Andrew Zimmer
In this paper we prove: if a bounded domain with $C^2$ boundary covers a manifold which has finite volume with respect to either the Bergman volume, the Kähler-Einstein volume, or the Kobayashi-Eisenman volume, then the domain is biholomorphic to the unit ball. This answers an old question of Yau. Further, when the domain is convex we can assume that the boundary only has $C^{1,ε}$ regularity.
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We show that if a bounded domain in complex Euclidean space with $\mathcal{C}^{1,1}$ boundary covers a compact manifold, then the domain is biholomorphic to the unit ball.
We show that every bounded domain $D$ in $\mathbb R^n$ with smooth $p$-convex boundary for $2\le p < n$ admits a smooth defining function $ρ$ which is $p$-plurisubharmonic on $\overline D$; if in addition $bD$ has no $p$-flat points then $ρ$ can be chosen strongly $p$-plurisubharmonic on $D$. If $bD$ is $2$-convex then for any open connected conformal surface $M$ and conformal harmonic map $f:M\to \overline D$, either $f(M)\subset D$ or $f(M)\subset bD$. In particular, every conformal harmonic map $\mathbb D^*\to D$ from the punctured disc extends to a conformal harmonic map $\mathbb D\to D$.
A Stein covering of a complex manifold may be used to realise its analytic cohomology in accordance with the Cech theory. If, however, the Stein covering is parameterised by a smooth manifold rather than just a discrete set, then we construct a cohomology theory in which an exterior derivative replaces the usual combinatorial Cech differential. Our construction is motivated by integral geometry and the representation theory of Lie groups.
Let S be a generic C-infinity smooth CR manifold in C^n, n > 1, and let M be a generic C-infinity CR submanifold of S X C^m. We prescribe conditions on M so that it is the disjoint union of graphs of CR maps f:S-->C^m. We also consider the special case where S is the boundary of a bounded, smoothly bounded open set D in C^n, n > 1. In this case we obtain conditions which guarantee that such an f above extends to be analytic on D. This provides a solution to a particular nonlinear Riemann-Hilbert boundary value problem for analytic functions.
We prove two density theorems for quadrature domains in $\mathbb{C}^n$, $n \geq 2$. It is shown that quadrature domains are dense in the class of all product domains of the form $D \times Ω$, where $D \subset \mathbb{C}^{n-1}$ is a smoothly bounded domain satisfying Bell's Condition R and $Ω\subset \mathbb{C}$ is a smoothly bounded domain and also in the class of all smoothly bounded complete Hartogs domains in $\mathbb{C}^2$.
In this paper we establish a gap theorem for the complex geometry of smoothly bounded convex domains which informally says that if the complex geometry near the boundary is close to the complex geometry of the unit ball, then the domain must be strongly pseudoconvex. One consequence of our general result is the following: for any dimension there exists some $ε> 0$ so that if the squeezing function on a smoothly bounded convex domain is greater than $1-ε$ outside a compact set, then the domain is strongly pseudoconvex (and hence the squeezing function limits to one on the boundary). Another con…
Sobolev Regularity of the Bergman Projection on a Smoothly Bounded Stein Domain that is not Hyperconvex
本文构造非超凸Stein域，证明Bergman投影在特定Sobolev空间中的正则性临界行为，并给出光滑函数投影非光滑的反例。
We consider proper holomorphic mappings of equidimensional pseudoconvex domains in complex Euclidean space, where both source and target can be represented as Cartesian products of smoothly bounded domains. It is shown that such mappings extend smoothly up to the closures of the domains, provided each factor of the source satisfies Condition R. It also shown that the number of smoothly bounded factors in the source and target must be the same, and the proper holomorphic map splits as product of proper mappings between the factor domains.
We show that two smoothly bounded, strongly pseudoconvex domains which are diffeomorphic may be smoothly deformed into each other, with all intermediate domains being strongly pseudoconvex. This result relates to Lempert's ideas about Kobayashi extremal discs, and also has intrinsic interest.
Constructing bounded pseudoconvex domains of finite D'Angelo type with prescribed weak loci
研究在\(\mathbb{C}^n\)中构造特定域，核心方法是利用规定闭集，贡献为得到相关有界伪凸或凸域。
In this work we explore the theme of $L^p$-boundedness of Bergman projections of domains that can be covered, in the sense of ramified coverings, by "nice" domains (e.g. strictly pseudoconvex domains with real analytic boundary). In particular, we focus on two-dimensional normal ramified coverings whose covering group is a finite unitary reflection group. In an infinite family of examples we are able to prove $L^p$-boundedness of the Bergman projection for every $p\in(1,\infty)$.
