Regularity of free boundary in optimal transport
Free boundary arises in optimal transport problem when only a portion of mass is transported. In an important paper Caffarelli and McCann established the regularity of the free boundary assuming the domains are strictly convex and disjoint. In this talk I will present our recent slight improvement on this result by relaxing the "strict convexity" condition on the domains to the usual convexity, thus making this regularity result sharp. This is based on a joint work with Jiakun Liu.
A fractional curvature flow on
We first recall some techniques of scalar curvature flow on developed in my previous paper joint with Professor Xingwang Xu (Invent. Math. 2012). Next, we extend such a flow approach to prove a perturbation result of the fractional Nirenberg problem, i.e. prescribing fractional Q-curvature problem on for and the fractional exponent . This is joint with Pak Tung Ho and Jingang Xiong.
Minkowski problems of convex bodies
黄 勇 湖南大学
In this talk, I will report briefly on the development of Minkowski problems of convex bodies. In particular, I will also discuss some new geometric measures and their characterisation problems in the Brunn-Minkowski theory, which is a joint work with Prof. E. Lutwak, D. Yang and G. Zhang.
A variational analysis of the planar dual Minkowski problem
We give a variational analysis to the planar dual Minkowski problem in Sobolev space. With the new variational characterization, we can deal with existence results for prescribed not necessarily positive data. Meanwhile, functional inequalities and multiple solutions are also obtained.
Kaehler hyperbolic manifolds and Chern number inequalities
In this talk we review a well-known conjecture due to Hopf and Chern, and explain their solution in the Kaehler case due to Gromov via the notion of "Kaehler hyperbolicity". Then we shall report our recent work around Kaehler hyperbolic manifolds.
A class of Monge-Ampere equations on the sphere
There are a number of geometric problems which can be reduced to the study of Monge-Ampere equations on the sphere, including the Aleksandrov problem, the Minkowski problem, and the dual Minkowski problem introduced recently. In this talk we give a brief discussion on these problems.
Steiner symmetrization and its applications in convex geometry
Steiner symmetrization was introduced by Steiner in the 18th century. Many (affine) isoperimetric inequalities in convex geometry that characterize ellipsoids can be established by using this approach. In this talk, we will present some new developments and applications of Steiner’s approach, including the affine inequalities for sets of finite perimeter, and general affine invariances related to Mahler volume.
Alexandrov-Fenchel’ s inequality for free boundary convex hypersurfaces in a ball
In this talk, we study free boundary convex hypersurfaces in a ball. A suitablequermassintegral will be introduced in this setting. In particular, the highest oder quermassintegral is a topological constant by Gauss-Bonnet-Chern’s formula. We will establish a family of Alexandrov-Fenchel’s inequalities by using a locally constraint inverse type curvature flow. This is a joint work with Julian Scheuer and Guofang Wang.
On the polar Orlicz-Minkowski problems
We will talk about the polar Orlicz-Minkowski problems: under what conditions on a nonzero finite measure and a continuous functionthere exists a convex body such that is an optimizer of the following optimization problems:
The solvability of the polar Orlicz-Minkowski problems is discussed under different conditions. In particular, under certain conditions on , the existence of a solution is proved for a nonzero finite measureon unit sphere which is not concentrated on any hemisphere of.