DIFFERENTIAL GEOMETRY WORKSHOP

at LUND 2014

in connection with

THE 10TH DIFFERENTIAL GEOMETRY DAY


ABSTRACTS

Thursday the 15th of May 2014

Martin Svensson (SDU Odense, DK) Timelike CMC surfaces and the geometric Cauchy problem I will describe the loop group construction of timelike CMC surfaces in Minkowski 3-space. I will show how to use this construction to solve the associated geometric Cauchy problem: given a curve and a vector field along this curve, find a timelike CMC surface that contains this curve and is everywhere tangent to the given vector field. Finally, I will discuss the surface singularities that arise when the boundary of the big Birkhoff cell is approached, and how to construct surfaces with prescribed singularities. The talk is based on joint work with David Brander (DTU). Sigmundur Gudmundsson (Lund University, SE) Harmonic Morphisms from Homogeneous Spaces - Some Existence Theory Harmonic morphisms $\phi:(M,g)\to(N,h)$ between Riemannian manifolds are solutions to over-determined non-linear systems of partial differential equations determined by the geometric data of the manifolds involved. For this reason, harmonic morphisms are difficult to find and have no general existence theory, not even locally. In this talk we discuss recent existence results for complex-valued harmonic morphisms from various Riemannian homogeneous spaces, in particular, Lie groups and symmetric spaces. Marina Ville (University of Tours, FR) Harmonic morphisms from 4-manifolds to 2-surfaces We investigate the singular points of a harmonic morphism F from a Riemannian 4-manifold M to a surface. As an illustration we will recall a construction by Jean-Marie Burel on S^4. Then we will show that under extra assumptions (an isolated singular point or a compact M), F is holomorphic near the singular point w.r.t. a local almost complex structure on an open set of M. Given our knowledge of J-holomorphic curves, this gives us a good description of the singular points. (joint work with Ali Makki) Eric Loubeau (University of Brest, FR) Surfaces in spheres: biharmonic and CMC CMC surfaces in spheres are investigated under the extra condition of biharmonicity. From the work of Miyata, especially in the flat case, we give a complete description of such immersions and show that for any $h\in (0,1)$ there exist CMC planes and cylinders in $S^5$ with $|H|=h$, while a necessary and sufficient condition on $h$ is found for the existence of CMC tori in $S^5$.

Friday the 16th of May 2014

Jonas Nordström (Lund University, SE) Harmonic morphisms from Riemannian Lie groups Radu Slobodeanu (University of Bucharest, RO) Shear-free fluids and p-harmonic morphisms. A local classification Radu Pantilie (Romanian Academy, RO) The Penrose transform in quaternionic geometry I shall describe the Penrose transform, in the setting of quaternionic geometry, with a view towards `harmonic' morphisms (work in progress). Steen Markvorsen (Danish Technical University, DK) Fields of Pointed Ovaloids - A Geometric Analysis In a given chart the r-distance spheres of a Riemannian manifold are - for a sufficiently small r - represented by a pointed field of strongly convex ovaloids. In the talk we will motivate and discuss the natural questions concerning firstly the reconstruction of the metric from the ovaloid field and secondly the extension of the ovaloid field to any other radius of the distance spheres.