Kyle Broder, Queensland. The Schwarz Lemma in Kähler and Non-Kähler Geometry

Giovedì 1 Giugno ore 16.30 , Aula Dal Passo

Abstract:

A folklore conjecture generalizing a conjecture made by Kobayashi half a century ago predicts that a compact hyperbolic manifold is projective and canonically polarized. There have been a significant number of achievements in this direction recently, the most notable being what has come to be known as the Wu—Yau theorem: A compact Kähler manifold with a Kähler metric of negative holomorphic sectional curvature is projective and canonically polarized. One of the primary stumbling blocks to further developments is the Schwarz Lemma in the Hermitian non-Kähler category. We will present some recent improvements on the Schwarz Lemma and related results. If time permits, we will discuss a conjectural positive analog of the Wu—Yau theorem.

Giuseppe Pipoli, L'Aquila. Constant mean curvature hypersurfaces in H^n x R with small planar boundary

Martedì 9 Maggio ore 16, Aula Dal Passo

Abstract:

Denoting with H^n the n-dimensional hyperbolic space, we show that constant mean curvature hypersurfaces in H^n x R with small boundary contained in a horizontal slice P are topological disks, provided they are contained in one of the two half-spaces determined by P.

This is the analogous in H^n x R of a result in R^3 by A. Ros and H. Rosenberg.

 The proof is based on geometric and analytic methods : from one side the constant mean curvature equation is a quasilinear elliptic PDE on manifolds, to the other the specific geometry of the ambient space produces some peculiar phenomena.

 This talk is based on a joint work with Barbara Nelli.

Nikolay Shcherbina, Wuppertal. Hyperbolicity of some unbounded domains in C^n

Giovedì 30 Marzo ore 16, Aula D'Antoni

Abstract:

The main topic of the talk is Kobayashi and Bergman hyperbolicity of some natural classes of unbounded domains in C^n. In particular, the question of dependence of Kobayashi hyperbolicity of Model domains (i.e. pseudoconvex domains of the form

$\mathfrak{A} = \{(z, w) \in \mathbb C^{n-1}_z \times \mathbb C_w : v >F(z, u)\}, w = u +iv$) on the dimension n of the ambient space will be

discussed. If the time permit, we will also indicate the construction of a smoothly bounded strongly pseuodoconvex domain in C^2 which is

Kobayashi and Bergman hyperbolic and complete, but has no nonconstant bounded holomorphic functions.

Giulio Tiozzo, Toronto. The harmonic measure for random walks on cocompact Fuchsian groups

Martedì 31 Gennaio ore 16, Aula Dal Passo

Abstract:

We consider random walks on groups of isometries of the hyperbolic plane, known as Fuchsian groups. It is well-known since Furstenberg that such random walks converge to the boundary at infinity, and the probability to reach a given subset of the boundary defines a hitting, or harmonic, measure on the circle. It has been a long-standing question whether this harmonic measure is absolutely continuous with respect to the Lebesgue measure. Conjecturally, this is never the case for random walks on cocompact, discrete groups. In the talk, based on joint work with Petr Kosenko, we settle the conjecture for nearest neighbour random walks on hyperelliptic groups. In fact, we show that the dimension of the harmonic measure for such walks is strictly less than one. This is also related to an inequality between entropy and drift.

Georg Schumacher, Marburg. Analytic GIT spaces vs. Coarse Moduli Spaces – Application to Polystable Vector Bundles

Mercoledì 30 Novembre ore 14, Aula Dal Passo

Abstract:

We introduce analytic GIT quotients as local models for analytic GIT spaces. These are classifying spaces for isomorphisms classes of polystable points. GIT spaces are compared to coarse moduli spaces. For polystable holomorphic vector bundles on compact Kähler manifolds the existence of classifying spaces is shown and the existence of coarse moduli spaces is discussed. (Joint with Nicholas Buchdahl)

Mårten Nilsson, Lund. Unbounded envelopes of plurisubharmonic functions

Giovedì 13 ottobre ore 16 aula Dal Passo

Abstract:

In this presentation, we study envelopes (point-wise suprema) of families of plurisubharmonic functions, i.e. subharmonic functions whose compositions with biholomorphic mappings are still subharmonic. Such envelopes occupy a central position within pluripotential theory as they for example (under suitable assumptions) constitute the unique solution to the Dirichlet problem for the complex Monge-Ampère operator. Specifically, we study families defined on a B-regular domain in C^n, bounded from above by a function continuous in the extended reals. Given some assumptions on the singularities, we establish a set where the envelope is guaranteed to be continuous. As an application of the methods involved, we are able to construct unique solutions to certain complex Monge-Ampère equations where the boundary data is unbounded.