Translating solitons of mean curvature flow of noncompact spacelike hypersurfaces in Minkowski space

In this paper, we study the existence, uniqueness and asymptotic behavior of rotationally symmetric translating solitons of the mean curvature flow in Minkowski space. We also study the asymptotic behavior and the strict convexity of general solitons of such flows.Comment: 16page

The Anisotropic Convexity of Domains and the Boundary Estimate for Two Monge-Amp\`ere Equations

We study the exact effect of the anisotropic convexity of domains on the boundary estimate for two Monge-Amp\`ere Equations: one is singular which is from the proper affine hyperspheres with constant mean curvature; the other is degenerate which is from the Monge-Amp\`ere eigenvalue problem. As a result, we obtain the sharp boundary boundary estimates and the optimal global H\"older regularity for the two equations

A priori estimates and existence of solutions to the prescribed centroaffine curvature problem

In this paper we study the prescribed centroaffine curvature problem in the Euclidean space Rⁿ⁺¹. This problem is equivalent to solving a Monge–Ampère equation on the unit sphere. It corresponds to the critical case of the Blaschke–Santaló inequality. By approximation from the subcritical case, and using an obstruction condition and a blow-up analysis, we obtain sufficient conditions for the a priori estimates, and the existence of solutions up to a Lagrange multiplier