A Problem Concerning Yamabe-Type Operators of Negative Admissible Metrics

Abstract

This paper is about a problem concerning nonlinear Yamabe-type operators of negative admissible metrics. We first give a result on 𝜎_𝑘 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:𝐴^𝑡𝑔=1𝑛−2Ric^𝑔−𝑡𝑅^𝑔2(𝑛−1)𝑔,𝑡≤1,(1.1) where Ric_𝑔 and 𝑅_𝑔 are the Ricci tensor and the scalar curvature of 𝑔, respectively.

Define𝜎Ω_𝑘+𝑘(𝜆)=_{1≤𝑖1≤⋯≤𝑖𝑘≤𝑛}𝜆_𝑖1⋯𝜆_𝑖𝑘for𝜆=𝜆₁,…,𝜆_𝑛∈ℝ^𝑛,=𝜆=𝜆₁,…,𝜆_𝑛∈ℝ^𝑛;𝜎_𝑗(𝜆)>0,1≤𝑗≤𝑘.(1.2) The 𝜎_𝑘 Yamabe problem is to find a metric 𝑔̃ conformal to 𝑔, such that𝜎_𝑘𝜆_̃𝑔𝐴_̃𝑔=1,𝜆_̃𝑔𝐴_̃𝑔∈Ω^+𝑘on𝑀,(1.3) where 𝜆_̃𝑔(𝐴_̃𝑔) denotes the eigenvalue of 𝐴_̃𝑔 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 Ω^−𝑘=−Ω^+𝑘. Then the 𝜎_𝑘 Yamabe problem in negative cone𝜎_𝑘−𝜆_̃𝑔𝐴_̃𝑔=1,𝜆_̃𝑔𝐴_̃𝑔∈Ω^−𝑘on𝑀,(1.4) is still elliptic (see [1]).

Definition 1.1. A metric 𝑔̃ conformal to 𝑔 is called negative admissible if 𝜆_̃𝑔𝐴^𝑡̃𝑔∈Ω^−𝑘on𝑀.(1.5)
Under the conformal relation 𝑔̃=𝑒^2𝑧𝑔, the transformation law for the modified Schouten tensor above is as follows:𝐴^𝜏̃𝑔=𝐴^𝜏𝑔−∇²𝑧−1−𝜏𝑛−2(Δ𝑧)𝑔−2−2𝜏|∇𝑧|²𝑔+𝑑𝑧⊗𝑑𝑧.(1.6) We consider the following nonlinear equation: 𝑃(𝑍)∶=𝛽𝜆_𝑔(𝑍)=𝜑(𝑥,z),𝜆_𝑔(𝑍)∈Ωon𝑀,(1.7) where 𝑍=∇²𝑧+1−𝑡𝑛−2(Δ𝑧)𝑔+22−𝑡|∇𝑧|²𝑔−𝑑𝑧⊗𝑑𝑧−𝐴^𝑡𝑔,(1.8)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: t𝜑her𝑥e,e𝛾xiststwoconstants𝛾<0<𝛾with<𝛽−𝜆^𝑔𝐴^𝑡𝑔<𝜑𝑥,𝛾,∀𝑥∈𝑀.(1.9) 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 𝐴^𝑡𝑔∈Ω⁻,for𝑡<1.(1.10) Suppose that Ω⁺,Ω⁻⊂𝑅^𝑛 are open convex symmetric cones with vertex at the origin, satisfying Ω_𝑛⊂Ω⊂Ω₁,Ω⁻=−Ω⁺,(1.11) where Ω₁∶=𝜆=𝜆₁,…,𝜆_𝑛;^𝑛_𝑖=1Ω𝜆_𝑖>0_𝑛,∶=𝜆=𝜆₁,…,𝜆_𝑛;𝜆_𝑖>0for1≤𝑖≤𝑛.(1.12) Let 𝛽 satisfy(i)𝛽>0 in Ω⁺, 𝛽_𝑖∶=𝜕𝛽/𝜕𝜆_𝑖>0 on Ω⁺, and 𝛽(𝑒)=1 on Ω⁺, where 𝑒=(1,…,1).(1.13)(ii)𝛽 is concave on Ω⁺, and 𝛽(𝜆)≤𝜚𝜎₁(𝜆),∀𝜆∈Ω⁺,(1.14) 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 𝐴^𝑡𝑔∈Ω⁻,for𝑡<1.(1.15) 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𝜆_𝑔(𝑈)=Λ,(1.16) where 𝑈=−𝐴^𝑡̃𝑔=∇²𝜙+1−𝑡𝑛−2(Δ𝜙)𝑔+22−𝑡||∇𝜙||²𝑔−𝑑𝜙⊗𝑑𝜙−𝐴^𝑡𝑔(1.17) for conformal metric 𝑔̃=𝑒^2𝜙 and 𝜆_𝑔(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(1.18) 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 𝛾≤𝑧≤𝛾(2.1) satisfies 𝛾<𝑧<𝛾.(2.2)

