Qi-Rui Li

Surfaces expanding by the power of the Gauss curvature flow

In this paper, we describe the flow of 2-surfaces in for some negative power of the Gauss curvature. We show that strictly convex surfaces expanding with normal velocity , when , converge to infinity in finite time. After appropriate rescaling, they converge to spheres. In the 2-dimensional case, our results close an apparent gap in the powers considered by previous authors, that is, for by Urbas and Huisken and for by Schnurer.

Two-dimensional Monge–Ampère equations under incomplete Hölder assumptions

in a bounded convex domain Ω in R2. We are concerned with the anisotropic regularity when f is merely partially or piecewise Hölder continuous. The Monge-Ampere equation has many applications in differential geometry and optimal transportation. The regularity problem for solutions of (1.1) has received a great deal of attention. In [1] Caffarelli established the interior C2,α estimates for the solution u when f is Hölder continuous, and later Jian and Wang [12] obtained the continuity estimates for D2u for Dini continuous f . In the two dimensional case, the interior C2,α estimates were established by Heinz, and these estimates were used for his solution of Weyl’s embedding problem with the aid of a continuity method. In particular, following Lewy’s work [17], Heinz developed a characteristic theory for the Monge-Ampère equations in two dimensions [9–11]. See Schulz’s book [23] for an excellent exposition to this characteristic theory. The main ingredient of the theory is the so-called partial Legendre transform, which was later used by many authors in their study of the local regularity of certain