complete the proof by using the continuity method.Consider the following equation:

where

Clearly,Tβand Ψβsatisfy the following conditions:

(1)where the constantCis independent ofβ;

(2)and

(3)whereCis independent ofβ.

It follows from Section 2 and Section 3 that for eachβ,the admissible solution of(4.2)has uniform a prioriC2estimates(independent ofβ).Then we obtain the uniformC2,αestimates by the Evans-Krylov’s theory.Define

Clearly,u≡0 is the unique admissible solution of(4.2).Hence,I≠ ∅.Then by Proposition 2.1,I⊂[0,1]is open.By the uniform a prioriC2,αestimates and the standard degree theory,we conclude thatIis also closed.Forβ=1,the equation(1.7)is solvable.

5 Proof of Theorem 1.2

The argument of the proofis similar to the one in[4,10].To solve the infinite boundary data Dirichlet problem(1.10),we consider a family of equations below

wheremis any positive integer andθis a positive constant which will be chosen later.

For any fixedm,it follows from Theorem 1.1 that(5.1)has a unique admissible solutionThe maximum principle implies that

Next,for anymand a smallδ>0,define a local barrier function by

whereρ(x)=distg(x,∂M),.Then,and

Therefore,we can chooseδsmall enough such thaton the boundary ofMδinM.By a direct calculation,we have

where the constantsC′andC′′depend only onand other known data.

ChooseandByinMδand(5.3),we have

If we require,then

Therefore,the maximum principle implies that

In order to control the upper bound ofum,we consider the following equation:

where

By(5.1),everyumis a subsolution of(5.5).Thus,we only need to construct a local supsolution of(5.5).For any fixed point,letx0be the nearest point ofy0on∂M.Choose the geodesic fromx0toy0,passing throughy0and going out a small distance to another pointz0.Denoteandfor any pointx∈.Whenρ0andδare small enough,we can assume thatr(x)is smooth in the ballBz0(R).Choose a local orthonormal frame{ei,i=1,···,n}atz0.Note thatA,CandDare positive constants.

Case(a)If 2s−nt>0,thenB>0.Consider the function∈Bz0(R)defined by

whereτandεare two positive constants to be chosen later.By a direct calculation,we have

and

Note that|∇r|=1.Then we have

Since△r2(z0)=2n,we can assumen≤△r2≤3ninBz0(R).Also note thatB>0,and then by(5.6)we obtain

ChooseandIt follows from(5.7)that

Note thatis in finite onFor anym≥1,applying the maximum principle on this ball,we conclude thatThus,

So

By(5.2),(5.4)and(5.8),we have

and

near the boundary∂M.

For any compact subsetby the boundary control(5.9)and the a priori estimates in Section 2 and Section 3,we obtain

where 0<α<1,andC=C(K)is independent ofm.Hence,the standard compactness argument and the Schauder regularity theory imply thatis an admissible solution of(1.10).

Case(b)If 2s−nt≤0,thenB≤0.This case is much simpler than Case(a).Set bv∈Bz0(R)defined by

whereεis a positive constant to be decided.Then

ChooseεandRas above,i.e.,ε

The remaining argument is similar to the part in Case(a),and we omit it here.

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