On the CR Analogue of Reilly Formula and Yau Eigenvalue Conjecture

TL;DR

This paper develops a CR analogue of Reilly's formula and applies it to estimate the first eigenvalues in CR geometry, advancing understanding of eigenvalue bounds in pseudohermitian manifolds and p-minimal hypersurfaces.

Contribution

It introduces a CR Reilly's formula and derives new eigenvalue estimates for CR Dirichlet problems and embedded p-minimal hypersurfaces, extending classical results to CR geometry.

Findings

First eigenvalue estimate for CR Dirichlet problem in compact pseudohermitian manifolds.

Eigenvalue estimate for tangential sublaplacian on p-minimal hypersurfaces.

Extension of Reilly's formula to CR geometric setting.

Abstract

In this paper, we derive the CR Reilly's formula and its applications to studying of the first eigenvalue estimate for CR Dirichlet eigenvalue problem and embedded p-minimal hypersurfaces. In particular, we obtain the first Dirichlet eigenvalue estimate in a compact pseudohermitian (2n+1)-manifold with boundary and the first eigenvalue estimate of the tangential sublaplacian on closed oriented embedded p-minimal hypersurfaces in a closed pseudohermitian (2n+1)-manifold of vanishing torsion.