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Journal of the American Mathematical Society

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

   
 

 

Tian's properness conjectures and Finsler geometry of the space of Kähler metrics


Authors: Tamás Darvas and Yanir A. 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
Published electronically: December 8, 2016
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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
Email: tdarvas@math.umd.edu

Yanir A. Rubinstein
Affiliation: Department of Mathematics, University of Maryland, 4176 Campus Drive, College Park, Maryland 20742-4015
Email: yanir@umd.edu

DOI: https://doi.org/10.1090/jams/873
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.
Article copyright: © Copyright 2016 Tamás Darvas and Yanir Rubinstein