The Obata sphere theorems on a quaternionic contact manifold of\n dimension bigger than seven

2013/03/02 by Stefan Ivanov, Ivanov, Stefan, Alexander Petkov +3
Mathematics · #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Geometric and Algebraic Topology

paper · pdf · doi:10.48550/arxiv.1303.0409

Abstract

We prove a quaternionic contact versions of the Obata's sphere theorems. We\nshow that if the first positive eigenvalue of the sub-Laplacian on a compact qc\nmanifold of dimension bigger than seven takes the smallest possible value then,\nup to a homothety of the qc structure, the manifold is qc equivalent to the\nstandard 3-Sasakian sphere. We also give a version of the theorem on\nnon-compact qc manifold which is complete with respect to the associated\nRiemannian metric using the existence of a function with traceless horizontal\nHessian. A qc version of the Liouville theorem is shown for qc-conformal maps\nbetween open connected sets of the 3-Sasakian sphere.\n

Related

• The Obata first eigenvalue theorems on a seven dimensional quaternionic\n contact manifold
• The qc Yamabe problem on non-spherical quaternionic contact manifolds
• The Lichnerowicz and Obata first eigenvalue theorems and the Obata uniqueness result in the Yamabe problem on CR and quaternionic contact manifolds
• The sharp lower bound of the first eigenvalue of the sub-Laplacian on a\n quaternionic contact manifold in dimension seven
• Non-umbilical quaternionic contact hypersurfaces in hyper-K "ahler\n manifolds