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Mei-Chi Shaw

Publications and source records attributed to Mei-Chi Shaw.

18 recordsLinked to original sources
arXiv2025

$L^2$-Sobolev Theory for $\bar\partial$ on Domains in $\Bbb {CP}^n$

In this article, we study the range of the Cauchy-Riemann operator $\bar\partial$ on domains in the complex projective space $\Bbb{CP}^n$. In particular, we show that $\bar\partial$ does not have closed range in $L^2$ for (2,1)-forms on the Hartogs triangle in $\Bbb{CP}^2$. We also study the $\bar\partial$-Cauchy problem on pseudoconvex domains and use it to prove the Sobolev estimates for $\bar\partial$ on pseudoconcave domains in $\Bbb{CP}^n$.

math.CV↗
arXiv2022

Extendability and the $\overline \partial$ Operator on the Hartogs Triangle

In this paper it is shown that the Hartogs triangle $\mathbf T$ in $\mathbf C^2$ is a uniform domain. This implies that the Hartogs triangle is a Sobolev extension domain. Furthermore, the weak and strong maximal extensions of the Cauchy-Riemann operator agree on the Hartogs triangle. These results have numerous applications. Among other things, they are used to study the Dolbeault cohomology groups with Sobolev coefficients on the complement of $\mathbf T$.

math.CV↗
arXiv2020

Holomorphic Approximation via Dolbeault Cohomology

The purpose of this paper is to study holomorphic approximation and approximation of $\bar\partial$-closed forms in complex manifolds of complex dimension $n\geq 1$. We consider extensions of the classical Runge theorem and the Mergelyan property to domains in complex manifolds for the smooth and the $L^2$ topology. We characterize the Runge or Mergelyan property in terms of certain Dolbeault cohomology groups and some geometric sufficient conditions are given.

math.CV↗
arXiv2020

Holomorphic Approximation and Mixed Boundary Value Problems for $\partial$

In this paper we study holomorphic approximation using boundary value problems for $\bar\partial$ on an annulus in the Hilbert space setting. The associated boundary conditions for $\bar\partial$ are the mixed boundary problems on an annulus. We characterize pseudoconvexity and Runge type property of the domain by the vanishing of related $L^2$ cohomology groups.

math.CV↗
arXiv2017

Hearing pseudoconvexity in Lipschitz domains with holes via $\overline\partial$

Let $Ω=\widetildeΩ\setminus \overline{D}$ where $\widetildeΩ$ is a bounded domain with connected complement in $\mathbb C^n$ (or more generally in a Stein manifold) and $D$ is relatively compact open subset of $\widetildeΩ$ with connected complement in $\widetildeΩ$. We obtain characterizations of pseudoconvexity of $\widetildeΩ$ and $D$ through the vanishing or Hausdorff property of the Dolbeault cohomology groups on various function spaces. In particular, we show that if the boundaries of $\widetildeΩ$ and $D$ are Lipschitz and $C^2$-smooth respectively, then both $\widetildeΩ$ and $D$ are pseudoconvex if and only if $0$ is not in the spectrum of the $\overline\partial$-Neumann Laplacian on $(0, q)$-forms for $1\le q\le n-2$ when $n\geq 3$; or $0$ is not a limit point of the spectrum of the $\overline\partial$-Neumannn Laplacian on $(0, 1)$-forms when $n=2$.

math.CV↗
arXiv2016

Solving $\overline\partial$ with prescribed support on Hartogs triangles in $\mathbb C^2$ and $\mathbb C\mathbb P^2$