A (bounded) manifold of circular type is a complex manifold M of dimension n admitting a (bounded) exhaustive real function u, defined on M minus a point x_o, so that: a) it is a smooth solution on $M\setminus {x_o}$ to the Monge-Ampère equation $(d d^c u)^n = 0$; b) x_o is a singular point for u of logarithmic type and e^u extends smoothly on the blow up of M at x_o; c) $d d^c (e^u) >0$ at any point of $M\setminus {x_o}$. This class of manifolds naturally includes all smoothly bounded, strictly linearly convex domains and all smoothly bounded, strongly pseudoconvex circular domains of $\bC^n$…
Bounded domains on Kobayashi hyperbolic manifolds covering compact complex manifolds
本文通过局部化技术证明，具有C^{1,1}边界且覆盖紧复流形的Kobayashi超复流形的universal cover必为欧几里得球。
Let $G \subset \mathbb{C}^2$ be a smoothly bounded pseudoconvex domain and assume that the Bergman kernel of $G$ is algebraic of degree $d$. We show that the boundary $\partial G $ is of finite type and the type $r$ satisfies $r\leq 2d$. The inequality is optimal as equality holds for the egg domains $\{|z|^2+|w|^{2s}<1\},$ $s \in \mathbb{Z}_+$, by D'Angelo's explicit formula for their Bergman kernels. Our results imply, in particular, that a smoothly bounded pseudoconvex domain $G \subset \mathbb{C}^2$ cannot have rational Bergman kernel unless it is strongly pseudoconvex and biholomorphic to…
We prove a Fatou type theorem for bounded functions with d_J -bar differential of a controled growth on smoothly bounded domains in an almost complex manifold.
Under a plausible geometric hypothesis, we show that a biholomorphic mapping of smoothly bounded, pseudoconvex domains extends to a diffeomorphism of the closures.
We review some recent results on existence and regularity of Monge-Ampère exhaustions on the smoothly bounded strongly pseudoconvex domains, which admit at least one such exhaustion of sufficiently high regularity. A main consequence of our results is the fact that the Kobayashi pseudo-metric k on an appropriare open subset of each of the above domains is actually a smooth Finsler metric. The class of domains to which our result apply is very large. It includes for instance all smoothly bounded strongly pseudoconvex complete circular domains and all their sufficiently small deformations.
We construct a smoothly bounded pseudoconvex domain such that every boundary point has a p.s.h. peak function but some boundary point admits no (local) holomorphic peak function.
We begin by giving an example of a smoothly bounded convex domain that has complex geodesics that do not extend continuously up to $\partial\mathbb{D}$. This example suggests that continuity at the boundary of the complex geodesics of a convex domain $Ω\Subset \mathbb{C}^n$, $n\geq 2$, is affected by the extent to which $\partialΩ$ curves or bends at each boundary point. We provide a sufficient condition to this effect (on $\mathcal{C}^1$-smoothly bounded convex domains), which admits domains having boundary points at which the boundary is infinitely flat. Along the way, we establish a Hardy--…
Let $π:\mathbb{C}^n\times\mathbb{C}\rightarrow\mathbb{C}$ be the projection map onto the second factor and let $D$ be a domain in $\mathbb{C}^{n+1}$ such that for $y\inπ(D)$, every fiber $D_y:=D\capπ^{-1}(y)$ is a smoothly bounded strongly pseudoconvex domain in $\mathbb{C}^n$ and is diffeomorphic to each other. By Chau's theorem, the Kähler-Ricci flow has a long time solution $ω_y(t)$ on each fiber $D_y$. This family of flows induces a smooth real (1,1)-form $ω(t)$ on the total space $D$ whose restriction to the fiber $D_y$ satisfies $ω(t)\vert_{D_y}=ω_y(t)$. In this paper, we prove that $ω(t…
We give an explicit description of smoothly bounded Reinhardt domains with noncompact automorphism groups. In particular, this description confirms a special case of a conjecture of Greene/Krantz.
