Course overview -- Sommersemester 2012 (Archive)

I will discuss the geometric problem of finding KE metrics which are bent along a divisor, and the equivalent analytic problem of solving the associated singular Monge-Ampere equation. There are several interesting applications for these metrics. Furthermore, the classical Aubin-Yau estimates do not work in this setting and a new route to the solvability of this equation must be found. This is joint work with Jeffres and Rubinstein.

Aufbauend auf den Ergebnissen meines letzten Vortrags werde ich (nach einer kurzen Wiederholung) die Unabhängigkeit von getroffenen Wahlen und die Natürlichkeit der Konstruktion untersuchen.

We discuss how to show that the flat solid torus is scattering rigid. We will consider compact Riemannian manifolds M with boundary N. We let IN be the unit vectors to M whose base point is on N and point inwards towards M. Similarly we define OUT. The scattering data (loosely speaking) of a Riemannian manifold with boundary is map from IN to OUT which assigns to each unit vector V of IN a unit vector W in OUT. W will be the tangent vector to the geodesic determined by V when that geodesic first hits the boundary N again. This may not be defined for all V since the geodesic might be trapped (i.e. never hits the boundary again). A manifold is said to be scattering rigid if any other Riemannian manifold Q with boundary isometric to N and with the same scattering data must be isometric to M. In this talk we will discuss the scattering rigidity problem and related inverse problems. There are a number of manifolds that are known to be scattering rigid and there are examples that are not scattering rigid. All of the known examples of non-rigidity have trapped geodesics in them. In particular, we will see that the flat solid torus is scattering rigid. This is the first scattering rigidity result for a manifold that has a trapped geodesic. The main issue is to show that the unit vectors tangent to trapped geodesics in any such Q have measure 0 in the unit tangent bundle of Q. We will also consider scattering rigidity of a number of two dimensional manifolds (joint work with Pilar Herreros) which have trapped geodesics..

A complete Riemannian manifold has no conjugate points if each of its geodesics is homotopically minimizing, i.e. if each lift of each geodesic to the universal Riemannian covering is minimizing. A theorem of E. Hopf from 1948 states that 2-tori with no conjugate points are flat. We prove flatness in case of the plane and the cylinder under optimal conditions on the area growth. The area growth of a surface is defined as the limit inferior, as r tends to infinity, of the quotient of the area of the metric r-ball about an arbitrarily fixed point and the area of a metric r-ball in the Euclidian plane. We prove that a complete plane with no conjugate points has area growth greater than or equal to one, and that equality holds only in the Euclidian case. The results were obtained in joint work with Victor Bangert.

For manifolds Poincaré duality is one of the most important properties of singular (co)homology theory. However proceeding to singular spaces, in general ordinary singular (co)homology does not satisfy
Poincaré duality no more. But there are several generalized
(co)homology theories for pseudomanifolds that satisfy Poincaré duality. One of those theories is M. Banagl's (co)homology theory of Intersection Spaces. In [Ban11] M. Banagl derived an alternate description of Intersection Space cohomology of a stratified pseudomanifold X, in cases where one has a singular stratum with flat link bundle endowed with a Riemannian metric such that the structure group of the bundle is contained in the isometries of the link. For that purpose he makes use of a certain subcomplex of the complex of differential forms on M, the non-singular part of X. In the isolated-singularity case the existence of an isomorphism between the two descriptions was shown. We want to generalize this De Rham isomorphism to the non-isolated
singularity case where we have a trivial link bundle. We therefore
make use of the Künneth-theorem.