On energy functionals, Kähler–Einstein metrics, and the Moser–Trudinger–Onofri neighborhood

Rubinstein, Yanir A.

doi:10.1016/j.jfa.2007.10.009

Cited by 24 publications

(33 citation statements)

References 40 publications

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“…These works are mostly based on the continuity method, the Ricci flow and the J-flow. To cite a few papers from a rapidly growing literature, we mention [16,43,52,19,53,46,69,68] and references therein.…”

Section: Sketch Of the Proofsmentioning

confidence: 99%

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

Darvas

Rubinstein

2016

J. Amer. Math. Soc.

133

237

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Well-known conjectures of Tian predict that existence of canonical Kähler metrics should be equivalent to various notions of properness of Mabuchi's K-energy functional. In some instances this has been verified, especially under restrictive assumptions on the automorphism group. We provide counterexamples to the original conjecture in the presence of continuous automorphisms. 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 versions of Tian's 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 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 Kenergy in the Finsler metric completion. Finally, our results also resolve Tian's conjecture on the Moser-Trudinger inequality for Fano manifolds with Kähler-Einstein metrics.

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Section: Sketch Of the Proofsmentioning

confidence: 99%

“…Examining (46) and recalling (45), one can show the existence of C > 1 such that for any w, v ∈ E 1 with w ≤ v one has…”

Section: Observe Thatmentioning

confidence: 99%

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

Darvas

Rubinstein

2016

J. Amer. Math. Soc.

133

237

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“…We now observe the following monotonicity result. Its proof can be deduced from some of our previous results [11]. Lemma 2.5.…”

Section: The Ricci Iterationmentioning

confidence: 89%

The Ricci iteration and its applications

Rubinstein

2007

Comptes Rendus. Mathématique

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Abstract.In this Note we introduce and study dynamical systems related to the Ricci operator on the space of Kähler metrics as discretizations of certain geometric flows. We pose a conjecture on their convergence towards canonical Kähler metrics and study the case where the first Chern class is negative, zero or positive. This construction has several applications in Kähler geometry, among them an answer to a question of Nadel and a construction of multiplier ideal sheaves. Résumé.Dans cette Note nous introduisons etétudions des systèmes dynamiques reliéesà l'opérateur de Ricci sur l'espace des métriques kählériennes comme discrétisations des certains flots géométriques. Nous posons une conjecture concernant leurs convergence vers des métriques kählériennes canoniques and nousétudions le cas où la première classe de Chern est négative, zéro ou positive. Cette construction a plusieurs applications en géométrie kählérienne, parmi elles une réponseà une question de Nadel et une construction des faisceaux d'idéaux multiplicateurs.

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“…In [65], Rubinstein gave a proof of the Onofri inequality that does not use symmetrization/rearrangement arguments. Also see [66] and in particular [66, Corollary 10.12] which contains a reinforced version of the inequality.…”

Section: A Review Of the Literaturementioning

confidence: 99%

The Moser-Trudinger-Onofri inequality

Dolbeault

Esteban

Jankowiak

2015

Chin. Ann. Math. Ser. B

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This paper is devoted to results on the Moser-Trudinger-Onofri inequality, or the Onofri inequality for brevity. In dimension two this inequality plays a role similar to that of the Sobolev inequality in higher dimensions. After justifying this statement by recovering the Onofri inequality through various limiting procedures and after reviewing some known results, the authors state several elementary remarks.Various new results are also proved in this paper. A proof of the inequality is given by using mass transportation methods (in the radial case), consistently with similar results for Sobolev inequalities. The authors investigate how duality can be used to improve the Onofri inequality, in connection with the logarithmic Hardy-Littlewood-Sobolev inequality. In the framework of fast diffusion equations, it is established that the inequality is an entropy-entropy production inequality, which provides an integral remainder term. Finally, a proof of the inequality based on rigidity methods is given and a related nonlinear flow is introduced.

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On energy functionals, Kähler–Einstein metrics, and the Moser–Trudinger–Onofri neighborhood

Cited by 24 publications

References 40 publications

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

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

The Ricci iteration and its applications

The Moser-Trudinger-Onofri inequality

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