# Zhuang-Dan Guan UC Riverside

## Extremal metrics and geodesic stability principles

Similar to the situation of the existence of Hermitian-Einstein metrics
on vector bundles, one might expect a relation between the existence of
K\"ahler-Einstein metrics and certain stability on a K\"ahler manifolds.
With many examples at hand we should see that there is a relation between
the existence of an extremal metric(following Calabi)
in a K\"ahler class and the geodesic stability of the given K\"ahler class. The uniqueness is always expected.