Curvature bound for Lp Minkowski problem

Abstract

We establish curvature estimates for anisotropic Gauss curvature flows. By using this, we show that given a measure μ with a positive smooth density f, any solution to the $L_p$ Minkowski problem in $R^{n+1}$ with $p\le -n+2$ is a hypersurface of class $C^{1,1}$. This is a sharp result because for each $p\in [-n+2,1)$ there exists a convex hypersurface of class $C^{1,\frac{1}{n+p-1}}$ which is a solution to the $L_p$ Minkowski problem for a positive smooth density f. In particular, the $C^{1,1}$ regularity is optimal in the case $p=-n+2$ which includes the logarithmic Minkowski problem in $R^3$.