This paper is about a problem concerning nonlinear Yamabe-type operators of negative admissible metrics. We first give a result on σk Yamabe problem of negative admissible metrics by virtue of the degree theory in nonlinear functional analysis and the maximum principle and then establish an existence and uniqueness theorem for the solutions to the problem.
1. Introduction
Let (M,g) be a compact closed, connected Riemannian manifold of dimension n≥3. In 2003, Gursky-Viaclovsky [1] introduced a modified Schouten tensor as follows:Agt=1n-2(Ricg-tRg2(n-1)g),t≤1,
where Ricg and Rg are the Ricci tensor and the scalar curvature of g, respectively.
Defineσk(λ)=∑1≤i1≤⋯≤ik≤nλi1⋯λikforλ=(λ1,…,λn)∈Rn,Ωk+={λ=(λ1,…,λn)∈Rn;σj(λ)>0,1≤j≤k}.
The σk Yamabe problem is to find a metric g̃ conformal to g, such thatσk(λg̃(Ag̃))=1,λg̃(Ag̃)∈Ωk+onM,
where λg̃(Ag̃) denotes the eigenvalue of Ag̃ with respect to the metric g̃. This problem has attracted great interest since the work of Viaclovsky in [2] (cf., e.g., [2–7] and references therein).
Assume Ωk-=-Ωk+. Then the σk Yamabe problem in negative coneσk(-λg̃(Ag̃))=1,λg̃(Ag̃)∈Ωk-onM,
is still elliptic (see [1]).
Definition 1.1.
A metric g̃ conformal to g is called negative admissible if
λg̃(Ag̃t)∈Ωk-onM.
Under the conformal relation g̃=e2zg, the transformation law for the modified Schouten tensor above is as follows:Ag̃τ=Agτ-∇2z-1-τn-2(Δz)g-2-τ2|∇z|2g+dz⊗dz.
We consider the following nonlinear equation:
P(Z):=β(λg(Z))=φ(x,z),λg(Z)∈ΩonM,
where
Z=∇2z+1-tn-2(Δz)g+2-t2|∇z|2g-dz⊗dz-Agt,β∈C∞(Ω+)∩C0(Ω+¯) is a symmetric function and is homogeneous of degree one normalized, and φ is a positive C∞ function satisfying the monotone condition:
thereexiststwoconstantsγ̲<0<γ¯withφ(x,γ̲)<β(-λg(Agt))<φ(x,γ¯),∀x∈M.
For this equation, we have the following.
Theorem 1.2.
Let (M,g) be a compact, closed, connected Riemannian manifold of dimension n≥3 and
Agt∈Ω-,fort<1.
Suppose that Ω+,Ω-⊂Rn are open convex symmetric cones with vertex at the origin, satisfying
Ωn⊂Ω⊂Ω1,Ω-=-Ω+,
where
Ω1:={λ=(λ1,…,λn);∑i=1nλi>0},Ωn:={λ=(λ1,…,λn);λi>0for1≤i≤n}.
Let β satisfy
β>0 in Ω+, βi:=∂β/∂λi>0 on Ω+, and β(e)=1 on Ω+, where
e=(1,…,1).
β is concave on Ω+, and
β(λ)≤ϱσ1(λ),∀λ∈Ω+,
where ϱ is a positive constant.
Moreover, assume that φ(x,z) is a positive C∞ satisfying condition (1.9). Then there exists a solution to (1.7).
Theorem 1.3.
Let (M,g) be a compact, closed, connected Riemannian manifold of dimension n≥3 and
Agt∈Ω-,fort<1.
Let (β,Ω+) be those as in Theorem 1.2. Then there exist a function ϕ and a positive number λ, such that ϕ is a solution to the eigenvalue problem
P(U):=β(λg(U))=Λ,
where
U=-Ag̃t=∇2ϕ+1-tn-2(Δϕ)g+2-t2|∇ϕ|2g-dϕ⊗dϕ-Agt
for conformal metric g̃=e2ϕ and λg(U) denotes the eigenvalue of U with respect to metric g.
Remark 1.4.
(1) (ϕ,Λ) is unique in Theorem 1.3 under the sense that, if there is another solution (ϕ′,Λ′) satisfying (1.16), then
Λ=Λ′,ϕ=ϕ′+c
for some constant c.
(2) Λ is called the eigenvalue related to fully nonlinear Yamabe-type operators of negative admissible metrics, and ϕ is called an eigenfunction with respect to Λ.
2. Proof of Theorem 1.2
To prove Theorem 1.2, firstly, let us give the following proposition.
Proposition 2.1.
Suppose all the conditions in Theorem 1.2 are satisfied. Then every C2 solution z to (1.7) with
γ̲≤z≤γ¯
satisfies
γ̲<z<γ¯.
Proof.
