Vol. 14, No. 6, 2021

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On sharp lower bounds for Calabi-type functionals and destabilizing properties of gradient flows

Mingchen Xia

Vol. 14 (2021), No. 6, 1951–1976

Let X be a compact Kähler manifold with a given ample line bundle L. Donaldson proved an inequality between the Calabi energy of a Kähler metric in c1(L) and the negative of normalized Donaldson–Futaki invariants of test configurations of (X,L). He also conjectured that the bound is sharp.

We prove a metric analogue of Donaldson’s conjecture; we show that if we enlarge the space of test configurations to the space of geodesic rays in 2 and replace the Donaldson–Futaki invariant by the radial Mabuchi K-energy M, then a similar bound holds and the bound is indeed sharp. Moreover, we construct explicitly a minimizer of M. On a Fano manifold, a similar sharp bound for the Ricci–Calabi energy is also derived.

destabilizing property, weak Calabi flow, inverse Monge–Ampère flow, geodesic ray
Mathematical Subject Classification 2010
Primary: 32Q15, 32U05, 51F99
Secondary: 35K96
Received: 3 October 2019
Revised: 27 February 2020
Accepted: 10 April 2020
Published: 7 September 2021
Mingchen Xia
Department of Mathematical Sciences
Chalmers Tekniska Högskola