The classical isoperimetric inequality states that a curve in the plane of length L bounds a disk whose area is at most L^2/4\pi. This inequality was generaized to curves in R^3 in the early 1900's. Such a curve bounds an immersed disk whose area is at most L^2/4\pi. It also bounds an embedded surface satisfying the same area bound.

An unknotted curve bounds an embedded disk in R^3. We show, in contrast to the above, that given any positive constant A, there are unknotted smooth curves of length 1 that do not bound embedded disks of area less than A. If we control the size of a tubular neighborhood of a curve then we do get explicit isoperimetric bounds.
(joint work with J. Lagarias and W. Thurston)

In this talk, I will demonstrate several ways
to characterize a pseudoconvex domain to be a ball by using
the potential function of Kahler-Einstein metric, pseudo scalar
curvature. Problems and theorems will be presented in this
talk are related to a conjecture of Yau and CR Yamabe problem.

In this talk, I will discuss the relative $K$-stability and modified $K$-energy associated to the Calabi's extremal metrics on toric manifolds. I will show a sufficient condition in the sense of polyhedrons associated to toric manifolds for both relative $K$-stability and modified $K$-energy. In particular, our result holds for toric Fano manifolds with vanishing Futaki invariant. We also verify our result on toric Fano surfaces.