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Li-Chun Liang

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Preprint Sep 2026

Interior Curvature Estimates of Semi-convex Solutions for the scalar curvature equation with Lipschitz Right-Hand Sides

In this paper, let $u\in C^4(B_{10})$ with $D^2u\geq -KI$ define a $2$-admissible graph $M=\{(x,u(x)):x\in B_{10}\}\subset\R^{n+1}$ satisfying \[ \sigma_2(\kappa[u])=f(x). \] We prove an interior curvature estimate depending on the Lipschitz norm of the right-hand sides. The proof combines a shifted Jacobi inequality f...

Li-Chun Liang · 0 citations

Preprint Aug 2026

Interior Hessian Estimates for Semi-convex Solutions of the $\sigma_2/\sigma_1$ Equation with Lipschitz Right-Hand Sides

Let $n\ge2$ and let $u$ be a smooth 2-convex and semi-convex solution of \[ \frac{\sigma _2(D^2u)}{\sigma _1(D^2u)}=f(x). \] We prove an interior Hessian estimate depending on the Lipschitz norm of $f$. The proof combines the integral approach of Chen--Jian--Zhou with the algebraic reduction of the quotient equation to...

Ke Ji, Li-Chun Liang · 0 citations