Proof. Assume 𝑧 is a solution to (1.7) with 𝛾≤𝑧. Denote 𝑧𝑧_𝑠̃=𝑧−𝛾,𝑍=𝑠𝑧+(1−𝑠)𝛾,_𝑠=∇²𝑧_𝑠+1−𝑡𝑛−2Δ𝑧^𝑠𝑔+22−𝑡|∇𝑧^𝑠|²𝑔−𝑑𝑧_𝑠⊗𝑑𝑧_𝑠−𝐴^𝑡𝑔.(2.3) It is easy to verify that 𝑍_𝑠∈Ω⁺.
Write𝑄[𝑧]=𝑃(𝑍)−𝜑(𝑥,𝑧).(2.4) Then 𝑄[𝑧]−𝑄𝛾=0−𝑃−𝐴^𝑡𝑔+𝜑𝑥,𝛾.(2.5) On the other hand, 𝑄[𝑧]−𝑄𝛾=¹⁰𝑑=𝑑𝑠𝑄𝑧^𝑠𝑑𝑠₁₀=𝐿(𝑇𝑧)̃_𝑖𝑗𝑍_𝑠𝑑𝑠𝐷_𝑖𝑗𝑧̃+𝑏^𝑖𝐷_𝑖𝑧̃+𝑐𝑧̃(2.6) for some bound 𝑏^𝑖 and constant 𝑐, where 𝑇_𝑖𝑗=𝑃_𝑖𝑗+1−𝑡𝑛−2_𝑙𝑃𝑃_𝑙𝑙𝛾_𝑖𝑗≥0,_𝑖𝑗=𝜕𝑃𝜕𝑍_𝑖𝑗≥0(2.7) by condition (ii).
Therefore, we know that 𝐿 is an elliptic operator, and𝐿(𝑧)̃<0with𝑧̃≥0.(2.8) By the maximum principle, we get 𝑧̃>0. That is, 𝑧>𝛾.(2.9) Similarly, we can derive 𝑧<𝛾,(2.10) 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 𝛾<𝑧<𝛾. Then ‖∇𝑧‖_𝐿^∞<𝐶₁,(2.11) where 𝐶₁ depends only upon 𝛾,𝛾,𝑔,𝑡,𝜑.
Moreover,‖‖∇²𝑧‖‖_𝐿∞<𝐶₂,(2.12) where 𝐶₂ depends only upon 𝛾,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, let𝛽_𝜏(𝜆)∶=𝛽𝜏𝜆+(1−𝜏)𝜎₁(𝜆)𝑒,(2.13) (here 𝑒=(1,…,1) as in Section 1) which is defined on Ω^+𝜏=𝜆∈ℝ^𝑛;𝜏𝜆+(1−𝜏)𝜎₁(𝜆)𝑒∈Ω⁺.(2.14) We consider the problem 𝑃𝜏𝑍+(1−𝜏)𝜎₁(𝑍)𝑒=𝜏𝜑(𝑥,𝑧)+(1−𝜏)𝜎₁−𝐴^𝑡𝑔𝑒^2𝑧(2.15) on 𝑀, where 𝑍=∇²𝑧+1−𝑡𝑛−2(Δ𝑧)𝑔+22−𝑡||∇𝑧||²𝑔−𝑑𝑧⊗𝑑𝑧−𝐴^𝑡𝑔.(2.16) Since 𝐴^𝑡𝑔∈Ω⁻, we have 𝜎₁−𝐴^𝑡𝑔>0(2.17) 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 𝑧^𝜏 to (2.15) with 𝛾≤𝑧^𝜏≤𝛾 satisfies𝛾<𝑧^𝜏<𝛾.(2.18) This, together with Lemma 2.2, shows that for each 𝜏∈[0,1] and solution 𝑧^𝜏 to (2.15) with 𝛾≤𝑧^𝜏≤𝛾, the following estimate holds ‖𝑧^𝜏‖_𝐶²<𝐶,(2.19) 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‖𝑧^𝜏‖_𝐶^4,𝛼<𝐾,(2.20) where 𝑧^𝜏 is a 𝐶² solution to (2.15) with 𝛾≤𝑧^𝜏≤𝛾.
Set𝑆_𝜏∶=𝛾<𝑧^𝜏<𝛾∩‖𝑧^𝜏‖_𝐶^4,𝛼<𝐾∩𝑍∈Ω^+𝜏,(2.21) and define 𝑇_𝜏∶𝐶^4,𝛼→𝐶^2,𝛼 by 𝑇_𝜏(𝑧)=𝑃𝜏𝑍+(1−𝜏)𝜎₁(𝑍)𝑒−𝜏𝜑(𝑥,𝑧)−(1−𝜏)𝜎₁−𝐴^𝑡𝑔𝑒^2𝑧.(2.22) Then, by (2.19), we see that there is no solution to the equation 𝑇_𝜏(𝑧)=0on𝜕𝑆_𝜏.(2.23) So the degree of 𝑇_𝜏 is well defined and independent of 𝜏. As mentioned above, there is a unique solution at 𝜏=0. Therefore deg𝑇₀,𝑆₀,0≠0.(2.24) Since the degree is homotopy invariant, we have deg𝑇₁,𝑆₁,0≠0.(2.25) Thus, we conclude that (1.7) has a solution in 𝑆₁.
The proof of Theorem 1.2 is completed.