In this paper we study the solvability of the Cauchy-Riemann equation with prescibed support in different spaces of forms. The unbounded Hartogs triangle in $\mathbb C^2$ and the Hartogs domains in $\mathbb C\mathbb P^2$ provide us new unexpected phenomena. In particular we prove that the Dolbeault isomorphism fails to hold for the Dolbeault cohomology with prescribed support

math.CV↗
arXiv2016

On the $L^2$-Dolbeault cohomology of annuli

For certain annuli in $\mathbb{C}^n$, $n\geq 2$, with non-smooth holes, we show that the $\bar{\partial}$-operator from $L^2$ functions to $L^2$ $(0,1)$-forms has closed range. The holes admitted include products of pseudoconvex domains and certain intersections of smoothly bounded pseudoconvex domains. As a consequence, we obtain estimates in the Sobolev space $W^1$ for the $\bar{\partial}$-equation on the non-smooth domains which are the holes of these annuli.

math.CV↗
arXiv2016

Bounded holomorphic functions on negatively curved Kähler manifolds of dimension $\ge 3$

Let M be a simply-connected complete Kahler manifold whose sectional curvature is bounded between two negative numbers. In this paper we prove the existence of non-constant bounded holomorphic functions on M if the complex dimension of M is greater or equal to three. Our proof uses bounded plurisubharmonic exhaustion functions, the Cauchy-Riemann equations and uniform Holder estimates for CR functions on geodesic spheres.

math.CV↗
arXiv2015

The $L^2$-cohomology of a bounded smooth Stein Domain is not necessarily Hausdorff

We give an example of a pseudoconvex domain in a complex manifold whose $L^2$-Dolbeault cohomology is non-Hausdorff, yet the domain is Stein. The domain is a smoothly bounded Levi-flat domain in a two complex-dimensional compact complex manifold. The domain is biholomorphic to a product domain in $\mathbb{C}^2$, hence Stein. This implies that for $q>0$, the usual Dolbeault cohomology with respect to smooth forms vanishes in degree $(p,q)$. But the $L^2$-Cauchy-Riemann operator on the domain does not have closed range on $(2,1)$-forms and consequently its $L^2$-Dolbeault cohomology is not Hausdorff.

math.CV↗
arXiv2012

Sobolev regularity of the $\bar{\partial}$-equation on the Hartogs triangle

The regularity of the $\bar{\partial}$-problem on the domain $\{|{z_1}|<|{z_2}|<1\}$ in $\mathbb{C}^2$ is studied using $L^2$ methods. Estimates are obtained for the canonical solution in weighted $L^2$-Sobolev spaces with a weight that is singular at the point $(0,0)$. The canonical solution for $\dbar$ with weights is exact regular in the weighted Sobolev spaces away from the singularity $(0,0)$. In particular, the singularity of the Bergman projection for the Hartogs triangle is contained at the singular point and it does not propagate.

math.CV↗
arXiv2010

$L^2$ Serre Duality on Domains in Complex Manifolds and Applications

An $L^2$ version of the Serre duality on domains in complex manifolds involving duality of Hilbert space realizations of the $\bar{\partial}$-operator is established. This duality is used to study the solution of the $\bar{\partial}$-equation with prescribed support. Applications are given to $\bar{\partial}$-closed extension of forms, as well to Bochner-Hartogs type extension of CR functions.

math.CV↗
arXiv2010

The Cauchy-Riemann equations on product domains

We establish the $L^2$ theory for the Cauchy-Riemann equations on product domains provided that the Cauchy-Riemann operator has closed range on each factor. We deduce regularity of the canonical solution on $(p,1)$-forms in special Sobolev spaces represented as tensor products of Sobolev spaces on the factors of the product. This leads to regularity results for smooth data.

math.CV↗
arXiv2006

The d-bar-Cauchy problem and nonexistence of Lipschitz Levi-flat hypersurfaces in CP^n with n>= 3

A Lipschitz hypersurface is a hypersurface which locally is the graph of a Lipschitz function. A Lipschitz (or C^1) hypersurface is said to be Levi-flat if it is locally foliated by complex manifolds of complex dimension (n-1). We shall prove that there exist no Lipschitz Levi-flat hypersurfaces in CP^n with n >= 3. Our new estimates on the d-bar-Cauchy problems are different from the earlier Siu's integral kernal method.

math.DG↗