Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

The $σ_{2}$-curvature equation on a compact manifold with boundary

Xuezhang Chen, Wei Wei

TL;DR

This work addresses the boundary $\sigma_2$-curvature equation on compact manifolds by establishing local boundary $C^2$-estimates for solutions under a nonlinear Neumann boundary condition, with crucial reliance on the nonnegativity of the boundary mean curvature. It then analyzes the associated boundary $\sigma_2$-eigenvalue problem and uses a Leray–Schauder strategy, supplemented by a detailed blow-up analysis in dimensions $n=3,4$, to obtain existence of conformal metrics with prescribed positive $\sigma_2$-curvature and prescribed nonnegative boundary mean curvature. The local estimates are achieved via a two-step barrier approach: controlling the trace of tangential derivatives and bounding the double normal derivative, exploiting the structure of the interior $\sigma_2$-equation and the boundary mean curvature equation; counterexamples in the non-umbilic boundary case highlight the necessity of $h_g\ge0$. The blow-up analysis rules out interior blow-up under totally non-umbilic boundary by leveraging rescaled nonnegative Ricci curvature and Liouville-type classifications, enabling a degree-theoretic existence proof for the prescribed boundary-value problem. Overall, the paper extends the boundary $\sigma_2$-curvature theory, clarifies the role of boundary geometry in a fully nonlinear Neumann problem, and provides a rigorous pathway from a priori estimates to existence of conformal metrics with controlled curvature data.

Abstract

We first establish local $C^2$ estimates of solutions to the $σ_2$-curvature equation with nonlinear Neumann boundary condition. Then, under assumption that the mean curvature of a background metric is nonnegative on totally non-umbilic boundary, for dimensions three and four there exists a conformal metric having a prescribed positive $σ_2$-curvature and a prescribed nonnegative boundary mean curvature. The local estimates play an important role in the blow up analysis for the latter existence result.

The $σ_{2}$-curvature equation on a compact manifold with boundary

TL;DR

This work addresses the boundary -curvature equation on compact manifolds by establishing local boundary -estimates for solutions under a nonlinear Neumann boundary condition, with crucial reliance on the nonnegativity of the boundary mean curvature. It then analyzes the associated boundary -eigenvalue problem and uses a Leray–Schauder strategy, supplemented by a detailed blow-up analysis in dimensions , to obtain existence of conformal metrics with prescribed positive -curvature and prescribed nonnegative boundary mean curvature. The local estimates are achieved via a two-step barrier approach: controlling the trace of tangential derivatives and bounding the double normal derivative, exploiting the structure of the interior -equation and the boundary mean curvature equation; counterexamples in the non-umbilic boundary case highlight the necessity of . The blow-up analysis rules out interior blow-up under totally non-umbilic boundary by leveraging rescaled nonnegative Ricci curvature and Liouville-type classifications, enabling a degree-theoretic existence proof for the prescribed boundary-value problem. Overall, the paper extends the boundary -curvature theory, clarifies the role of boundary geometry in a fully nonlinear Neumann problem, and provides a rigorous pathway from a priori estimates to existence of conformal metrics with controlled curvature data.

Abstract

We first establish local estimates of solutions to the -curvature equation with nonlinear Neumann boundary condition. Then, under assumption that the mean curvature of a background metric is nonnegative on totally non-umbilic boundary, for dimensions three and four there exists a conformal metric having a prescribed positive -curvature and a prescribed nonnegative boundary mean curvature. The local estimates play an important role in the blow up analysis for the latter existence result.
Paper Structure (16 sections, 41 theorems, 654 equations)

This paper contains 16 sections, 41 theorems, 654 equations.

Table of Contents

  1. Introduction
  2. Local estimates
  3. Trace of second tangential derivatives
  4. Double normal derivative
  5. Counter-examples to boundary local $C^2$ estimates
  6. First boundary $\sigma_{2}$-eigenvalue problem
  7. Blow up analysis
  8. Prior results
  9. Basic results for all dimensions
  10. Isolated singularity behavior
  11. Isolated blow-up points are simple
  12. Compactness
  13. Existence of conformal metrics: Proof of Theorem \ref{['Thm:existence']}
  14. Appendix: Possible umbilic points of higher dimensional ellipsoids
  15. Appendix: Proof of Lemmas \ref{['one critical point']} and \ref{['lem:four-dim lower bound-boundary']}
  16. ...and 1 more sections

Key Result

Theorem 1.1

Let $(M,g)$ be a smooth compact Riemannian manifold of dimension $n\geq3$ with boundary $\partial M$ and assume $0\leq c \in C^\infty(\partial M)$. Suppose $f(x,z): M \times \mathbb{R} \to \mathbb{R}_+$ is a smooth function. Let $\mathcal{O}$ and $\mathcal{O}'$ be two neighborhoods near $\partial M$ Here $\tilde{\nabla}$ denotes the Levi-Civita connection with respect to the induced metric on $\pa

Theorems & Definitions (81)

  • Theorem 1.1
  • Theorem 1.2
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Claim
  • Lemma 2.3
  • proof
  • Corollary 2.1
  • ...and 71 more