A new characterization of the CR sphere and the sharp eigenvalue estimate for the Kohn Laplacian

Abstract

In Riemannian geometry, the well-known Lichnerowicz-Obata theorem gives a sharp estimate and a characterization of equality case (rigidity theorem) for first positive eigenvalue of Laplacian. In CR geometry, analogous problems are more delicate due to the presence of the torsion of Tanaka-Webster connection. Estimates and rigidity theorems for the first positive eigenvalue of the sub-Laplacian have been studied by many authors (see [11] and the reference therein), e.g., Li and Wang proved an Obata-type theorem for sub-Laplacian without any additional assumption on the torsion [11]. In [6], Chanillo, Chiu and Yang gave a sharp eigenvalue estimate for the Kohn Laplacian on three-dimensional manifolds. This poses the question whether a version of rigidity theorem holds in this case. A partial answer was given by Chang and Wu in 2012. They generalized the eigenvalue estimate to general dimension and proved, under additional assumptions on the torsion, that the equality holds only when the manifold is the CR sphere. In the present paper, we completely resolve the rigidity question for manifolds of dimension at least five by establishing a new characterization of CR sphere in terms of the existence of a (non-trivial) function satisfying a certain

An upper bound for the first positive eigenvalue of the Kohn Laplacian on Reinhardt real hypersurfaces

“…So, although the sphere is Reinhardt, the hypersurfaces of interest here are not diffeomorphic to the sphere. By [6] and [2], the aforementioned estimates for the first eigenvalue of the Kohn Laplacian are not sharp for manifolds satisfying Definition 1.1, and this motivates the present paper.…”

“…The spectrum Spec( b ) in general, and estimates for the first positive eigenvalue (in the embeddable case) in particular, have been studied in several papers. For example, in [3] a Lichnerowicz-type estimate for the first positive eigenvalue in terms of the Webster scalar curvature (or Ricci curvature) was established, while the characterization of the equality case in this estimate was treated in [6] and [2], for the higher-dimensional and the three-dimensional cases respectively. In [5], several upper bounds for the first positive eigenvalue of the Kohn Laplacian on the boundaries of certain domains in C 2 were obtained; the bounds are extrinsic, depending on the realization of the CR manifolds as embedded compact real hypersurfaces in C 2 .…”

The holomorphic sectional curvature and “convex” real hypersurfaces in Kähler manifolds

“…A motivation for studying lower estimates of the scalar curvature is to bound the first positive eigenvalue λ 1 ( b ) of the Kohn Laplacian b := ∂ * b ∂b + ∂b ∂ * b from below. Such a lower bound was first obtained in [3] under the nonnegativity of the CR Paneitz operator (see [3] and [15] for more details on the CR Paneitz operator and [8] for higher-dimensional version of this estimate). Combining Corollary 1.4, Chanillo, Chiu, and Yang's lower bound [3], and the recent result of Takeuchi [15], we obtain Corollary 1.5.…”

CR-analogue of the Siu-∂\overline{∂}-formula and applications to the rigidity problem for pseudo-Hermitian harmonic maps

“…For basic notions in pseudohermitian geometry, we refer the reader to [7,8] or [17], [9], [12] and [11]. Let (M 2m+1 , θ) be a (2m + 1)-dimensional, strictly pseudoconvex pseudo-Hermitian manifold and let (N n , g) be a Kähler manifold.…”

“…(However, our notion of pseudo-Hermitian harmonic maps defined below does not coincide with the notion of pseudoharmonic maps into Riemannian manifolds as defined in [4]). Motivated by these research and our own work with X. Wang [11] on Kohn-Laplacian, we define the notion of pseudo-Hermitian harmonic maps, using Kohn-Laplacian on CR manifolds, and study the rigidity analogous to Siu's strong rigidity.…”