Let （M, θ） be a compact strictly pseudoconvex pseudonermitian manifold winch is CR embedded into a complex space. In an earlier paper, Lin and the authors gave several sharp upper bounds for the first positive eige...Let （M, θ） be a compact strictly pseudoconvex pseudonermitian manifold winch is CR embedded into a complex space. In an earlier paper, Lin and the authors gave several sharp upper bounds for the first positive eigenvalue λ1 of the Kohn-Laplacian □b on （M, θ）. In the present paper, we give a sharp upper bound for λ1, generalizing and extending some previous results. As a corollary, we obtain a Reilly-type estimate when M is embedded into the standard sphere. In another direction, using a Lichnerowicz-type estimate by Chanillo, Chiu, and Yang and an explicit formula for the Webster scalar curvature, we give a lower bound for λ1 when the pseudohermitian structure θ is volume-normalized.展开更多
基金supported by the Austrian Science Fund FWF,Pro ject No.I01776
文摘Let （M, θ） be a compact strictly pseudoconvex pseudonermitian manifold winch is CR embedded into a complex space. In an earlier paper, Lin and the authors gave several sharp upper bounds for the first positive eigenvalue λ1 of the Kohn-Laplacian □b on （M, θ）. In the present paper, we give a sharp upper bound for λ1, generalizing and extending some previous results. As a corollary, we obtain a Reilly-type estimate when M is embedded into the standard sphere. In another direction, using a Lichnerowicz-type estimate by Chanillo, Chiu, and Yang and an explicit formula for the Webster scalar curvature, we give a lower bound for λ1 when the pseudohermitian structure θ is volume-normalized.
