Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

The Eigenvalue Problem for the Complex Hessian Operator on $m$-Pseudoconvex Manifolds

Jianchun Chu, Yaxiong Liu, Nicholas McCleerey

Abstract

We establish $C^{1,1}$-regularity and uniqueness of the first eigenfunction of the complex Hessian operator on strongly $m$-pseudoconvex manifolds, along with a variational formula for the first eigenvalue. From these results, we derive a number of applications, including a bifurcation-type theorem and geometric bounds for the eigenvalue.

The Eigenvalue Problem for the Complex Hessian Operator on $m$-Pseudoconvex Manifolds

Abstract

We establish -regularity and uniqueness of the first eigenfunction of the complex Hessian operator on strongly -pseudoconvex manifolds, along with a variational formula for the first eigenvalue. From these results, we derive a number of applications, including a bifurcation-type theorem and geometric bounds for the eigenvalue.
Paper Structure (25 sections, 36 theorems, 323 equations)

This paper contains 25 sections, 36 theorems, 323 equations.

Table of Contents

  1. Introduction
  2. Main Results
  3. Remarks on the Proofs
  4. Previous Works
  5. Further Directions
  6. Outline
  7. Background Results on Strongly m-Pseudoconvex Manifolds
  8. Definitions
  9. Dirichlet problem
  10. First eigenvalue of the complex Laplacian
  11. Stability estimate
  12. Finite Energy Classes
  13. Subextenstion Theorem
  14. A Priori Estimates
  15. Interior Hessian estimate
  16. ...and 10 more sections

Key Result

Theorem 1.1

Suppose that $\Omega$ is an strongly $m$-pseudoconvex manifold $($see Section Background for a definition$)$, $\omega$ is a Kähler metric on $\Omega$, and $0 < f \in C^\infty(\overline{\Omega})$. Then there exists some $\lambda_1 := \lambda_1(\Omega, f) > 0$ and $u_1\in m\mathrm{SH}(\Omega)\cap C^\i

Theorems & Definitions (67)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Definition 2.1
  • Definition 2.2
  • Theorem 2.3: Theorem 1.1 of CP22
  • Corollary 2.4
  • proof
  • Theorem 2.5
  • ...and 57 more