On the strict convexity of the K-energy

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 c1(L), endowed with the Mabuchi-Donaldson-Semmes 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.

Strict convexity of the Mabuchi functional for energy minimizers

“…Finally, we would like to compare our result with a recently result by Berman [3]. He constructed a counter-example of the Conjecture 1.1, when the geodesic has degenerate boundaries.…”