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Journal of the American Mathematical Society

Published by the American Mathematical Society, the Journal of the American Mathematical Society (JAMS) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6834 (online) ISSN 0894-0347 (print)

The 2020 MCQ for Journal of the American Mathematical Society is 4.79.

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Tian’s properness conjectures and Finsler geometry of the space of Kähler metrics
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by Tamás Darvas and Yanir A. Rubinstein PDF
J. Amer. Math. Soc. 30 (2017), 347-387

Abstract:

Well-known conjectures of Tian predict that the existence of canonical Kähler metrics should be equivalent to various notions of properness of Mabuchi’s K-energy functional. First, we provide counterexamples to Tian’s first conjecture in the presence of continuous automorphisms. Second, we resolve Tian’s second conjecture, confirming the Moser–Trudinger inequality for Fano manifolds. The construction hinges upon an alternative approach to properness that uses in an essential way the metric completion with respect to a Finsler metric and its quotients with respect to group actions. This approach also allows us to formulate and prove new optimal replacements for Tian’s first conjecture in the setting of smooth and singular Kähler–Einstein metrics, with or without automorphisms, as well as for Kähler–Ricci solitons. Moreover, we reduce both Tian’s original first conjecture (in the absence of automorphisms) and our modification of it (in the presence of automorphisms) in the general case of constant scalar curvature metrics to a conjecture on regularity of minimizers of the K-energy in the Finsler metric completion.
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Additional Information
  • Tamás Darvas
  • Affiliation: Department of Mathematics, University of Maryland, 4176 Campus Drive, College Park, Maryland 20742-4015
  • MR Author ID: 1016588
  • Email: tdarvas@math.umd.edu
  • Yanir A. Rubinstein
  • Affiliation: Department of Mathematics, University of Maryland, 4176 Campus Drive, College Park, Maryland 20742-4015
  • MR Author ID: 795645
  • Email: yanir@umd.edu
  • Received by editor(s): August 5, 2015
  • Published electronically: December 8, 2016
  • Additional Notes: The authors were supported by BSF grant 2012236, NSF grants DMS-1206284 and 1515703, and a Sloan Research Fellowship.
  • © Copyright 2016 Tamás Darvas and Yanir Rubinstein
  • Journal: J. Amer. Math. Soc. 30 (2017), 347-387
  • MSC (2010): Primary 32Q20, 58E11; Secondary 14J50, 32W20, 32U05
  • DOI: https://doi.org/10.1090/jams/873
  • MathSciNet review: 3600039