In this paper, we characterize $C^2$-smooth totally geodesic isometric embeddings $f\colon Ω\toΩ'$ between bounded symmetric domains $Ω$ and $Ω'$ which extend $C^1$-smoothly over some open subset in the Shilov boundaries and have nontrivial normal derivatives on it. In particular, if $Ω$ is irreducible, there exist totally geodesic bounded symmetric subdomains $Ω_1$ and $Ω_2$ of $Ω'$ such that $f = (f_1, f_2)$ maps into $Ω_1\times Ω_2\subset Ω$ where $f_1$ is holomorphic and $f_2$ is anti-holomorphic totally geodesic isometric embeddings. If $\text{rank}(Ω')<2\text{rank}(Ω)$, then either $f$ o…
In this paper we prove that, if $p$ is a boundary point of a smoothly bounded pseudoconvex Reinhardt domain in $\C^n$, then the variety type at $p$ is identical to the regular type.
We prove that the Bass stable rank of the algebra of bounded holomorphic functions on an unbranched covering of a finite bordered Riemann surface is equal to one.
Boundary behaviour of the Fefferman--Szegö metric in strictly pseudoconvex domains
本文研究严格伪凸域中 Fefferman--Szegö 度量及相关不变量的边界行为。
We prove the subsolution theorem for the complex Hessian equations in a smoothly bounded strongly $m$-pseudoconvex domain, $1 < m < n$, in $\bC^n$.
Let $D$ be a smoothly bounded pseudoconvex domain in $\mathbf C^n$, $n > 1$. Using $G(z, p)$, the Green function for $D$ with pole at $p \in D$ associated with the standard sum-of-squares Laplacian, N. Levenberg and H. Yamaguchi had constructed a Kähler metric (the so-called $\La$-metric) using the Robin function $\La(p)$ arising from $G(z, p)$. The purpose of this article is to study this metric by deriving its boundary asymptotics and using them to calculate the holomorphic sectional curvature along normal directions. It is also shown that the $\La$-metric is comparable to the Kobayashi (and…
In this paper continuing our work started in our earlier papers we prove the corona theorem for the algebra of bounded holomorphic functions defined on an unbranched covering of a Caratheodory hyperbolic Riemann surface of finite type.
We relate the existence and regularity of a solution operator to on smoothly bounded pseudoconvex domains to the existence and regularity of a projection operator onto the kernel of dbar.
We review and apply Cheeger-Gromov theory on $l^2$-cohomology of infinite coverings of complete manifolds with bounded curvature and finite volume. Applications focus on $l^2$-cohomology of (pullback of) harmonic Higgs bundles on some covering of Zariski open sets of Kähler manifolds. The $l^2-$Dolbeault to DeRham spectral sequence of these Higgs bundles is seen to degenerate at $E_2$.
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 Hausd…
In the present paper continuing our previous work we prove an extension theorem for matrices with entries in the algebra of bounded holomorphic functions defined on an unbranched covering of a Caratheodory hyperbolic Riemann surface of finite type.
We exhibit a smoothly bounded domain $Ω$ with the property that for suitable $K\subset\partial Ω$ and $z\in Ω$ the "Sadullaev boundary relative extremal functions" satisfy the inequality $ω_1(z,K,Ω)<ω_2(z,K,Ω)\le ω(z,K,Ω)$.
Given a pair of smoothly bounded domains $D_1, D_2 \subset \mathbb C$, the purpose of this note is to obtain an inequality that relates the Carathéodory metrics on $D_1, D_2, D_1 \cap D_2$ and $D_1 \cup D_2$.
In this paper, we prove that the identity map for the smoothly bounded pseudoconvex domain of finite type in $\mathbb{C}^2$ extends to a bi-Hölder map between the Euclidean boundary and Gromov boundary. As an application, we show the bi-Hölder boundary extensions for quasi-isometries between these domains. Moreover, we get a more accurate index of the Gehring-Hayman type theorem for the bounded $m$-convex domains with Dini-smooth boundary.
We show that biholomorphic mappings between two bounded, pseudoconvex domains with smooth boundary extend smoothly to the boundaries of the domains, under a regularity condition on a family of twisted Bergman-like projections. This result is inspired by and generalizes a sufficient condition for extension due to Bell and Ligocka.
