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.