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
TBA
John D'Angelo, University of Illinois at Urbana-Champaign
TBA
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 $n=2$.