Uniform K-stability and asymptotics of energy functionals in K\"ahler geometry

TL;DR

This paper establishes a link between the asymptotic behavior of energy functionals in Kähler geometry and algebraic stability notions, showing that coercivity of the Mabuchi functional implies uniform K-stability.

Contribution

It proves that the slope at infinity of common Kähler functionals equals their non-Archimedean counterparts, connecting geometric analysis with algebraic stability.

Findings

Slope at infinity of functionals equals non-Archimedean evaluation

Coercivity of Mabuchi functional implies uniform K-stability

Asymptotic analysis bridges differential geometry and algebraic stability

Abstract

Consider a polarized complex manifold (X,L) and a ray of positive metrics on L defined by a positive metric on a test configuration for (X,L). For most of the common functionals in K\"ahler geometry, we prove that the slope at infinity along the ray is given by evaluating the non-Archimedean version of the functional (as defined in our earlier paper) at the non-Archimedean metric on L defined by the test configuration. Using this asymptotic result, we show that coercivity of the Mabuchi functional implies uniform K-stability.