Extremal metrics for the Q0-curvature in three dimensions

Jeffrey S. Case; Chin Yu Hsiao; Paul Yang

doi:10.4171/JEMS/845

Extremal metrics for the Q⁰-curvature in three dimensions

Jeffrey S. Case, Chin Yu Hsiao, Paul Yang

Mathematics

Research output: Contribution to journal › Article › peer-review

Abstract

We construct contact forms with constant Q⁰-curvature on compact three-dimensional CR manifolds which admit a pseudo-Einstein contact form and satisfy some natural positivity conditions. These contact forms are obtained by minimizing the CR analogue of the II-functional from conformal geometry. Two crucial steps are to show that the P⁰-operator can be regarded as an elliptic pseudodifferential operator and to compute the leading order terms of the asymptotic expansion of the Green function for P⁰

Original language	English (US)
Pages (from-to)	585-626
Number of pages	42
Journal	Journal of the European Mathematical Society
Volume	21
Issue number	2
DOIs	https://doi.org/10.4171/JEMS/845
State	Published - 2019

All Science Journal Classification (ASJC) codes

Mathematics(all)
Applied Mathematics

Keywords

Beckner–Onofri inequality
CR manifolds
Green function asymptotics
Pseudo-Einstein structures
Pseudodifferential operator
Q-prime curvature

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10.4171/JEMS/845

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@article{dcddc4a96bb0435e9235b1436843c062,

title = "Extremal metrics for the Q0-curvature in three dimensions",

abstract = "We construct contact forms with constant Q0-curvature on compact three-dimensional CR manifolds which admit a pseudo-Einstein contact form and satisfy some natural positivity conditions. These contact forms are obtained by minimizing the CR analogue of the II-functional from conformal geometry. Two crucial steps are to show that the P0-operator can be regarded as an elliptic pseudodifferential operator and to compute the leading order terms of the asymptotic expansion of the Green function for P0 ",

keywords = "Beckner–Onofri inequality, CR manifolds, Green function asymptotics, Pseudo-Einstein structures, Pseudodifferential operator, Q-prime curvature",

author = "Case, {Jeffrey S.} and Hsiao, {Chin Yu} and Paul Yang",

note = "Funding Information: The authors thank the Academia Sinica in Taipei and Princeton University for warm hospitality and generous support while this work was being completed. They also thank Po-Lam Yung for his careful reading and the anonymous referees for their helpful comments on early versions of this article. JSC was partially supported by NSF Grant DMS-1004394 and a grant from the Simons Foundation (Grant No. 524601). CYH was supported by Taiwan Ministry of Science of Technology project 103-2115-M-001-001, 104-2628-M-001-003-MY2 and the Golden-Jade fellowship of Kenda Foundation. PY was partially supported by NSF Grant DMS-1509505. Publisher Copyright: {\textcopyright} European Mathematical Society 2019",

year = "2019",

doi = "10.4171/JEMS/845",

language = "English (US)",

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pages = "585--626",

journal = "Journal of the European Mathematical Society",

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T1 - Extremal metrics for the Q0-curvature in three dimensions

AU - Case, Jeffrey S.

AU - Hsiao, Chin Yu

AU - Yang, Paul

N1 - Funding Information: The authors thank the Academia Sinica in Taipei and Princeton University for warm hospitality and generous support while this work was being completed. They also thank Po-Lam Yung for his careful reading and the anonymous referees for their helpful comments on early versions of this article. JSC was partially supported by NSF Grant DMS-1004394 and a grant from the Simons Foundation (Grant No. 524601). CYH was supported by Taiwan Ministry of Science of Technology project 103-2115-M-001-001, 104-2628-M-001-003-MY2 and the Golden-Jade fellowship of Kenda Foundation. PY was partially supported by NSF Grant DMS-1509505. Publisher Copyright: © European Mathematical Society 2019

PY - 2019

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AB - We construct contact forms with constant Q0-curvature on compact three-dimensional CR manifolds which admit a pseudo-Einstein contact form and satisfy some natural positivity conditions. These contact forms are obtained by minimizing the CR analogue of the II-functional from conformal geometry. Two crucial steps are to show that the P0-operator can be regarded as an elliptic pseudodifferential operator and to compute the leading order terms of the asymptotic expansion of the Green function for P0

KW - Beckner–Onofri inequality

KW - CR manifolds

KW - Green function asymptotics

KW - Pseudo-Einstein structures

KW - Pseudodifferential operator

KW - Q-prime curvature

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