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Andrew Zimmer

Publications and source records attributed to Andrew Zimmer.

At least 55 records · Page 3Linked to original sources
arXiv2020

The automorphism group and limit set of a bounded domain I: the finite type case

For bounded pseudoconvex domains with finite type we give a precise description of the automorphism group: if an orbit of the automorphism group accumulates on at least two different points of the boundary, then the automorphism group has finitely many components and is the almost direct product of a compact group and connected Lie group locally isomorphic to ${ \rm Aut}(\mathbb{B}_k)$. Further, the limit set is a smooth submanifold diffeomorphic to the sphere of dimension $2k-1$. As applications we prove a new finite jet determination theorem and a Tits alternative theorem. The geometry of the Kobayashi metric plays an important role in the paper.

math.CV↗
arXiv2020

Compactness of the $\bar{\partial}$-Neumann problem on domains with bounded intrinsic geometry

By considering intrinsic geometric conditions, we introduce a new class of domains in complex Euclidean space. This class is invariant under biholomorphism and includes strongly pseudoconvex domains, finite type domains in dimension two, convex domains, $\mathbb{C}$-convex domains, and homogeneous domains. For this class of domains, we show that compactness of the $\bar{\partial}$-Neumann operator on $(0,q)$-forms is equivalent to the boundary not containing any $q$-dimensional analytic varieties (assuming only that the boundary is a topological submanifold). We also prove, for this class of domains, that the Bergman metric is equivalent to the Kobayashi metric and that the pluricomplex Green function satisfies certain local estimates in terms of the Bergman metric.

math.CV↗
arXiv2019

Homeomorphic extension of quasi-isometries for convex domains in $\mathbb C^d$ and iteration theory

We study the homeomorphic extension of biholomorphisms between convex domains in $\mathbb C^d$ without boundary regularity and boundedness assumptions. Our approach relies on methods from coarse geometry, namely the correspondence between the Gromov boundary and the topological boundaries of the domains and the dynamical properties of commuting 1-Lipschitz maps in Gromov hyperbolic spaces. This approach not only allows us to prove extensions for biholomorphisms, but for more general quasi-isometries between the domains endowed with their Kobayashi distances.

math.CV↗
arXiv2018

Two boundary rigidity results for holomorphic maps

In this paper we establish two boundary versions of the Schwarz lemma. The first is for general holomorphic self maps of bounded convex domains with $C^2$ boundary. This appears to be the first boundary Schwarz lemma for general holomorphic self maps that requires no strong pseudoconvexity or finite type assumptions. The second is for biholomorphisms of domains who have an invariant Kähler metric with bounded sectional curvature. This second result applies to holomorphic homogeneous regular domains and appears to be the first boundary Schwarz lemma that makes no assumptions on the regularity of the boundary.

math.CV↗
arXiv2018

Asymptotic behavior of orbits of holomorphic semigroups

Let $(ϕ_t)$ be a holomorphic semigroup of the unit disc (i.e., the flow of a semicomplete holomorphic vector field) without fixed points in the unit disc and let $Ω$ be the starlike at infinity domain image of the Koenigs function of $(ϕ_t)$. In this paper we completely characterize the type of convergence of the orbits of $(ϕ_t)$ to the Denjoy-Wolff point in terms of the shape of $Ω$. In particular we prove that the convergence is non-tangential if and only if the domain $Ω$ is `quasi-symmetric with respect to vertical axes'. We also prove that such conditions are equivalent to the curve $[0,\infty)\ni t\mapsto ϕ_t(z)$ being a quasi-geodesic in the sense of Gromov. Also, we characterize the tangential convergence in terms of the shape of $Ω$.

math.CV↗
arXiv2018

Characterizing strong pseudoconvexity, obstructions to biholomorphisms, and Lyapunov exponents

In this paper we consider the following question: For bounded domains with smooth boundary, can strong pseudoconvexity be characterized in terms of the intrinsic complex geometry of the domain? Our approach to answering this question is based on understanding the dynamical behavior of real geodesics in the Kobayashi metric and allows us to prove a number of results for domains with low regularity. For instance, we show that for convex domains with $C^{2,ε}$ boundary strong pseudoconvexity can be characterized in terms of the behavior of the squeezing function near the boundary, the behavior of the holomorphic sectional curvature of the Bergman metric near the boundary, or any other reasonable measure of the complex geometry near the boundary. The first characterization gives a partial answer to a question of Fornæss and Wold. As an application of these characterizations, we show that a convex domain with $C^{2,ε}$ boundary which is biholomorphic to a strongly pseudoconvex domain is also strongly pseudoconvex.

