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

Xia, Mingchen

doi:10.2140/apde.2021.14.1951

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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

DOI: 10.2140/apde.2021.14.1951

Abstract

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 $c_{1} (L)$ 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 $ℰ^{2}$ 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.

Keywords

destabilizing property, weak Calabi flow, inverse Monge–Ampère flow, geodesic ray

Mathematical Subject Classification 2010

Primary: 32Q15, 32U05, 51F99

Secondary: 35K96

Milestones

Received: 3 October 2019

Revised: 27 February 2020

Accepted: 10 April 2020

Published: 7 September 2021

Authors

Mingchen Xia
Department of Mathematical Sciences Chalmers Tekniska Högskola Göteborg Sweden

Vol. 14, No. 6, 2021