Plurisubharmonicity for the solution of the Fefferman equation and applications

Abstract

In this paper, the author introduces a concept of the super-pseudoconvex domain. He proves that the solution of the Fefferman equation on a smoothly bounded strictly pseudoconvex domain D in C n is plurisubharmonic in D if and only if D is super-pseudoconvex. As an application, when D is super-pseudoconvex, he gives the sharp lower bound for the bottom of the spectrum of the Laplace-Beltrami operators by using the result of Li and Wang (Int. Math. Res. Not. 2012).

The Webster scalar curvature and sharp upper and lower bounds for the first positive eigenvalue of the Kohn-Laplacian on real hypersurfaces

“…Then J[ρ] = 1 + O(ρ 2 ) and B(z) = 0 on M (see [11]). The unique volume-normalized structure is given by Θ This agrees with Hammond's result, except that the constant is different from the one in [6] due to a different normalization.…”

“…Following Li [11], we say that M is super-pseudoconvex if the Webster scalar curvature R Θ of Θ is positive. This condition is equivalent to the fact that the approximate solution to the Fefferman equation is strictly plurisubharmonic near M (see [11]). For any defining function ρ of M , we define…”

On the classification of normal Stein spaces and finite ball quotients with Bergman-Einstein metrics

“…Cheng [8] in 1979 asserts that the Bergman metric of a bounded, strongly pseudoconvex domain in C n with smooth boundary is Kähler-Einstein if and only if the domain is biholomorphic to the unit ball B n . There are also variations of this conjecture in terms of other canonical metrics; see, e.g., Li [23,24,25] and references therein.…”