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

Abstract

We prove quaternionic contact versions of two of Obata's sphere theorems. On a compact quaternionic contact (qc) manifold of dimension bigger than seven and satisfying a Lichnerowicz type lower bound estimate we show that if the first positive eigenvalue of the sub-Laplacian takes the smallest possible value then, up to a homothety of the qc structure, the manifold is qc equivalent to the standard 3-Sasakian sphere. The same conclusion is shown to hold on a non-compact qc manifold which is complete with respect to the associated Riemannian metric assuming the existence of a function with traceless horizontal Hessian. The third result of the paper is a qc version of Liouville's theorem showing that a qc-conformal diffeomorphism between open connected sets of the 3-Sasakian sphere is a restriction of an element of the qc-conformal automorphism group of the sphere.

Convexity of the entropy of positive solutions to the heat equation on quaternionic contact and CR manifolds

“…We conclude by mentioning another proof of the monotonicity of the energy in the recent preprint [10], which was the result of a past collaborative work with Ivanov and Petkov. Remarkably, [3] is also not acknowledged in [10] despite the fact that the calculations in [10] came after I introduced to Ivanov many of the interesting (sub-Riemannian) comparison problems and drew their attention to [3].…”

“…Remarkably, [3] is also not acknowledged in [10] despite the fact that the calculations in [10] came after I introduced to Ivanov many of the interesting (sub-Riemannian) comparison problems and drew their attention to [3]. While I can hardly wish to be associated with [10], a quick look shows the line for line substantial overlap of [10, Section 3] with Chang and Wu' proof [3,Lemma 3.3], the publication of collaborative work without a discussion with all sides is notable. Therefore, I decided to give my independent approach to the problem.…”

Non-Umbilical Quaternionic Contact Hypersurfaces in Hyper-Kähler Manifolds

“…Let us remark that the final step of the proof is similar to an argumentation that had been already used before in the proof of [10,Theorem 1.3].…”

“…By a qc-homothety, depending on the sign of the qc-scalar curvature, we can reduce the claim to one of the cases S = 2, S = 0 or S = −2. We recall that the model spaces (1.1) are qc-Einstein qc-conformally flat manifolds with positive qc-scalar curvature S = 2 in the case i) of the 3-Sasakian sphere [7,10], flat in the case of the quaternionic Heisenberg group iii) [7], and negative qc-scalar curvature S = −2, [9], for the hyperboloid ii).…”

The Obata first eigenvalue theorems on a seven dimensional quaternionic contact manifold

“…In fact, on the 3-Sasakian sphere the eigenspace of the sub-Laplacian with eigenvalue 4n is given by the restrictions to the sphere of all linear functions in Euclidean space. The rigidity result when the dimension of the qc manifold is at least eleven, i.e., the Obata type theorem characterizing the 3-Sasakian sphere as the only case in which the lowest possible eigenvalue is achieved was proven in [17]. In fact, [17] established a general result valid on any complete with respect to the associated Riemannian metric qc manifold, characterizing the 3-Sasakian sphere of dimension at least eleven through the existence of an eigenfunction whose Hessian with respect to the Biquard connection [3] is in the space generated by the metric and fundamental 2-forms of the quaternionic contact structure.…”