The Sharp Lower Bound of the First Eigenvalue of the Sub-Laplacian on a Quaternionic Contact Manifold

被引：18

作者：

Ivanov, S. ^{[1
]}

Petkov, A. ^{[1
]}

Vassilev, D. ^{[2
]}

机构：

[1] Univ Sofia, Fac Math & Informat, Sofia 1164, Bulgaria

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

来源：

JOURNAL OF GEOMETRIC ANALYSIS | 2014年 / 24卷 / 02期

关键词：

Sub-Laplacian; First eigenvalue estimate; Quaternionic contact; Bochner formula; POSITIVE EIGENVALUE; SUBLAPLACIAN; REGULARITY; EQUATIONS;

D O I：

10.1007/s12220-012-9354-9

中图分类号：

O1 [数学];

学科分类号：

0701 ; 070101 ;

摘要：

The main technical result of the paper is a Bochner type formula for the sub-Laplacian on a quaternionic contact manifold. With the help of this formula we establish a version of Lichnerowicz's theorem giving a lower bound of the eigenvalues of the sub-Laplacian under a lower bound on the Sp(n)Sp(1) components of the qc-Ricci curvature. It is shown that in the case of a 3-Sasakian manifold the lower bound is reached iff the quaternionic contact manifold is a round 3-Sasakian sphere. Another goal of the paper is to establish a priori estimates for square integrals of horizontal derivatives of smooth compactly supported functions. As an application, we prove a sharp inequality bounding the horizontal Hessian of a function by its sub-Laplacian on the quaternionic Heisenberg group.

引用

页码：756 / 778

页数：23

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