Pure and Applied Mathematics Quarterly

Volume 15 (2019)

Number 4

Special Issue in Honor of Simon Donaldson: Part 2 of 2

Guest Editor: Richard Thomas (Imperial College London)

On the strict convexity of the K-energy

Pages: 983 – 999

DOI: https://dx.doi.org/10.4310/PAMQ.2019.v15.n4.a1


Robert J. Berman (Mathematical Sciences, Chalmers University of Technology and the University of Gothenburg, Sweden)


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.

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

Received 11 November 2017

Published 20 March 2020