Lower bounds on the Calabi functional

TL;DR

This paper establishes new lower bounds on the Calabi functional for Kähler manifolds using test configurations and asymptotic analysis, drawing parallels with Yang-Mills theory on Riemann surfaces.

Contribution

It introduces a novel method to derive lower bounds on the Calabi functional via test configurations and asymptotic expansions, extending previous geometric analysis techniques.

Findings

Test configurations provide effective lower bounds on the L^2 norm of scalar curvature.

The approach uses the Tian-Zelditch-Lu expansion for asymptotic approximation.

The results draw an analogy with Yang-Mills functional analysis on Riemann surfaces.

Abstract

The main result of this paper shows that "test configurations" give new lower bounds on the $L^{2}$ norm of the scalar curvature on a Kahler manifold. This is closely analogous to the analysis of the Yang-Mills functional over Riemann surfaces by Atiyah and Bott. The proof uses asymptotic approximation by finite-dimensional problems: the essential ingredient being the Tian-Zelditch-Lu expansion of the "density of states" function.