The Webster Scalar Curvature and Sharp Upper and Lower Bounds for the First Positive Eigenvalue of the Kohn-Laplacian on Real Hypersurfaces  

作　　者：Song Ying LI Duong Ngoc SON 

机构地区：[1]Department of Mathematics, University of California, Irvine, CA 92697-3875 [2]Department of Mathematics, Fujian Normal University, Fujian 350108, P. R. China [3]Fakultiit fiir Mathematik, Universitiit Wien, Oskar-Morgenstern-Platz 1, 1090 Wien, Osterreich

出　　处：《Acta Mathematica Sinica,English Series》2018年第8期1248-1258,共11页数学学报（英文版）

基　　金：supported by the Austrian Science Fund FWF,Pro ject No.I01776

摘　　要：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.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.

关 键 词：CR manifold EIGENVALUE Kohn-Laplacian Webster curvature 

分 类 号：O186.1[理学—数学]