Assume z is a solution to (1.7) with γ̲≤z. Denote
z̃=z-γ̲,zs=sz+(1-s)γ,Zs=∇2zs+1-tn-2(Δzs)g+2-t2|∇zs|2g-dzs⊗dzs-Agt.
It is easy to verify that Zs∈Ω+.
WriteQ[z]=P(Z)-φ(x,z).
Then
Q[z]-Q[γ̲]=0-P(-Agt)+φ(x,γ̲).
On the other hand,
Q[z]-Q[γ̲]=∫01ddsQ[zs]ds=∫01Tij(Zs)dsDijz̃+biDiz̃+cz̃=L(z̃)
for some bound bi and constant c, where
Tij=Pij+1-tn-2∑lPllγij≥0,Pij=∂P∂Zij≥0
by condition (ii).
Therefore, we know that L is an elliptic operator, andL(z̃)<0withz̃≥0.
By the maximum principle, we get z̃>0. That is,
z>γ̲.
Similarly, we can derive
z<γ¯,
for solution z with z≤γ¯.
Thus, we have the following Gradient and Hessian estimates for solutions to (1.7).
Lemma 2.2.
Let z be a C3 solution to (1.7) for some t<1 satisfying γ̲<z<γ¯. Then
‖∇z‖L∞<C1,
where C1 depends only upon γ̲,γ¯,g,t,φ.
Moreover,‖∇2z‖L∞<C2,
where C2 depends only upon γ̲,γ¯,g,t,φ,C1.
Proof of Theorem 1.2.
We now prove Theorem 1.2 using a priori estimates in Lemma 2.2, the maximum principle in Proposition 2.1, and the degree theory in nonlinear functional analysis (cf., e.g., [8]).
For each 0≤τ≤1, letβτ(λ):=β(τλ+(1-τ)σ1(λ)e),
(here e=(1,…,1) as in Section 1) which is defined on
Ωτ+={λ∈Rn;τλ+(1-τ)σ1(λ)e∈Ω+}.
We consider the problem
P(τZ+(1-τ)σ1(Z)e)=τφ(x,z)+(1-τ)σ1(-Agt)e2z
on M, where
Z=∇2z+1-tn-2(Δz)g+2-t2|∇z|2g-dz⊗dz-Agt.
Since Agt∈Ω-, we have
σ1(-Agt)>0
by condition (ii). Hence for τ=0, it follows from the maximum principle that z=0 is the unique solution.
In view of Proposition 2.1, we see that, for each τ∈[0,1], every C2 solution zτ to (2.15) with γ̲≤zτ≤γ¯ satisfiesγ̲<zτ<γ¯.
This, together with Lemma 2.2, shows that for each τ∈[0,1] and solution zτ to (2.15) with γ̲≤zτ≤γ¯, the following estimate holds
‖zτ‖C2<C,
for some constant C independent of τ.
This estimate yields uniform ellipticity, and by virtue of the concavity condition (ii), the well-known theory of Evans-Krylov, and the standard Schauder estimate (cf. [9]), we know that there exists a constant K independent of τ such that‖zτ‖C4,α<K,
where zτ is a C2 solution to (2.15) with γ̲≤zτ≤γ¯.
SetSτ:={γ̲<zτ<γ¯}∩{∥zτ∥C4,α<K}∩{Z∈Ωτ+},
and define Tτ:C4,α→C2,α by
Tτ(z)=P(τZ+(1-τ)σ1(Z)e)-τφ(x,z)-(1-τ)σ1(-Agt)e2z.
Then, by (2.19), we see that there is no solution to the equation
Tτ(z)=0on∂Sτ.
So the degree of Tτ is well defined and independent of τ. As mentioned above, there is a unique solution at τ=0. Therefore
deg(T0,S0,0)≠0.
Since the degree is homotopy invariant, we have
deg(T1,S1,0)≠0.
Thus, we conclude that (1.7) has a solution in S1.
The proof of Theorem 1.2 is completed.
3. Proof of Theorem 1.3Proof of Theorem 1.3.
Take a look at the following equation:
P̃(u)=P(∇2u+1-tn-2(Δu)g+2-t2|∇u|2g-du⊗du-Agt)-eu=λ.
We will prove that, for small λ>0, (3.1) has a unique smooth solution.
Since ∂P̃/∂u<0, the uniqueness of the solution to (3.1) follows from the maximum principle.
Next, we show the existence of the solution to (3.1) by using Theorem 1.2.
It follows fromAgt∈Ω-
that, for λ>0 small enough, we can find two constants γ̲<0<γ¯, such that
eγ̲+λ<P(-Agt)<eγ¯+λ.
That is, condition (1.9) for φ(x,z) in Theorem 1.2 is satisfied. Therefore, by the result in Theorem 1.2, the existence of unique solution to (3.1) is established for small λ>0.
SetE:={λ>0;(3.1)hasasolution}.
