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Register a new userCR-invariance of the Steinness index
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Authors
Jihun Yum
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We characterize the Diederich-Fornaess index and the Steinness index in terms of a special 1-form, which we call DAngelo 1-form. We then prove that the Diederich-Fornaess and Steinness indices are invariant under CR-diffeomorphisms by showing CR-invariance of DAngelo 1-forms.
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We introduce the concept of Steinness index related to the Stein neighborhood basis. We then show several results: (1) The existence of Steinness index is equivalent to that of strong Stein neighborhood basis. (2) On the Diederich-Forn{ae}ss worm domains in particular, we present an explicit formula relating the Steinness index to the well-known Diederich-Forn{ae}ss index. (3) The Steinness index is 1 if a smoothly bounded pseudoconvex domain admits finitely many boundary points of infinite type.
We propose the concept of Diederich--Forn{ae}ss and Steinness indices on compact pseudoconvex CR manifolds of hypersurface type in terms of the DAngelo 1-form. When the CR manifold bounds a domain in a complex manifold, under certain additional non-degeneracy condition, those indices are shown to coincide with the original Diederich--Forn{ae}ss and Steinness indices of the domain, and CR invariance of the original indices follows.
In this paper, we prove the semi-continuity theorem of Diederich-Forn{ae}ss index and Steinness index under a smooth deformation of pseudoconvex domains in Stein manifolds.
In this paper we characterize sums of CR functions from competing CR structures in two scenarios. In one scenario the structures are conjugate and we are adding to the theory of pluriharmonic boundary values. In the second scenario the structures are related by projective duality considerations. In both cases we provide explicit vector field-based characterizations for two-dimensional circular domains satisfying natural convexity conditions.
A smooth, strongly $mathbb{C}$-convex, real hypersurface $S$ in $mathbb{CP}^n$ admits a projective dual CR structure in addition to the standard CR structure. Given a smooth function $u$ on $S$, we provide characterizations for when $u$ can be decomposed as a sum of a CR function and a dual CR function. Following work of Lee on pluriharmonic boundary values, we provide a characterization using differential forms. We further provide a characterization using tangential vector fields in the style of Audibert and Bedford.
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