Lecture Course :
Claude LeBrun (Stony Brook)
Title: Zoll Metrics, Holomorphic Disks, and Twistor Correspondences
Content: Claude LeBrun will deliver a series of lectures on the topic "Zoll Metrics, Holomorphic Disks, and Twistor Correspondences".
- Zoll manifolds and complex surfaces. J. Differential Geom. 61 (2002), no. 3, 453–535.
- Twistors, holomorphic disks, and Riemann surfaces with boundary. Perspectives in Riemannian geometry, 209–221, CRM Proc. Lecture Notes, 40, Amer. Math. Soc., Providence, RI, 2006.
- Nonlinear gravitons, null geodesics, and holomorphic disks. Duke Math. J. 136 (2007), no. 2, 205–273.
- The Einstein-Weyl equations, scattering maps, and holomorphic disks. Math. Res. Lett. 16 (2009), no. 2, 291–301.
- Zoll metrics, branched covers, and holomorphic disks. Comm. Anal. Geom. 18 (2010), no. 3, 475–502.
Slides are available on Claude's webpage.
Research talks :
Lionel Mason (Oxford)
Title: Transparent connections on the sphere and for self-dual Yang-Mills in split signature
Abstract: This talk treats the analogue for bundles of the problem of constructing Zoll metrics on a surface. This is taken to be the problem of constructing connections on bundles on the sphere with trivial holonomy around great circles. These are shown to be in correspondence with certain holomorphic vector bundles on the complex projective plane equipped with a Hermitian metric on the real plane inside the complex plane. These nicely relate to soliton constructions in integrable systems. There is a similar construction for solutions to the self-dual Yang-Mills equations in split signature on S^2 x S^2.
Thomas Mettler (Frankfurt)
Title: Metrisability of projective surfaces and pseudo-holomorphic curves
Abstract: A projective structure is said to be metrisable if it arises from the Levi-Civita connection of some Riemannian metric. I will explain that metrisability of a two-dimensional projective structure is equivalent to the existence of a certain holomorphic curve and a pseudo-holomorphic curve. Some related (global) non-existence result for holomorphic curves will also be discussed. Joint with Gabriel Paternain.
Hans-Bert Rademacher (Leipzig)
Title: Critical values of homology classes of loops and positive curvature
Stefan Suhr (Bochum)
Title: A Morse theoretic Characterization of Zoll metrics
Abstract: From the Morse theoretic point of view Zoll metrics are rather peculiar. All critical sets of the energy on the loop space are nondegenerate critical manifolds diffeomorphic to the unit tangent bundle. This especially implies that min-max values associated to certain homology classes coincide. In my talk I will explain that the coincidence of these min-max values characterises Zoll metrics in any dimension. A specially focus will lie on the case of the 2-sphere. This is work in collaboration with Marco Mazzucchelli (ENS Lyon).
Christian Lange (Cologne)
Title: Introduction to Zoll surfaces
Abstract: We show that a surface with a metric all of whose geodesics are closed is either a sphere or a projective plane. On the sphere we describe a construction of an infinite family of such metrics due to Zoll. On the projective plane we prove that such a metric has constant curvature.
Thomas Mettler (Frankfurt)
Title: The model case
Abstract: I will describe the construction of a certain non-compact complex surface fibering over an oriented projective surface whose fibres are holomorphic disks. In the model case of the standard projective structure on the 2-sphere the complex surface is identified with an open orbit of the natural SL(3,R) action on the dual complex projective plane. I will explain how the intersection pattern of the boundaries of these holomorphic disks nicely encodes the projective structure and discuss the geometric significance of holomorphic curves into this complex surface.
Roberta Maccheroni (Parma)
Title: Holomorphic disc fillings of totally real tori
Abstract: In the first part of the seminar I will recall some classical results about a fundamental problem in complex geometry: the existence of a holomorphic disc with boundary on a totally real submanifold. Then I will define the notion of a holomorphic disc filling and present Zehmisch's existence result for totally real tori in C^2.
The second part will be dedicated to a non-existence result for holomorphic disc fillings of minimal Lagrangian 2-tori. I will provide also a generalization in higher dimensions.
Kevin Wiegand (Gießen)
Title: Compactness for holomorphic discs - an example
Abstract: We take a look at the moduli space of holomorphic discs with boundary on a 2-sphere inside the 3-sphere. Compactness for this moduli space is connected to questions about the filling of said sphere with holomorphic discs.