math.CV↗
arXiv2018

Smoothly bounded domains covering finite volume manifolds

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.

math.CV↗
arXiv2018

Proper quasi-homogeneous domains in flag manifolds and geometric structures

In this paper we study domains in flag manifolds which are bounded in an affine chart and whose projective automorphism group acts co-compactly. In contrast to the many examples in real projective space, we will show that no examples exist in many flag manifolds. Moreover, in the cases where such domains can exist, we show that they satisfy a natural convexity condition and have an invariant metric which generalizes the Hilbert metric. As an application we give some restrictions on the developing map for certain $(G,X)$-structures.

math.DG↗
arXiv2018

The automorphism group and limit set of a bounded domain II: the convex case

For convex domains with $C^{1,ε}$ boundary we give a precise description of the automorphism group: if an orbit of the automorphism group accumulates on at least two different closed complex faces of the boundary, then the automorphism group has finitely many components and the connected component of the identity is the almost direct product of a compact group and a non-compact connected simple Lie group with real rank one and finite center. In this case, we also show the limit set is homeomorphic to a sphere and prove a gap theorem: either the domain is biholomorphic to the unit ball (and the limit set is the entire boundary) or the limit set has co-dimension at least two in the boundary.

math.CV↗
arXiv2017

A gap theorem for the complex geometry of convex domains

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 consequence is the following: for any dimension $d$ there exists some $ε> 0$ so that if the holomorphic sectional curvature of the Bergman metric on a smoothly bounded convex domain is within $ε$ of $-4/(d+1)$ outside a compact set, then the domain is strongly pseudoconvex (and hence the holomorphic sectional curvature limits to $-4/(d+1)$ on the boundary).

math.CV↗
arXiv2017

Goldilocks domains, a weak notion of visibility, and applications

In this paper we introduce a new class of domains in complex Euclidean space, called Goldilocks domains, and study their complex geometry. These domains are defined in terms of a lower bound on how fast the Kobayashi metric grows and an upper bound on how fast the Kobayashi distance grows as one approaches the boundary. Strongly pseudoconvex domains and weakly pseudoconvex domains of finite type always satisfy this Goldilocks condition, but we also present families of Goldilocks domains that have low boundary regularity or have boundary points of infinite type. We will show that the Kobayashi metric on these domains behaves, in some sense, like a negatively curved Riemannian metric. In particular, it satisfies a visibility condition in the sense of Eberlein and O'Neill. This behavior allows us to prove a variety of results concerning boundary extension of maps and to establish Wolff-Denjoy theorems for a wide collection of domains.

math.CV↗
arXiv2016

Entropy rigidity of Hilbert and Riemannian metrics

In this paper we provide two new characterizations of real hyperbolic $n$-space using the Poincaré exponent of a discrete group and the volume growth entropy. The first characterization is in the space of Hilbert metrics and generalizes a result of Crampon. The second is in the space of Riemannian metrics with Ricci curvature bounded below and generalizes a result of Ledrappier and Wang.

math.DG↗
arXiv2016

Generic analytic polyhedron with non-compact automorphism group

In this paper we prove the following rigidity theorem: a generic analytic polyhedron with non-compact automorphism group is biholomorphic to the product of a complex manifold with compact automorphism group and a polydisk. Moreover, this complex manifold and the dimension of this polydisk can be explicitly described in terms of the limit set of the automorphism group.

math.CV↗
arXiv2015

Rigidity of convex divisible domains in flag manifolds

In contrast to the many examples of convex divisible domains in real projective space, we prove that up to projective isomorphism there is only one convex divisible domain in the Grassmannian of $p$-planes in $\mathbb{R}^{2p}$ when $p > 1$. Moreover, this convex divisible domain is a model of the symmetric space associated to the simple Lie group SO$(p, p)$.

math.DG↗