The purpose of this paper is to organize some results on the local geometry of CR singular real-analytic manifolds that are images of CR manifolds via a CR map that is a diffeomorphism onto its image. We find a necessary (sufficient in dimension 2) condition for the diffeomorphism to extend to a finite holomorphic map. The multiplicity of this map is a biholomorphic invariant that is precisely the Moser invariant of the image when it is a Bishop surface with vanishing Bishop invariant. In higher dimensions, we study Levi-flat CR singular images and we prove that the set of CR singular points must be large, and in the case of codimension 2, necessarily Levi-flat or complex. We also show that there exist real-analytic CR functions on such images that satisfy the tangential CR conditions at the singular points, yet fail to extend to holomorphic functions in a neighborhood. We provide many examples to illustrate the phenomena that arise.Comment: 21 pages, accepted to Arkiv for Mathemati

In this paper, we generalize several results about the ∂-complex on the Segal-Bargmann space of C n to weighted Bergman spaces on Hermitian manifolds. We also study in detail the ∂-complex on the unit ball with the complex hyperbolic metric and a non-Kähler metric. The former case turns out to have duality properties similar to the Segal-Bargmann space while the later exhibits a different behavior. We apply these results to solve the ∂-equation on the Bergman spaces in the unit ball of C n with the "exponential" and "standard" weights.

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We derive an explicit formula for the well-known Chern-Moser-Weyl tensor for nondegenerate real hypersurfaces in complex space in terms of their defining functions. The formula is considerably simplified when applying to "pluriharmonic perturbations" of the sphere or to a Fefferman approximate solution to the complex Monge-Ampère equation. As an application, we show that the CR invariant one-form Xα constructed recently by Case and Gover is nontrivial on each real ellipsoid of revolution in C 3 , unless it is equivalent to the sphere. This resolves affirmatively a question posed by these two authors in 2017 regarding the (non-) local CR invariance of the I ′ -pseudohermitian invariant in dimension five and provides a counterexample to a recent conjecture by Hirachi.

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