The intrinsic geometry on bounded pseudoconvex domains

Liu, Bingyuan

doi:10.48550/arxiv.1610.06530

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2019

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“…In the 2000s, Fornaess-Herbig (see [10] and [11]) proved that the domain is of trivial index if it admits a plurisubharmonic defining function on the boundary. Recently, Krantz-Liu-Peloso [20] improved Fornaess-Herbig's condition in C 2 and exhibited more pseudoconvex domains of trivial index (see also [23]). Just a while ago, an index associated to boundary was found by the author [22].…”

Section: Introductionmentioning

confidence: 99%

The Diederich–Fornæss index II: For domains of trivial index

Liu¹

2019

Advances in Mathematics

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We study bounded pseudoconvex domains in complex Euclidean spaces. We find analytical necessary conditions and geometric sufficient conditions for a domain being of trivial Diederich-Fornaess index (i.e. the index equals to 1). We also connect a differential equation to the index. This reveals how a topological condition affects the solution of the associated differential equation and consequently obstructs the index being trivial. The proofs relies on a new method of study of the complex geometry of the boundary. The method was motivated by geometric analysis of Riemannian manifolds. We also generalize our main theorems under the context of de Rham cohomology.

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