Since E≠∅, we can define
Λ=supλ∈Eλ.
We claim Λ is finite. Actually,
λ<P(∇2u+1-tn-2(Δu)g+2-t2|∇u|2g-du⊗du-Agt).
If we assume that at x0, u achieves its maximum, then ∇2u≤0, and so
λ<P(∇2u+1-tn-2(Δu)g-Agt)≤P(-Agt).
This means that
Λ≤P(-Agt).
For any sequence λi⊂E with λi→Λ, let uλi be the corresponding solution to (3.1) with λ=λi.
First, we claim thatinfMuλi⟶-∞asi⟶∞.
Suppose this is not true, that is,
infMuλi≥-C0
for a positive constant C0. Then, by (3.1), at any maximum point x0 of uλi,
maxMuλi≤C
for some constant C depending only on P(-Agt). Then the apriori estimates imply that uλi (by taking a subsequence) converges to a smooth function u0 in C∞, such that u0 satisfies (3.1) for λ=λ0. Since the linearized operator of (3.1) is invertible, by the standard implicit function theorem, we have a solution to (3.1) for
λ=λ0+δwithδ>0small enough.
This is a contradiction. Hence (3.9) holds.
Next, we prove thatmaxMuλi⟶-∞asi⟶∞.
We divided our proof into two steps.
Step 1.
Let
Λ=P(-Agt).
Then, following the above argument,
uλi→ϕ0inC∞,
and (Λ,u0) is a solution to (3.1). Assume u0 attains its maximum at y0. Then at y0,
∇2u0≤0,∇u0=0.
Therefore,
eu0(y0)≤P(-Agt)-Λ=0.
So
u0(y0)=-∞.
That means that (3.13) holds.
Step 2.
Let
P(-Agt)-Λ=ϖ>0.
Then, if (3.13) is not true, that is,
maxMuλi≥-C0
for a positive constant C0, write
zλi:=uλi-maxMuλi.
Then we have
maxMzλi⟶0,infMzλi⟶-∞,
as i→∞.
On the other hand, zλi satisfiesP(∇2zλi+1-tn-2(Δzλi)g+2-t2|∇zλi|2-dzλi⊗dzλi-Agt)=emaxMuλiezλi+λi.
Since at any minimum point z0 of zλi,
∇2zλi≥0,∇zλi=0.
Consequently, at z0, we obtain
emaxMuλiezλi≥P(-Agt)-Λ>0.
Thus, it is easy to verify that zλi is bounded from below as i→∞. This is a contradiction. So we see that (3.13) is true.
By a priori estimates results again, we deduce that zλi converges to a smooth function z in C∞ and z satisfies (1.16) with λ=Λ.
Finally, let us prove the uniqueness.
DenoteZ:=∇2z+1-tn-2(Δz)g+2-t2|∇z|2g-dz⊗dz-Agt,
and for any smooth functions z0 and z1, set
v=z1-z0,zs=sz1+(1-s)z0,Zs=∇2zs+1-tn-2(Δzs)g+2-t2|∇zs|2g-dzs⊗dzs-Agt.
Then we get
P(Z1)-P(Z0)=∫01ddsP(Zs)=∫01[Pij+1-tn-2∑lPllγij](Zs)dsvij+blvl
for some bounded bl. Thus, if
z0=ϕ,z1=ϕ′
are two solutions to (1.16) for some λ and λ′, respectively, then aij is positive definite. Therefore,
ϕ=ϕ′+c
for some constant c by the maximum principle.
Acknowledgment
The authors acknowledge support from the NSF of China (11171210) and the Chinese Academy of Sciences.
GurskyM.ViaclovskyJ.Fully nonlinear equations on Riemannian manifolds with negative curvature200352239941910.1512/iumj.2003.52.23131976082ViaclovskyJ.Conformal geometry, contact geometry and the calculus of variations20001012283316BransonT. P.GoverA. R.Variational status of a class of fully nonlinear curvature prescription problems200832225326210.1007/s00526-007-0141-62389992ChangA.GurskyM.YangP.An equation of Monge-Ampère type in conformal geometry, and four-manifolds of positive Ricci curvature2002155370978710.2307/30621311923964GurskyM.ViaclovskyJ.Prescribing symmetric functions of the eigenvalues of the Ricci tensor2007166247553110.4007/annals.2007.166.4752373147LiA.ZhuH.Some fully nonlinear problems on manifolds with boundary of negative admissible curvatureAnalysis and Partial Differential Equations. In press, http://arxiv.org/abs/1102.3308v3ViaclovskyJ.Estimates and existence results for some fully nonlinear elliptic equations on Riemannian manifolds20021048158461925503DeimlingK.1985Berlin, GermanySpringer787404GilbargD.TrudingerN. S.19832242ndBerlin, GermanySpringer737190