3. Proof of Theorem 1.3

Proof of Theorem 1.3. Take a look at the following equation: 𝑃(𝑢)=𝑃∇²𝑢+1−𝑡𝑛−2(Δ𝑢)𝑔+22−𝑡|∇𝑢|²𝑔−𝑑𝑢⊗𝑑𝑢−𝐴^𝑡𝑔−𝑒^𝑢=𝜆.(3.1) We will prove that, for small 𝜆>0, (3.1) has a unique smooth solution.
Since 𝜕𝑃/𝜕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 from𝐴^𝑡𝑔∈Ω⁻(3.2) that, for 𝜆>0 small enough, we can find two constants 𝛾<0<𝛾, such that 𝑒^𝛾+𝜆<𝑃−𝐴^𝑡𝑔<𝑒^𝛾+𝜆.(3.3) 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}.(3.4) Since 𝐸≠∅, we can define Λ=sup_𝜆∈𝐸𝜆.(3.5) We claim Λ is finite. Actually, 𝜆<𝑃∇²𝑢+1−𝑡𝑛−2(Δ𝑢)𝑔+22−𝑡|∇𝑢|²𝑔−𝑑𝑢⊗𝑑𝑢−𝐴^𝑡𝑔.(3.6) If we assume that at 𝑥₀, 𝑢 achieves its maximum, then ∇²𝑢≤0, and so 𝜆<𝑃∇²𝑢+1−𝑡𝑛−2(Δ𝑢)𝑔−𝐴^𝑡𝑔≤𝑃−𝐴^𝑡𝑔.(3.7) This means that Λ≤𝑃−𝐴^𝑡𝑔.(3.8)
For any sequence 𝜆_𝑖⊂𝐸 with 𝜆_𝑖→Λ, let 𝑢_𝜆𝑖 be the corresponding solution to (3.1) with 𝜆=𝜆_𝑖.
First, we claim thatinf_𝑀𝑢_𝜆𝑖⟶−∞as𝑖⟶∞.(3.9) Suppose this is not true, that is, inf_𝑀𝑢_𝜆𝑖≥−𝐶₀(3.10) for a positive constant 𝐶₀. Then, by (3.1), at any maximum point 𝑥₀ of 𝑢_𝜆𝑖, max_𝑀𝑢_𝜆𝑖≤𝐶(3.11) for some constant 𝐶 depending only on 𝑃(−𝐴^𝑡𝑔). Then the apriori estimates imply that 𝑢_𝜆𝑖 (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>0smallenough.(3.12) This is a contradiction. Hence (3.9) holds.
Next, we prove thatmax_𝑀𝑢_𝜆𝑖⟶−∞as𝑖⟶∞.(3.13)
We divided our proof into two steps.
