[Paper Review] Some fully nonlinear problems on manifolds with boundary of negative admissible curvature

Aobing Li, Huan Zhu|arXiv (Cornell University)|Feb 16, 2011

Nonlinear Partial Differential Equations15 references3 citations

TL;DR

This paper establishes the existence of solutions to a fully nonlinear elliptic PDE on compact Riemannian manifolds with boundary, where the equation involves the eigenvalues of a modified Schouten tensor and prescribes both scalar curvature and mean curvature. By conformally deforming the metric near the boundary to ensure negative Ricci curvature and using a priori estimates and the method of continuity, the authors prove the existence of a $ C^{4,eta} $ conformal metric solving the fully nonlinear curvature problem with given boundary data.

ABSTRACT

In this paper, we consider a fully nonlinear problem on manifolds with boundaries of negative admissible curvatures. As a consequence, we conclude the existence of certain types of metrics on the general differential manifolds with boundaries.

Motivation & Objective

• To establish the existence of solutions to a fully nonlinear curvature problem on compact Riemannian manifolds with boundary, where both scalar and mean curvature are prescribed.
• To extend previous results on the boundary Yamabe problem to the fully nonlinear setting using the modified Schouten tensor $ A_g^t $ for $ t < 1 $.
• To construct a conformal metric near the boundary such that the negative eigenvalues of $ A_g^t $ lie in a given convex cone $ \Gamma $, ensuring ellipticity and solvability.
• To prove that for any smooth compact Riemannian manifold with boundary, a conformal metric can be found near the boundary such that the modified Schouten tensor satisfies the required curvature conditions and the mean curvature is controlled.

Proposed method

• Introduce the modified Schouten tensor $ A_g^t = \frac{1}{n-2}\left(\text{Ric}_g - \frac{tR_g}{2(n-1)}g\right) $ for $ t < 1 $, generalizing the classical Schouten tensor.
• Use a conformal deformation of the metric near the boundary to ensure $ \text{Ric}_{g_3} < 0 $ on $ M $, which implies $ -\lambda_{g_3}(A_{g_3}^t) \in \Gamma_n \subset \Gamma $ for $ t < 1 $.
• Apply the method of continuity and a priori estimates to solve the fully nonlinear PDE $ f(-\lambda_{\tilde{g}}(A^{t}_{\tilde{g}})) = \phi $ on $ M $, with $ \tilde{g} \in [g] $ conformal to $ g $ near $ \partial M $.
• Preserve the mean curvature condition $ h_{\tilde{g}} = \psi $ on $ \partial M $ by ensuring the conformal factor is constant on the boundary in the deformation process.
• Use the maximum principle and boundary estimates to control the behavior of solutions near $ \partial M $, leveraging the fact that $ h_{g_3} = 0 $ on $ \partial M $.
• Leverage results from [15] to extend a metric $ g_2 $ with $ \text{Ric}_{g_2} < 0 $ near $ \partial M $ to a global metric $ g_3 $ with $ \text{Ric}_{g_3} < 0 $ on all of $ M $, ensuring the required curvature conditions.

Experimental results

Research questions

• RQ1 Can a fully nonlinear curvature equation involving the modified Schouten tensor be solved on a compact Riemannian manifold with boundary, with prescribed scalar curvature and mean curvature?
• RQ2 Under what conditions on the background metric and the curvature operator can a solution be guaranteed to exist via the method of continuity?
• RQ3 Is it possible to conformally deform a given metric near the boundary to ensure negative Ricci curvature and preserve the desired curvature cone condition for the modified Schouten tensor?
• RQ4 How can the boundary mean curvature be controlled during the conformal deformation process while maintaining the ellipticity of the PDE?
• RQ5 What is the regularity of the solution to the fully nonlinear curvature problem under minimal assumptions on the data $ \phi $ and $ \psi $?

Key findings

• For any smooth compact Riemannian manifold $ (M^n, g) $ with $ n \geq 3 $ and non-empty boundary, there exists a $ C^{4,\alpha_0} $ conformal metric $ \tilde{g} $ near $ \partial M $ such that $ f(-\lambda_{\tilde{g}}(A^{t}_{\tilde{g}})) = \phi $ and $ h_{\tilde{g}} = \psi $ on $ \partial M $, provided $ f \in C^{2,\alpha_0}(\Gamma) $ satisfies the required convexity and ellipticity conditions.
• The construction ensures $ -\lambda_{g_3}(A_{g_3}^t) \in \Gamma_n \subset \Gamma $ on $ M $, which guarantees the ellipticity of the PDE and allows the use of the method of continuity.
• The paper proves that $ \text{Ric}_{g_3} < 0 $ on $ M $, which implies that the eigenvalues of $ A_{g_3}^t $ satisfy the required negativity condition for the fully nonlinear operator.
• The solution $ \tilde{g} $ is unique in the class of metrics conformal to $ g $ near $ \partial M $, under the given curvature and boundary conditions.
• The result holds for any $ t < 1 $, and in particular, for $ t = 0 $, the equation reduces to $ f(-\lambda_{\tilde{g}}(\text{Ric}_{\tilde{g}})) = \phi $, showing the solvability of a Ricci curvature-type PDE.
• The existence is established via a priori estimates and the method of continuity, with the key step being the construction of a background metric $ g_3 $ with negative Ricci curvature and zero mean curvature on $ \partial M $.