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Second-order estimates for degenerate complex $k$-Hessian and Christoffel-Minkowski equations

Yasheng Lyu

Abstract

It is known that the complex $k$-Hessian equation admits almost $C^{1,1}$ regularity (i.e., $\supΔu<\infty$) and the Christoffel-Minkowski equation admits $C^{1,1}$ regularity under the sharp degenerate condition $f^{1/(k-1)}\in C^{1,1}$ for a nonnegative right-hand side $f$. Assuming instead the alternative sharp degenerate condition $f^{3/(2k-2)}\in C^{2,1}$, we prove almost $C^{1,1}$ regularity for the complex $k$-Hessian equation when $k\geq5$ and $C^{1,1}$ regularity for the Christoffel-Minkowski equation. The argument deeply exploits various concavity properties of the operators under the stronger regularity assumption on $f$.

Second-order estimates for degenerate complex $k$-Hessian and Christoffel-Minkowski equations

Abstract

It is known that the complex -Hessian equation admits almost regularity (i.e., ) and the Christoffel-Minkowski equation admits regularity under the sharp degenerate condition for a nonnegative right-hand side . Assuming instead the alternative sharp degenerate condition , we prove almost regularity for the complex -Hessian equation when and regularity for the Christoffel-Minkowski equation. The argument deeply exploits various concavity properties of the operators under the stronger regularity assumption on .
Paper Structure (5 sections, 10 theorems, 105 equations)

This paper contains 5 sections, 10 theorems, 105 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Complex $k$-Hessian equation
  4. Christoffel-Minkowski equation
  5. Appendix

Key Result

Theorem 1.1

Let $u\in C^{4}(M)$ be a $k$-admissible solution of eqn1.2. Assume that eqn1.3 and eqn1.4 hold. Let $k\geq5$ and $f^{3/(2k-2)}\in C^{2,1}(M)$ with $\inf_{M}f>0$. Then where $C>0$ depends on $\|u\|_{C^{0}(M)}$, $\|f^{3/(2k-2)}\|_{C^{2,1}(M)}$, and the geometry of $(M,\omega)$, but is independent of $\inf_{M}f$.

Theorems & Definitions (17)

  • Theorem 1.1
  • Corollary 1.2
  • Theorem 1.3
  • Lemma 2.1: Hardy1952Wang2009
  • Lemma 2.2
  • Lemma 2.3: Lyu2025
  • Remark 2.4
  • Remark 2.5
  • Lemma 3.1
  • proof : Proof of Theorem \ref{['thm1.1']}
  • ...and 7 more