The Obata First Eigenvalue Theorem on a Seven-Dimensional Quaternionic Contact Manifold

被引：0

作者：

Mohamed, Abdelrahman ^{[1
]}

Vassilev, Dimiter ^{[2
]}

机构：

[1] Univ North Alabama, Dept Math, Florence, AL 35630 USA

[2] Univ New Mexico, Dept Math & Stat, Albuquerque, NM 87131 USA

来源：

JOURNAL OF GEOMETRIC ANALYSIS | 2023年 / 33卷 / 01期

关键词：

Sub-Riemannian geometry; CR and quaternionic contact structures; Sobolev inequality; Yamabe equation; Lichnerowicz eigenvalue estimate; Obata theorem; SUB-LAPLACIAN; YAMABE PROBLEM; CR; SPHERE;

D O I：

10.1007/s12220-022-01072-1

中图分类号：

O1 [数学];

学科分类号：

0701 ; 070101 ;

摘要：

We prove an Obata type rigidity result for the first eigenvalue of the sub-Laplacian on a compact seven-dimensional quaternionic contact manifold which satisfies a Lichnerowicz-type bound on its quaternionic contact Ricci curvature and has a non-negative Paneitz P-function. In particular, under the stated conditions, the lowest possible eigenvalue of the sub-Laplacian is achieved if and only if the manifold is qc-equivalent to the standard 3-Sasakian sphere.

引用

页数：30

共 14 条

[1] The Obata First Eigenvalue Theorem on a Seven-Dimensional Quaternionic Contact Manifold
Abdelrahman Mohamed
Dimiter Vassilev
The Journal of Geometric Analysis, 2023, 33
[2] On seven-dimensional quaternionic contact solvable Lie groups
Conti, Diego
Fernandez, Marisa
Santisteban, Jose A.
FORUM MATHEMATICUM, 2014, 26 (02) : 547 - 576
[3] The Obata sphere theorems on a quaternionic contact manifold of dimension bigger than seven
Ivanov, Stefan
Petkov, Alexander
Vassilev, Dimiter
JOURNAL OF SPECTRAL THEORY, 2017, 7 (04) : 1119 - 1170
[4] Extremals for the Sobolev inequality on the seven-dimensional quaternionic Heisenberg group and the quaternionic contact Yamabe problem
Ivanov, Stefan
Minchev, Ivan
Vassilev, Dimiter
JOURNAL OF THE EUROPEAN MATHEMATICAL SOCIETY, 2010, 12 (04) : 1041 - 1067
[5] The Lichnerowicz and Obata first eigenvalue theorems and the Obata uniqueness result in the Yamabe problem on CR and quaternionic contact manifolds
Ivanov, Stefan
Vassilev, Dimiter
NONLINEAR ANALYSIS-THEORY METHODS & APPLICATIONS, 2015, 126 : 262 - 323
[6] The sharp lower bound of the first eigenvalue of the sub-Laplacian on a quaternionic contact manifold in dimension seven
Ivanov, S.
Petkov, A.
Vassilev, D.
NONLINEAR ANALYSIS-THEORY METHODS & APPLICATIONS, 2013, 93 : 51 - 61
[7] The Sharp Lower Bound of the First Eigenvalue of the Sub-Laplacian on a Quaternionic Contact Manifold
S. Ivanov
A. Petkov
D. Vassilev
The Journal of Geometric Analysis, 2014, 24 : 756 - 778
[8] The Sharp Lower Bound of the First Eigenvalue of the Sub-Laplacian on a Quaternionic Contact Manifold
Ivanov, S.
Petkov, A.
Vassilev, D.
JOURNAL OF GEOMETRIC ANALYSIS, 2014, 24 (02) : 756 - 778
[9] AN OBATA-TYPE THEOREM ON A THREE-DIMENSIONAL CR MANIFOLD
Ivanov, S.
Vassilev, D.
GLASGOW MATHEMATICAL JOURNAL, 2014, 56 (02) : 283 - 294
[10] Yang-Mills instantons on seven-dimensional Manifold of G2 holonomy
Miyagi, S
MODERN PHYSICS LETTERS A, 1999, 14 (37) : 2595 - 2604

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