On the strict convexity of the K-energy

TL;DR

This paper investigates the convexity properties of the K-energy functional on complex manifolds, showing it is not strictly convex in general but can be under certain conditions, impacting the uniqueness of energy minimizers.

Contribution

It demonstrates the non-strict convexity of the K-energy in general and establishes conditions under which strict convexity and uniqueness of minimizers hold in the toric case.

Findings

K-energy is not strictly convex modulo automorphisms in general.

Strict convexity can hold in the toric case under additional assumptions.

Unique minimizer exists modulo automorphisms when certain positivity conditions are met.

Abstract

Let (X,L) be a polarized projective complex manifold. We show, by a simple toric one-dimensional example, that Mabuchi's K-energy functional on the geodesically complete space of bounded positive (1,1)-forms in the first Chern class of L, endowed with the Mabuchi metric, is not strictly convex modulo automorphisms. However, under some further assumptions the strict convexity in question does hold in the toric case. This leads to a uniqueness result saying that a finite energy minimizer of the K-energy (which exists on any toric polarized manifold (X,L) which is uniformly K-stable) is uniquely determined modulo automorphisms under the assumption that there exists some minimizer with strictly positive curvature current.