Fundamental groups of 3-manifolds are known to satisfy strong
properties, and in recent years there have been several advances in their
study. In this talk I will discuss how some of these properties can be
exploited to give us insight (and results) in the study of 4-manifolds.

In 2001, Donaldson proved that the existence of cscK metrics on a
polarized manifold (X,L) with discrete automorphism group implies the
existence of balanced metrics on L^k for k large enough. We show that the
similar statement holds if one twists the line bundle L with a simple
stable vector bundle E. More precisely we show that if E is a simple
stable bundle over a polarized manifold (X,L), (X,L) admits cscK metric
and have discrete automorphism group, then (PE^*, \O(d) \otimes L^k)
admits a balanced metric for k large enough.

Recently, using the desingularization technique, a new family of complete properly embedded self-shrinkers asymptotic to cones in three dimensional Euclidean space has been constructed by Kapouleas-Kleene-Moeller and independently by Nguyen.

In this talk, we present the uniqueness of self-shrinking ends asymptotic to any given cone in general Euclidean space. The feature of our uniqueness result is that we do not require the control on the boundaries of self-shrinking ends or the rate of convergence to cones at infinity. As applications, we show that, there do not exist complete properly embedded self-shrinkers other than hyperplanes having ends asymptotic to rotationally symmetric cones.

A smooth metric space is a Riemanian manifold together with a weighted volume. It is naturally associated with a weighted Laplacian. In this talk, I will discuss some recent results about function theoretic and spectral propeties of the weighted Laplacian and volume estimates for the volume and weighted volume. The results can be applied to study the shrinking gradient Ricci solitons and self-shrinker for mean curvature flows.