On the existence of holomorphic embeddings of strictly pseudoconvex algebraic hypersurfaces into spheres

We show that there are strictly pseudoconvex, real algebraic hypersurfaces in \bC^{n+1} that cannot be locally embedded into a sphere in \bC^{N+1} for any $N$. In fact, we show that there are strictly pseudoconvex, real algebraic hypersurfaces in \bC^{n+1} that cannot be locally embedded into any compact, strictly pseudoconvex, real algebraic hypersurface

Eigenvalues of the Kohn Laplacian and deformations of pseudohermitian structures on compact embedded strictly pseudoconvex CR manifolds

We study the eigenvalues of the Kohn Laplacian on a closed embedded strictly pseudoconvex CR manifold as functionals on the set of positive oriented contact forms $\mathcal{P}_+$. We show that the functionals are continuous with respect to a natural topology on $\mathcal{P}_+$. Using a simple adaptation of the standard Kato-Rellich perturbation theory, we prove that the functionals are (one-sided) differentiable along 1-parameter analytic deformations. We use this differentiability to define the notion of critical contact forms, in a generalized sense, for the functionals. We give a necessary (also sufficient in some situations) condition for a contact form to be critical. Finally, we present explicit examples of critical contact form on both homogeneous and non-homogeneous CR manifolds.Comment: 19 pages. Comments are welcom

Remarks on subharmonic envelopes

We prove that the subharmonic envelope of a lower semicontinuous function on Ω is harmonic on a certain open subset of Ω, using a very classical method in potential theory. The result gives simple proofs of theorems on harmonic measures and Jensen measures obtained by Cole and Ransford

Semi-isometric CR immersions of CR manifolds into K\"ahler manifolds and applications

We study the second fundamental form of semi-isometric CR immersions from strictly pseudoconvex CR manifolds into K\"ahler manifolds. As an application, we give a precise condition for the CR umbilicality of real hypersurfaces, extending an well-known theorem by Webster on the nonexistence of CR umbilical points on generic real ellipsoids. As other applications, we extend the linearity theorem of Ji-Yuan for CR immersions into spheres with vanishing second fundamental form to the important case of three-dimensional manifolds, and prove the ``first gap'' theorem in the spirit of Webster, Faran, Cima-Suffridge, and Huang for semi-isometric CR immersions into a complex euclidean space of ``low'' codimension. Our new approach to the linearity theorem is based on the study of the first positive eigenvalue of the Kohn Laplacian.Comment: Final version, accepted in Ann. Sc. Norm. Super. Pisa Cl. Sci. (5

The holomorphic sectional curvature and "convex" real hypersurfaces in K\"ahler manifolds

We prove a sharp lower bound for the Tanaka-Webster holomorphic sectional curvature of strictly pseudoconvex real hypersurfaces that are "semi-isometrically" immersed in a K\"ahler manifold of nonnegative holomorphic sectional curvature under an appropriate convexity condition. This gives a partial answer to a question posed by Chanillo, Chiu, and Yang regarding the positivity of the Tanaka-Webster scalar curvature of the boundary of a strictly convex domain in $\mathbb{C}^2$ from 2012. In fact, the main result proves a stronger positivity property, namely the $\frac12$-positivity in the sense of Cao, Chang, and Chen, for compact "convex" real hypersurfaces in a K\"ahler manifold of nonnegative holomorphic sectional curvature. Our approach is rather simple and uses a version of the Gauss equation for semi-isometric CR immersions of pseudohermitian manifolds into K\"ahler manifolds.Comment: 20 pages, accepted in Colloquium Mathematicu