The 8th DIFFERENTIAL GEOMETRY DAY

at LUND

Wednesday the 16th of May 2012


ABSTRACTS

Anna Sakovich (KTH Stockholm) Penrose type inequalities for asymptotically hyperbolic graphs Penrose type inequalities is a collective term used to describe a class of inequalities giving a lower bound for the mass of an initial data set (M,g,K) (here (M,g) is a Riemannian manifold, K is a symmetric 2-tensor) with nonnegative energy density in terms of the area of suitable hypersurfaces representing black holes. This topic has remained an active area of research since 1973, when an inequality of this kind was first proposed by Penrose. If (M,g) is asymptotically Euclidean then taking K=0 (time symmetric case) one can formulate what is known as the Riemannian Penrose inequality, first proved in 2001 (Huisken&Ilmanen, Bray). A simplified proof of this result was obtained by Lam in 2010 for asymptotically Euclidean manifolds which are the graphs of a smooth function over R^n. After reviewing these results, we will consider time symmetric asymptotically hyperbolic initial data sets. Almost no results being available in this direction, we will present a proof of Penrose type inequalities for asymptotically hyperbolic manifolds which are the graphs of a smooth asymptotically constant function over H^n. This result is inspired by the aforementioned work of Lam, and is a joint work of Dahl, Gicquaud, and the speaker. Martin Svensson (SDU Odense) The geometric Cauchy problem for timelike CMC surfaces 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). Fran Burstall (University of Bath) Classical projective geometry and harmonic maps I shall use ideas of classical projective geometry to describe a uniform approach to several variational problems in parabolic geometries and to the harmonic Gauss maps associated with them. Hajime Urakawa (Tohoku University) Geometry of biharmonic maps --- Chen's conjecture, bubbling phenomena and symplectic geometry --- A harmonic map is a critical map of the energy functional, and the Euler-Lagrange equation is that the tension field vanishes identically. The bi-harmonic map is a critical map of the 2-energy which is by definition half of the integral of the square norm of the tension field. In my talk, I will give a brief survey of my recent works on harmonic maps and bi-harmonic maps: (1) After explaining basic notion of bi-harmonic maps, we will show the reduction theorems of bi-harmonic maps into compact Lie groups with the bi-invariant Riemannian metric and compact Riemannian symmetric space, and show many examples of bi-harmonic maps but not harmonic. (2) Then, B.Y. Chen's conjecture is: Any isometric bi-harmonic immersion into the Euclidean space is minimal (harmonic), and also the generalized Chen's conjecture is: Any isometric bi-harmonic immersion into a Riemannian manifold of non-positive curvature is harmonic. The Chen's conjecture is still open, but the generalized one already turned out false by given the counter-example due to Y-L. Ou and L. Tang, 2010. Even so, we will show that the generalized Chen's conjecture is true if we assume that the integral of square norm of the mean curvature tensor field is finite. (3) Finally, we will show our recent results of regularity theorems and bubbling phenomena of harmonic maps and bi-harmonic maps.