Workshop on Harmonic Maps and Curvature Properties of Submanifolds, 2

Main Speakers: Titles and Abstracts

A. Bobenko, "Bonnet Problem and Integrable Systems"

Abstract: A generic surface in Euclidean 3-space is determined uniquely by its metric and curvature. Classification of all special surfaces where this is not the case, i.e. of surfaces possessing isometries which preserve the mean curvature, is known as the Bonnet problem. These surfaces and their global properties are described and studied through methods from the theory of integrable systems.

F. Burstall, "Harmonic maps in conformal geometry"

Abstract: We introduce a quaternionic formalism for studying conformal geometry and give a concrete application in the form of a family of Backlund transformations of Willmore surfaces in R^4. These arise by exploiting a formal analogy between the Gauss maps of CMC surfaces in R^3 (viewed as isothermic surfaces) and those of Willmore surfaces.

T.E. Cecil, "Dupin Hypersurfaces"

Abstract: A hypersurface M in Euclidean space or the sphere is said to be (proper) Dupin if the number g of distinct principal curvatures is constant on M, and if each principal curvature is constant along each leaf of its corresponding principal foliation. Important examples are the cyclides of Dupin and those hypersurfaces obtained from isoparametric hypersurfaces in a sphere by Lie sphere transformation. Many results have been obtained about Dupin hypersurfaces over the past two decades, both local and global in nature. In this talk, we will survey the known results in the field, including recent local classifications of Dupin hypersurfaces with three principal curvatures, and those with four principal curvatures of multiplicity one and constant Lie curvature.

F. Helein, "Hamiltonian stationary Lagrangian surfaces in four-dimensional symplectic manifolds"

Abstract: We present a work done in collaboration with Pascal Romon. Hamiltonian stationary Lagrangian surfaces are Lagrangian surfaces which are critical points of the area functional assuming only Hamiltonian infinitesimal variations. The Euler-Lagrange equation of this constrained variational problem is a third order PDE which turns out to be completely integrable when the ambient symplectic space is homogeneous. It leads to a formulation using loop groups and Weierstrass type representations linked with quaternions and spinors.

R. Miyaoka, "Hypersurface Geometry and Integrable Systems",

Abstract: The classification of Isoparametric hypersurfaces, Dupin hypersurfaces, developable hypersurfaces, etc. is an old and new problem, which is related to the Lie sphere geometry, projective geometry, and recently, to the theory of integrable systems. We talk about the homogeneity of isoparametric hypersurfaces with six principal curvatures, the approach to the classification of Dupin hypersurfaces and developable hypersurfaces, as well as its relation to the Hamiltonian system of hydrodynamic type.

Y. Ohnita, "Geometry of the moduli spaces of harmonic maps into Lie groups"

Abstract: In this talk I shall discuss geometric structures on the moduli spaces of harmonic maps from compact Riemann surfaces into compact Lie groups and compact symmetric spaces. Our approach is to study the gauge-theoretic equations, which were investigated previously by Hitchin and Valli, coming from such harmonic maps and the moduli spaces of their solutions. This work is a joint work with Dr. Mariko Mukai (PD, Tokyo Metropolitan Univ.).

F. Pedit, "Quaternionic holomorphic geometry, Pluecker formulas and Dirac eigenvalue estimates I"

Abstract: We will develop the quaternionic holomorphic geometry of Riemann surfaces. This includes quaternionic holomorphic vector bundles, their Willmore energy, the Riemann-Roch formula, the Kodaira correspondence and quaternionic holomorphic curves in projective spaces. We will prove a quaternionic analogue of the classical Pluecker relations, which will give a lower bound on the Willmore energy of a quaternionic holomorphic line bundle. Applications of this result to eigenvalue estimates of the Dirac operator and an estimate of the spectral genus of minimal tori in the 3-sphere in terms of the volume will be discussed.

U. Pinkall, "Quaternionic holomorphic geometry, Pluecker formulas and Dirac eigenvalue estimates II"

Abstract: see above.

G. Thorbergsson, "Polar Actions on Rank One Symmetric Spaces"

Abstract: In this talk I will explain the classification of polar actions on rank one symmetric spaces obtained in collaboration with Fabio Podesta.