On the strict convexity of the K-energy

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 $c_1(L)$, 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.

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This work was supported by grants from the ERC and the KAW foundation.

Received 11 November 2017

Published 20 March 2020