Uniformization of spherical CR manifolds

Jih Hsin Cheng; Hung Lin Chiu; Paul Yang

doi:10.1016/j.aim.2014.01.002

Uniformization of spherical CR manifolds

Jih Hsin Cheng, Hung Lin Chiu, Paul Yang

Mathematics

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

24 Scopus citations

Abstract

Let M be a closed (compact with no boundary) spherical CR manifold of dimension 2n+1. Let M~ be the universal covering of M. Let Φ denote a CR developing mapΦ:M~→S2n+1 where S²ⁿ⁺¹ is the standard unit sphere in complex n+1-space Cⁿ⁺¹. Suppose that the CR Yamabe invariant of M is positive. Then we show that Φ is injective for n≥3. In the case n=2, we also show that Φ is injective under the condition: s(M)<1 where s(M) means the minimum exponent of the integrability of the Green's function for the CR invariant sublaplacian on M~. It then follows that M is uniformizable.

Original language	English (US)
Pages (from-to)	182-216
Number of pages	35
Journal	Advances in Mathematics
Volume	255
DOIs	https://doi.org/10.1016/j.aim.2014.01.002
State	Published - Apr 1 2014

All Science Journal Classification (ASJC) codes

General Mathematics

Keywords

CR developing map
CR sublaplacian
Green's function
Paneitz-like operator
Spherical CR manifold

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10.1016/j.aim.2014.01.002

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title = "Uniformization of spherical CR manifolds",

abstract = "Let M be a closed (compact with no boundary) spherical CR manifold of dimension 2n+1. Let M~ be the universal covering of M. Let Φ denote a CR developing mapΦ:M~→S2n+1 where S2n+1 is the standard unit sphere in complex n+1-space Cn+1. Suppose that the CR Yamabe invariant of M is positive. Then we show that Φ is injective for n≥3. In the case n=2, we also show that Φ is injective under the condition: s(M)<1 where s(M) means the minimum exponent of the integrability of the Green's function for the CR invariant sublaplacian on M~. It then follows that M is uniformizable.",

keywords = "CR developing map, CR sublaplacian, Green's function, Paneitz-like operator, Spherical CR manifold",

author = "Cheng, {Jih Hsin} and Chiu, {Hung Lin} and Paul Yang",

note = "Funding Information: The first author's research was supported in part by NSC 98-2115-M-001-008-MY3 , the second author's research was supported in part by CIZE Foundation and in part by NSC 96-2115-M-008-017-MY3 , and the third author's research was supported in part by DMS-0758601. ",

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doi = "10.1016/j.aim.2014.01.002",

language = "English (US)",

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T1 - Uniformization of spherical CR manifolds

AU - Cheng, Jih Hsin

AU - Chiu, Hung Lin

AU - Yang, Paul

N1 - Funding Information: The first author's research was supported in part by NSC 98-2115-M-001-008-MY3 , the second author's research was supported in part by CIZE Foundation and in part by NSC 96-2115-M-008-017-MY3 , and the third author's research was supported in part by DMS-0758601.

PY - 2014/4/1

Y1 - 2014/4/1

N2 - Let M be a closed (compact with no boundary) spherical CR manifold of dimension 2n+1. Let M~ be the universal covering of M. Let Φ denote a CR developing mapΦ:M~→S2n+1 where S2n+1 is the standard unit sphere in complex n+1-space Cn+1. Suppose that the CR Yamabe invariant of M is positive. Then we show that Φ is injective for n≥3. In the case n=2, we also show that Φ is injective under the condition: s(M)<1 where s(M) means the minimum exponent of the integrability of the Green's function for the CR invariant sublaplacian on M~. It then follows that M is uniformizable.

AB - Let M be a closed (compact with no boundary) spherical CR manifold of dimension 2n+1. Let M~ be the universal covering of M. Let Φ denote a CR developing mapΦ:M~→S2n+1 where S2n+1 is the standard unit sphere in complex n+1-space Cn+1. Suppose that the CR Yamabe invariant of M is positive. Then we show that Φ is injective for n≥3. In the case n=2, we also show that Φ is injective under the condition: s(M)<1 where s(M) means the minimum exponent of the integrability of the Green's function for the CR invariant sublaplacian on M~. It then follows that M is uniformizable.

KW - CR developing map

KW - CR sublaplacian

KW - Green's function

KW - Paneitz-like operator

KW - Spherical CR manifold

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DO - 10.1016/j.aim.2014.01.002

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SN - 0001-8708

VL - 255

SP - 182

EP - 216

JO - Advances in Mathematics

JF - Advances in Mathematics

ER -

Uniformization of spherical CR manifolds

Abstract

All Science Journal Classification (ASJC) codes

Keywords

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