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Interior curvature estimate for curvature quotient equations on convex hypersurfaces

Jianxiang Liu

TL;DR

This work establishes interior $C^2$ estimates for convex hypersurfaces in $\mathbb{R}^{n+1}$ that satisfy the curvature quotient equation $F(\lambda)=\frac{\sigma_n}{\sigma_{n-2}}(\lambda)=f(X)>0$ with $n\ge 3$. Building on the Hessian quotient framework, the authors develop a pointwise Jacobi inequality on Riemannian manifolds and combine it with Guan–Qiu-type auxiliary functions, performing a careful case analysis to bound principal curvatures without resorting to Legendre transforms. The main result is a local bound $\sup_{B_{r/2}}|\lambda_i|\le C$ depending on $n$, $r$, and data from $f$ and the hypersurface, contributing to the regularity theory for curvature quotient equations on convex graphs. The approach broadens the toolbox for interior curvature estimates and offers a pathway toward higher-dimensional results for Hessian quotient equations.

Abstract

We study interior curvature estimates for convex graphs which satisfy the quotient equation $\frac{σ_{n}}{σ_{n-2}}(λ)=f(X)>0$ in this paper.

Interior curvature estimate for curvature quotient equations on convex hypersurfaces

TL;DR

This work establishes interior estimates for convex hypersurfaces in that satisfy the curvature quotient equation with . Building on the Hessian quotient framework, the authors develop a pointwise Jacobi inequality on Riemannian manifolds and combine it with Guan–Qiu-type auxiliary functions, performing a careful case analysis to bound principal curvatures without resorting to Legendre transforms. The main result is a local bound depending on , , and data from and the hypersurface, contributing to the regularity theory for curvature quotient equations on convex graphs. The approach broadens the toolbox for interior curvature estimates and offers a pathway toward higher-dimensional results for Hessian quotient equations.

Abstract

We study interior curvature estimates for convex graphs which satisfy the quotient equation in this paper.
Paper Structure (4 sections, 8 theorems, 84 equations)

This paper contains 4 sections, 8 theorems, 84 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Jacobi inequality
  4. Proof of the Theorem \ref{['thm:1.1']}

Key Result

Theorem 1.1

For $n\geq3$, suppose that $M=(x,u(x))$ is a local $C^4$ graph over a ball $x\in B_r\subset \mathbb{R}^n$ with positive principal curvatures $\lambda=(\lambda_{1},\cdots,\lambda_{n})>0$ and that it is a solution to equation (eq:originaleq) in $B_r$. And $f\in C^{2}(B_r)$ is a positive function. Then where $C$ depends only on $n,r,\Vert f \Vert_{C^{2}(B_r)}, inf_{B_r}|f|, \Vert M \Vert_{C^{1}(B_r)}

Theorems & Definitions (13)

  • Theorem 1.1
  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • proof
  • Lemma 2.6
  • ...and 3 more