The sharp lower bound for the first positive eigenvalue of a sub-Laplacian on a pseudo-Hermitian manifold

Abstract

Abstract. This paper studies, using the Bochner technique, a sharp lower bound of the first eigenvalue of a subelliptic Laplace operator on a strongly pseudoconvex CR manifold in terms of its pseudo-Hermitian geometry. For dimensions greater than or equal to 7, the lower bound under a condition on the Ricci curvature and the torsion was obtained by Greenleaf. We give a proof for all dimensions greater than or equal to 5. For dimension 3, the sharp lower bound is proved under a condition which also involves a distinguished covariant derivative of the torsion.

Curvature Dimension Inequalities and Subelliptic Heat Kernel Gradient Bounds on Contact Manifolds

“…This result can be seen as a Lichnerowicz type estimate on contact Riemannian manifolds. We should mention that such estimates have already been obtained on CR manifolds (see for instance ( [5], [14], [16], [23]) and that our result is not sharp since the lower bound does not involve the dimension of the manifold (in the Riemannian case our bound writes λ 1 ≥ ρ where ρ is a lower bound on the Ricci curvature). However, the main point here, is that we do not assume the compactness of the manifold.…”

“…Bochner's type formulas on CR manfolds have been extensively studied in the literature (see for instance [5], [12], [14], [16], [18], [23]). The horizontal Bochner's formula we obtain in Theorem 3.3 is an extension of the CR Bochner fomula of the above mentioned works since we work in the more general framework of an abritary Riemannian contact manifold for which Tanno's tensor is not necessary zero.…”

The Bochner-type formula and the first eigenvalue of the sub-Laplacian on a contact Riemannian manifold

“…But due to a mistake in calculation pointed out in [6] and [12], the coefficient n+1 2 in (1.1) was mistaken to be n 2 . The corresponding results for n = 2 and n = 1 were obtained later in [18] and [8], respectively. The CR Obata-type theorem was conjectured in [6], which states that if nκ/(n + 1) is an eigenvalue of the sub-Laplacian on a pseudohermitian manifold, then it is the standard CR structure on the unit sphere in C n+1 .…”

The Lichnerowicz–Obata Theorem on Sub-Riemannian Manifolds with Transverse Symmetries

“…In that case, the story goes back at least to the work by Greenleaf [12] which has seen, since then, several improvements and variations. We mention in particular the works by Aribi-Dragomir-El Soufi [1], Barletta [2], Baudoin-Wang [7], Ivanov-Petkov-Vassilev [18,19], Li [21] and Li-Luk [22]. Some optimal lower bounds for the first eigenvalue of sub-Laplacians also have been obtained in the context of quaternionic contact manifolds by Ivanov-Petkov-Vassilev [15,16,17].…”