Let $M$ be a non-compact connected Riemann surface of finite type, and $R\subset\subset M$ be a relatively compact domain such that $H_{1}(M,\Z)=H_{1}(R,\Z)$. Let $\tilde R\longrightarrow R$ be a covering. We study the algebra $H^{\infty}(U)$ of bounded holomorphic functions defined in some domains $U\subset\tilde R$. Our main result is a Forelli type theorem on projections in $H^{\infty}(\Di)$.
Let $Ω\subset \mathbb{C}^n$ for $n\geq 2$ be a bounded pseudoconvex domain with a $C^2$-smooth boundary. We study the compactness of composition operators on the Bergman spaces of smoothly bounded convex domains. We give a partial characterization of compactness of the composition operator (with sufficient regularity of the symbol) in terms of the behavior of the Jacobian on the boundary. We then construct a counterexample to show the converse of the theorem is false.
A new proof of Oka's lemma is given for smoothly bounded, pseudoconvex domains $D\subset\mathbb{C}^n$. The method of proof is then also applied to other convexity-like hypotheses on the boundary of $D$.
We prove a generalization of the second variation formula of the Robin function associated to a smooth variation of domains in C^N to the case of the c-Robin function associated to a smooth variation of domains in a complex manifold M equipped with a Hermitian metric and a smooth, nonnegative function c. Our purpose is that, with this added flexibility, we are able to give a criterion for a bounded, smoothly bounded, pseudoconvex domain D in a complex homogeneous space to be Stein.
We consider second-order elliptic equations with oblique derivative boundary conditions, defined on a family of bounded domains in $\mathbb{C}$ that depend smoothly on a real parameter $λ\in [0,1]$. We derive sharp regularity properties of the solutions in all variables, including the parameter $λ$. More specifically we show that the solution and its derivatives are continuous in all variables, and the Hölder norms of the space variables are bounded uniformly in $λ$.
We give three proofs of the fact that a smoothly bounded, convex domain in R^n has smooth defining functions whose Hessians are non-negative definite in a neighborhood of the boundary of the domain.
Automorphisms of the Worm Domain
本文针对多复变中作为关键反例的Diederich-Fornæss蠕虫域开展研究，考察其自同构群，明确了该域边界除例外轨迹与端帽外处处局部球形的性质。
We study the boundary limits of the Carathéodory and Kobayashi-Eisenman volume elements on smoothly bounded convex finite type domains and also on Levi corank one domains.
It is proved that on any smoothly bounded domain $D$ in $\mathbb{R}^n$, $n>1$, the output of the harmonic Bergman projection belongs to the Sobolev space of order $k$ whenever all tangential derivatives of order up to k of the input function belong to $L^2(D)$.
Let D be a smoothly bounded domain in C^2. Suppose that D admits a smooth defining function which is plurisubharmonic on the boundary of D. Then the Diederich-Fornaess exponent can be chosen arbitrarily close to 1, and the closure of D admits a Stein neighborhood basis.
Let U be a universal covering of a connected nonsingular projective variety X with large and residually finite fundamental group. We construct metrics on U and provide another version of the uniformization theorem, namely: if the fundamental group of X is, in addition, nonamenable then U is a bounded Stein domain. The Appendix contains a proof of the Shafarevich conjecture provided the fundamental group is residually finite.
Functions that are holomorphic and Lipschitz in a smoothly bounded domain enjoy a gain in the order of Lipschitz regularity in the complex tangential directions near the boundary. We describe this gain explicitly in terms of the defining function near points of finite type in the boundary.
The Bergman-Fridman invariant on some classes of pseudoconvex domains
本文研究了由Bergman度量定义的Fridman不变函数变体在多种伪凸域（包括Levi余秩1域、强伪凸域、光滑有界凸域及多面体域）上的边界行为。
J. E. Fornaess has posed the question whether the boundary point of smoothly bounded pseudoconvex domain is strictly pseudoconvex, if the asymptotic limit of the squeezing function is 1. The purpose of this paper is to give an affirmative answer when the domain is in C^2 with smooth boundary of finite type in the sense of D'Angelo.
Let D be a smoothly bounded domain in complex space of dimension larger than 2. Suppose that D admits a smooth defining function which is plurisubharmonic on the boundary of D. Then the Diederich-Fornaess exponent can be chosen arbitrarily close to 1, and the closure of D admits a Stein neighborhood basis.
