Stable Pairs and Coercive Estimates for The Mabuchi Functional

TL;DR

This paper establishes a deep connection between the stability of projective manifolds and the properness of the Mabuchi energy, linking geometric stability to energy functionals in complex geometry.

Contribution

It proves that a projective manifold's stability is equivalent to the properness of the Mabuchi energy and that stability implies a finite automorphism group.

Findings

Stability iff Mabuchi energy is proper

Stability implies finite automorphism group

Provides criteria for geometric stability

Abstract

We show that a projective manifold is stable if and only if the Mabuchi energy is proper on the space of algebraic metrics. We show that stability implies finite automorphism group.