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On sharp lower bounds
for Calabi-type functionals and destabilizing properties of
gradient flows
Mingchen Xia |
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Vol. 14 (2021), No. 6, 1951–1976
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Abstract |
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Let be a compact Kähler manifold with a given ample line bundle . Donaldson proved an inequality between the Calabi energy of a Kähler metric in and the negative of normalized Donaldson–Futaki invariants of test configurations of . 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 and replace the Donaldson–Futaki invariant by the radial Mabuchi K-energy , then a similar bound holds and the bound is indeed sharp. Moreover, we construct explicitly a minimizer of . On a Fano manifold, a similar sharp bound for the Ricci–Calabi energy is also derived. |
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Keywords
destabilizing property, weak Calabi flow, inverse
Monge–Ampère flow, geodesic ray
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Mathematical Subject Classification 2010
Primary: 32Q15, 32U05, 51F99
Secondary: 35K96
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Milestones
Received: 3 October 2019
Revised: 27 February 2020
Accepted: 10 April 2020
Published: 7 September 2021
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Authors |
| Mingchen Xia | |
| Department of Mathematical
Sciences Chalmers Tekniska Högskola Göteborg Sweden |
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