MenuSeveral Complex Variables Seminar
Spring 2009
| Date: | February 19, 2009 |
| Time: | 11:10am |
| Location: | MILN 317 |
| Speaker: | Mark Agranovsky, Bar-Ilan University and TAMU |
| Title: | Parametric argument principle and detecting CR functions and manifolds |
| Date: | February 26, 2009 |
| Time: | 11:10am |
| Location: | MILN 317 |
| Speaker: | Robert Jacobson, Texas A&M University |
| Title: | Characterizing pseudoconvex domains |
| Abstract: | There are dozens of ways to characterize pseudoconvex domains which are also known as domains of holomorphy. For some characterizations, there exist formally weaker statements which nonetheless imply the same conclusion. This motivates exploring what happens when the hypotheses of some of the characterizations of pseudoconvex domains are slightly loosened. We explore some results in a recent paper by Nikolov, Pflug, Thomas, and Zwonek along these lines. In particular, a domain D is pseudoconvex if and only if for any $a \in \partial D$ there is a $u_a \in \mathcal{PSH}(D)$ with $\lim_{z \to a}u_a(z) = \infty$. We discuss what happens when we replace $\lim$ with $\limsup$ |
| Date: | March 26, 2009 |
| Time: | 11:10am |
| Location: | MILN 317 |
| Speaker: | Chris Hammond |
| Title: | The Isoperimetric Problem for Fefferman Hypersurface Measure |
| Abstract: | Fefferman introduced a scaled-version of surface-area measure on real hypersurfaces in $\mathbb{C}^{n}$ which is invariant under volume-preserving biholomorphisms. We derive the Euler equation for the associated isopermetric problem. We then use volume-preserving invariants to characterize the solutions. We show that under "natural" assumptions, spheres are the only solutions. In the process, we will characterize those hypersurfaces that are volume-preserving equivalent to the sphere. |
| Date: | April 9, 2009 |
| Time: | 11:10am |
| Location: | MILN 317 |
| Speaker: | Joe Perez, Texas A&M University, Kingsville |
| Title: | The Levi Problem On Strongly Pseudoconvex G-Bundles |
| Abstract: | Let $G$ be a unimodular Lie group, $X$ a compact manifold with boundary, and $M$ be the total space of a principal bundle $G\to M\to X$ so that $M$ is also a strongly pseudoconvex complex manifold and $G$ acts by holomorphic transformations. Our ultimate goal is to determine if, in this setting, every point of the boundary is a peak point for the space of square-integrable holomorphic functions on $M$. This question is related to the Levi problem, posed in 1911, which was to characterize domains of holomorphy in $\mathbb C^n$. First, we established that $\square_{0,1}$ on $M$ is a $G$-Fredholm operator (J Geom Analy, 19, 87--106, 2009). Now we exploit the previous result (roughly that the Laplacian is almost onto $L2(M,\Lambda^{0,1})$) by constructing large, closed, smooth invariant subspaces in $L2$ that the image of $\square$ must meet. A noncommutative generalization of the Paley-Wiener theorem is used. Finally, we show that if $G$ acts by transformations satisfying a local property, then the space of square-integrable holomorphic functions on $M$ is infinite $G$-dimensional and the functions in this space have the property that they lack holomorphic extensions beyond the boundary, almost as Levi desired. See http://arxiv.org/find/all/1/all:+AND+joe+perez/0/1/0/all/0/1 |
