The 8th DIFFERENTIAL GEOMETRY DAY
Wednesday the 16th of May 2012
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.
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).
(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.
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.