We construct a special plurisubharmonic defining function for a smoothly bounded strictly pseudoconvex domain so that the determinant of the complex Hessian vanishes to high order on the boundary. This construction, coupled with regularity of solutions of complex Monge-Ampere equation and the reflection principle, enables us to give a new proof of the Fefferman mapping theorem.
Let Y be a regular covering of a complex projective manifold M\hookrightarrow CP^{N} of dimension n\geq 2. Let C be intersection with M of at most n-1 generic hypersurfaces of degree d in CP^{N}. The preimage X of C in Y is a connected submanifold. Let H^{\infty}(Y) and H^{\infty}(X) be the Banach spaces of bounded holomorphic functions on Y and X in the corresponding supremum norms. We prove that the restriction H^{\infty}(Y)\longrightarrow H^{\infty}(X) is an isometry for d large enough. This answers the question posed in [L] by F. Larusson and strengthen his example of the Riemann surface w…
We define Szego coordinates on a finitely connected smoothly bounded planar domain which effect a holomorphic change of coordinates on the domain that can be as close to the identity as desired and which convert the domain to a quadrature domain with respect to boundary arc length. When these Szego coordinates coincide with Bergman coordinates, the result is a double quadrature domain with respect to both area and arc length. We enumerate a host of interesting and useful properties that such double quadrature domains possess, and we show that such domains are in fact dense in the realm of boun…
We establish basic results of complex function theory within certain algebras of holomorphic functions on coverings of Stein manifolds (such as algebras of Bohr's holomorphic almost periodic functions on tube domains or algebras of all fibrewise bounded holomorphic functions arising, e.g., in the corona problem for $H^\infty$). In particular, in this context we obtain results on holomorphic extension from complex submanifolds, properties of divisors, corona type theorems, holomorphic analogues of the Peter-Weyl approximation theorem, Hartogs type theorems, characterizations of uniqueness sets,…
We reduce the problem of constructing a linear solution operator to the $\bar{\partial}$-equation on smoothly bounded weakly pseudoconvex domains, $Ω$, in $\mathbb{C}^2$ to the problem of the boundary $\bar{\partial}_b$-equation. We show there is a solution operator to $\bar{\partial}$ which is bounded as a map $W^{s}_{(0,1)}(Ω)\cap{ker} \bar{\partial} \rightarrow W^{s}(Ω)$ for all $s\ge 0$ if there is a corresponding solution operator to the $\bar{\partial}_b$-problem with analogous regularity properties.
We develop complex function theory within certain algebras of holomorphic functions on coverings of Stein manifolds. This, in particular, includes the results on holomorphic extension from complex submanifolds, corona type theorems, properties of divisors, holomorphic analogs of the Peter-Weyl approximation theorem, Hartogs type theorems, characterization of uniqueness sets. The model examples of these algebras are: (1) Bohr's algebra of holomorphic almost periodic functions on tube domains; (2) algebra of all fibrewise bounded holomorphic functions (e.g., arising in the corona problem for $H^…
Let Y be an infinite covering space of a projective manifold M in P^N of dimension n geq 2. Let C be the intersection with M of at most n-1 generic hypersurfaces of degree d in P^N. The preimage X of C in Y is a connected submanifold. Let phi be the smoothed distance from a fixed point in Y in a metric pulled up from M. Let O_phi(X) be the Hilbert space of holomorphic functions f on X such that f^2 e^(-phi) is integrable on X, and define O_phi(Y) similarly. Our main result is that (under more general hypotheses than described here) the restriction O_phi(Y) to O_phi(X) is an isomorphism for d l…
Let $D$ be a smoothly bounded pseudoconvex domain in $\mathbf C^n$, $n > 1$. Using the Robin function $Λ(p)$ that arises from the Green function $G(z, p)$ for $D$ with pole at $p \in D$ associated with the standard sum-of-squares Laplacian, N. Levenberg and H. Yamaguchi had constructed a Kähler metric (the so-called $Λ$-metric) on $D$. In this article, we study the existence of geodesic spirals for this metric.
In this paper we consider the topological class, denoted by AV2, of universal covering maps from the plane to the sphere with two removed points. We prove that an element f in AV2 with finite post-singular set is combinatorially equivalent to a meromorphic transcendental map g with constant Schwarzian derivative if any only if f satisfies a Bounded Geometry condition.
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