Temple-Rutgers Global Analysis Seminar
Current contacts: Gerardo Mendoza (Temple) and Siqi Fu and Howard Jacobowitz (Rutgers)
The seminar takes place Friday 3:00 - 3:50 pm in Wachman 527 for talks at Temple, or 319 Cooper, Rm 110, for talks at Rutgers-Camden. Click on title for abstract.
Friedrich Haslinger, Universität Wien
We apply methods from complex analysis, in particular the $\overline\partial$-Neumann operator, to investigate spectral properties of Schrödinger operators with magnetic field (Pauli operators). For this purpose we consider the weighted $\overline\partial$-complex on $\mathbb C^n$ with a plurisubharmonic weight function.
We derive a necessary condition for compactness of the corresponding $\overline\partial$-Neumann operator (the inverse of the complex Laplacian) and a sufficient condition, both are not sharp. So far, a characterization can only be given in the complex 1-dimensional case.
The Pauli operators appear at the beginning and at the end of the weighted $\overline\partial$-complex. It is also of importance to know whether a related Bergman space of entire functions is of infinite dimension.
In addition we consider the $\partial$-complex, where the underlying Hilbert space is the Fock space - the space of entire functions with the Gaußian weight.
John D'Angelo, University of Illinois at Urbana-Champaign
I will begin with a short discussion of old results by many authors about rational sphere maps. I will then attempt to put these results into a general context. I will discuss work with Ming Xiao that associates various groups with proper holomorphic mappings. In the case of balls we proved that every finite subgroup of the source automorphism group arises as the Hermitian invariant group of a proper mapping between balls. I will discuss this result and some generalizations.
Nordine Mir, Texas A&M University-Qatar
We discuss recent joint results with B. Lamel on the $C^\infty$ regularity of CR maps between smooth CR submanifolds embedded in complex spaces of possibly different dimensions. Applications to the boundary regularity of proper holomorphic maps of positive codimension between domains with smooth boundaries of Dâ€™Angelo finite type will be given.
Vladimir Matveev, Friedrich Schiller University, Jena, Germany
With the help of a generalization of the Fermat principle in general relativity, we show that chains in CR geometry are geodesics of a certain Kropina metric constructed from the CR structure. We study the projective equivalence of Kropina metrics and show that if the kernel distributions of the corresponding 1-forms are non-integrable then two projectively equivalent metrics are trivially projectively equivalent. As an application, we show that sufficiently many chains determine the CR structure up to conjugacy, generalizing and reproving the main result of [J.-H. Cheng, 1988]. The talk is based on a joint paper with J.-H. Cheng, T. Marugame and R. Montgomery.
Fang Wang, Shanghai Jiao Tong University and Princeton University
I will talk about two types of Escobar-Yamabe compactification for Poincare-Einstein manifolds and derive some inequalities between the Yamabe constants for the conformal class of compactified metric and the Yamabe constant for the conformal infinity. Based on thoses inequalities, I will also give some rigidity theorems. This work was motivated by recent work of Gursky-Han, and collaborated with M. Lai and X. Chen.
Sonmez Sahutoglu, University of Toledo
Let $\Omega$ be a $C^2$-smooth bounded pseudoconvex domain in $\mathbb{C}^n$ for $n\geq 2$ and let $\varphi$ be a holomorphic function on $\Omega$ that is $C^2$-smooth on the closure of $\Omega$. We prove that if the Hankel operator $H_{\overline{\varphi}}$ is in Schatten $p$-class for $p\leq 2n$ then $\varphi$ is a constant function. As a corollary, we show that the $\overline{\partial}$-Neumann operator on $\Omega$ is not Hilbert-Schmidt. This is joint work with Nihat Gokhan Gogus
Stephen McKeown, Princeton University
This talk will concern cornered asymptotically hyperbolic spaces, which have a finite boundary in addition to the usual infinite boundary. After introducing a normal form near the corner for these spaces, I will discuss formal existence and uniqueness, near the corner, of asymptotically hyperbolic Einstein metrics, with a CMC-umbilic condition imposed on the finite boundary. This is analogous to the Fefferman-Graham construction for the ordinary, non-cornered setting. Finally, I will mention ongoing work regarding scattering on such spaces.
Purvi Gupta, Rutgers University
Starting with the observation that every continuous complex-valued function on the unit circle can be approximated by rational combinations of one function and polynomial combinations of two functions, we will discuss analogous approximation phenomena for compact manifolds of higher dimensions. On a related note, we will discuss some questions regarding the minimum embedding (complex) dimension of real manifolds within the context of polynomial convexity. In the special case of even-dimensional manifolds, we will present a technique that improves previously known bounds. This is joint work with Rasul Shafikov.
Siqi Fu, Rutgers University
In this talk, I will explain the proof of the following result due to C. Laurent-Thiebault, M.-C. Shaw and myself: Let $\Omega=\widetilde{\Omega}\setminus \overline{D}$ where $\widetilde{\Omega}$ is a bounded domain with connected complement in $\mathbb C^n$ and $D$ is relatively compact open subset of $\widetilde{\Omega}$ with connected complement in $\widetilde{\Omega}$. If the boundaries of $\widetilde{\Omega}$ and $D$ are Lipschitz and $C^2$-smooth respectively, then both $\widetilde{\Omega}$ and $D$ are pseudoconvex if and only if $0$ is not in the spectrum of the $\bar\partial $-Neumann Laplacian on $(0, q)$-forms for $1\le q\le n-2$ when $n\ge 3$; or $0$ is not a limit point for the spectrum of the $\bar\partial $-Neumannn Laplacian on $(0, 1)$-forms when $n=2$.