[Paper Review] A fully nonlinear version of the Yamabe problem on locally conformally flat manifolds with umbilic boundary

Yanyan Li, Luc Nguyen|ArXiv.org|Nov 17, 2009

Nonlinear Partial Differential Equations49 references3 citations

TL;DR

This paper establishes the existence of a solution to a fully nonlinear Yamabe-type problem on compact, locally conformally flat Riemannian manifolds with umbilic boundary, by proving the existence of a conformal metric with constant k-curvature (defined via a fully nonlinear symmetric function of the Schouten tensor) under natural geometric and analytic conditions on the cone $Γ$ and the function $f$. The key result is the solvability of the associated fully nonlinear PDE system via degree theory and a priori estimates, extending the classical Yamabe problem to the fully nonlinear and boundary setting.

ABSTRACT

In this paper we establish existence and compactness of solutions to a general fully nonlinear version of the Yamabe problem on locally conformally flat Riemannian manifolds with umbilic boundary.

Motivation & Objective

• To extend the classical Yamabe problem to the fully nonlinear setting on manifolds with boundary, specifically for locally conformally flat manifolds with umbilic boundary.
• To establish the existence of a conformal metric with constant $k$-curvature, defined by a fully nonlinear symmetric function $f$ of the Schouten tensor, under natural conditions on $f$ and the cone $\Gamma$.
• To prove the solvability of the associated fully nonlinear PDE system with Neumann-type boundary conditions via topological degree theory and a priori estimates.

Proposed method

• Formulate the fully nonlinear Yamabe problem as a PDE system involving the Schouten tensor and a symmetric function $f$ defined on a convex cone $\Gamma$ with specific positivity and concavity conditions.
• Use the Leray-Schauder degree theory to prove existence of a solution by constructing a homotopy between the original operator and a simpler one.
• Establish a priori $C^{4,\alpha}$ estimates for solutions via maximum principle and obliqueness arguments, ensuring compactness of the solution set.
• Apply the implicit function theorem and continuity methods to show that the degree is non-zero, implying existence of a solution.
• Use the fact that $f$ is homogeneous of degree one and concave to ensure the necessary ellipticity and boundary conditions are satisfied.
• Construct a smooth, strictly positive defining function for the cone $\Gamma$ to ensure the existence of a solution under general geometric assumptions.

Experimental results

Research questions

• RQ1 Does a fully nonlinear Yamabe problem admit a solution on a locally conformally flat manifold with umbilic boundary when the curvature is prescribed via a fully nonlinear symmetric function $f$ of the Schouten tensor?
• RQ2 Can the existence of a solution be established under the natural geometric and analytic conditions on $f$ and $\Gamma$, including positivity, concavity, and homogeneity?
• RQ3 What is the role of the cone $\Gamma$ and the function $f$ in ensuring the solvability of the associated PDE system with Neumann-type boundary conditions?
• RQ4 How does the degree-theoretic approach via homotopy and compactness arguments ensure the existence of a solution in the absence of a priori $L^\infty$ bounds?
• RQ5 Can the classical Yamabe problem's solvability be extended to the fully nonlinear setting on manifolds with boundary, particularly when the boundary is umbilic and the manifold is locally conformally flat?

Key findings

• The paper proves the existence of a solution to the fully nonlinear Yamabe problem on a compact, locally conformally flat Riemannian manifold with umbilic boundary, under the assumption that $f$ is smooth, positive, strictly increasing in each variable, and vanishes on $\partial\Gamma$, with $\sum \partial f / \partial \lambda_k \geq \delta > 0$.
• The solution exists for any $f$ satisfying the conditions (3)–(9), including the important case $f = \sigma_k^{1/k}$ on $\Gamma_k$, which corresponds to the $k$-curvature problem.
• The Leray-Schauder degree of the associated nonlinear operator is non-zero, implying the existence of at least one solution, via a homotopy argument connecting the original problem to a simpler one.
• A priori $C^{4,\alpha}$ estimates are established through the maximum principle and obliqueness, ensuring the solution set is compact and the degree is well-defined.
• The degree is computed as $(-1)^{\dim E^{-}(A_1)}$, where $A_1$ is a linearized operator, and this sign is non-zero, confirming existence.
• The result extends the classical Yamabe problem to the fully nonlinear and boundary setting, providing a complete existence theory under natural geometric and analytic conditions.