Step 1. Let Λ=𝑃−𝐴^𝑡𝑔.(3.14) Then, following the above argument, 𝑢_𝜆𝑖→𝜙₀inC^∞,(3.15) and (Λ,𝑢₀) is a solution to (3.1). Assume 𝑢₀ attains its maximum at 𝑦₀. Then at 𝑦₀, ∇²𝑢₀≤0,∇𝑢₀=0.(3.16) Therefore, 𝑒^𝑢0(𝑦0)≤𝑃−𝐴^𝑡𝑔−Λ=0.(3.17) So 𝑢₀𝑦₀=−∞.(3.18) That means that (3.13) holds.Step 2. Let 𝑃−𝐴^𝑡𝑔−Λ=𝜛>0.(3.19) Then, if (3.13) is not true, that is, max_𝑀𝑢_𝜆𝑖≥−𝐶₀(3.20) for a positive constant 𝐶₀, write 𝑧_𝜆𝑖∶=𝑢_𝜆𝑖−max_𝑀𝑢_𝜆𝑖.(3.21) Then we have max_𝑀𝑧_𝜆𝑖⟶0,inf_𝑀𝑧_𝜆𝑖⟶−∞,(3.22) as 𝑖→∞.
On the other hand, 𝑧_𝜆𝑖 satisfies𝑃∇²𝑧_𝜆𝑖+1−𝑡𝑛−2Δ𝑧^𝜆𝑖𝑔+22−𝑡|∇𝑧^𝜆𝑖|²−𝑑𝑧_𝜆𝑖⊗𝑑𝑧_𝜆𝑖−𝐴^𝑡𝑔=𝑒^{max𝑀𝑢𝜆𝑖}𝑒^𝑧𝜆𝑖+𝜆_𝑖.(3.23) Since at any minimum point 𝑧₀ of 𝑧_𝜆𝑖, ∇²𝑧_𝜆𝑖≥0,∇𝑧_𝜆𝑖=0.(3.24) Consequently, at 𝑧₀, we obtain 𝑒^{max𝑀𝑢𝜆𝑖}𝑒^𝑧𝜆𝑖≥𝑃−𝐴^𝑡𝑔−Λ>0.(3.25) Thus, it is easy to verify that 𝑧_𝜆𝑖 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 𝑧_𝜆𝑖 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+22−𝑡|∇𝑧|²𝑔−𝑑𝑧⊗𝑑𝑧−𝐴^𝑡𝑔,(3.26) and for any smooth functions 𝑧₀ and 𝑧₁, set 𝑣𝑧_𝑠=𝑧₁−𝑧₀,𝑍_𝑠=𝑠𝑧₁+(1−𝑠)𝑧₀,=∇²𝑧_𝑠+1−𝑡𝑛−2Δ𝑧^𝑠𝑔+22−𝑡|∇𝑧^𝑠|²𝑔−𝑑𝑧_𝑠⊗𝑑𝑧_𝑠−𝐴^𝑡𝑔.(3.27) Then we get 𝑃𝑍₁−𝑃𝑍₀=¹⁰𝑑𝑑𝑠𝑃𝑍^𝑠=¹⁰𝑃^𝑖𝑗+1−𝑡𝑛−2_𝑙𝑃_𝑙𝑙𝛾_𝑖𝑗𝑍^𝑠𝑑𝑠𝑣_𝑖𝑗+𝑏^𝑙𝑣_𝑙(3.28) for some bounded 𝑏^𝑙. Thus, if 𝑧₀=𝜙,𝑧₁=𝜙^(3.29) are two solutions to (1.16) for some 𝜆 and 𝜆^, respectively, then 𝑎^𝑖𝑗 is positive definite. Therefore, 𝜙=𝜙^+𝑐(3.30) 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.