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Interior Curvature Estimates of Semi-convex Solutions for the scalar curvature equation with Lipschitz Right-Hand Sides
Abstract
In this paper, let u∈ C4(B10) with D2u≥ −KI define a $2$-admissible graph M={(x,u(x)):x∈ B10}⊂Rn+1 satisfying σ2(κ[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 for b=log(H+J0), a parallel hypersurface transformation that makes the Newton tensor uniformly elliptic and local boundedness estimate then reduces the pointwise bound to a weighted L1 estimate, which is completed using the Jacobi energy inequality and integration by parts.
Related
- Interior curvature estimates for convex solutions of a prescribed curvature quotient equation
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- Gradient estimates for the Lagrangian mean curvature equation with Lipschitz critical phase
- Interior gradient estimates for mean curvature equations on Riemannian manifolds
BibTeX
@misc{liang2026interior,
title = {{Interior Curvature Estimates of Semi-convex Solutions for the scalar curvature equation with Lipschitz Right-Hand Sides}},
author = {Lichun Liang},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.03958},
eprint = {2609.03958},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.03958},
}
RIS
TY - UNPB
TI - Interior Curvature Estimates of Semi-convex Solutions for the scalar curvature equation with Lipschitz Right-Hand Sides
AU - Lichun Liang
PY - 2026
DA - 2026/09/03
PB - arXiv
DO - 10.48550/arxiv.2609.03958
UR - https://arxiv.org/abs/2609.03958
AB - In this paper, let u∈ C^4(B_10) with D^2u≥ −KI define a $2$-admissible graph M={(x,u(x)):x∈ B_10}⊂R^n+1 satisfying σ_2(κ[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 for b=log(H+J_0), a parallel hypersurface transformation that makes the Newton tensor uniformly elliptic and local boundedness estimate then reduces the pointwise bound to a weighted L^1 estimate, which is completed using the Jacobi energy inequality and integration by parts.
ER -