Infimum of the spectrum of Laplace–Beltrami operator on a bounded pseudoconvex domain with a Kähler metric of Bergman type

Plurisubharmonicity for the solution of the Fefferman equation and applications

“…since D is strictly super-pseudoconvex, there is a strictly plurisubharmonic function [19] and [20]. Therefore, the proof of Theorem 1.3 is complete.…”

“…Then the problem of estimating the upper bound and lower bound for λ 1 have studied by many authors, including Cheng [4], Lee [9], Li and Wang [12,13], Munteanu [22], Li and Tran [19] and Li and Wang [20], Wang [24], etc... When the Ricci curvature is super Einstein: R i j ≥ −(n + 1)g i j , Munteanu [22] proves that λ 1 ≤ n 2 .…”

“…When the Ricci curvature is super Einstein: R i j ≥ −(n + 1)g i j , Munteanu [22] proves that λ 1 ≤ n 2 . For the lower bound estimate of λ 1 , Li and Tran [19] and Li and Wang [20] consider a smoothly bounded pseudoconvex domain in C n with defining function r ∈ C 4 (D) such that u =: − log(−r ) is strictly plurisubharmonic in D. When r is plurisubharmonic in D, Li and Tran [19] prove that λ 1 = n 2 . When g [u] is super asymptotic Einstein and det H (r ) ≥ 0 on ∂ D, Li and Wang [20] prove λ 1 = n 2 .…”

“…Many researches [14,15,19,20] indicate that the following problem is very interesting and very important. It is well known that ρ(z) = |z| 2 − 1 is strictly plurisubharmonic when D = B n , the unit ball in C n .…”

Bottom of Spectrum of Kähler Manifolds with a Strongly Pseudoconvex Boundary

“…It should be noted that we use different normalization here from ones appeared in [LW2,M,LT,Li3]. Munteanu [M] proved the following improved estimate for Kähler manifolds.…”