We develop new results on global existence and convergence
of solutions to the gradient flow equation for the Yang-Mills energy
functional on a principal bundle, with compact Lie structure group, over
a closed, four-dimensional, Riemannian, smooth manifold, including the
following. If the initial connection is close enough to a minimum of the
Yang-Mills energy functional, in a norm or energy sense, then the
Yang-Mills gradient flow exists for all time and converges to a
Yang-Mills connection. If the initial connection is allowed to have
arbitrary energy but we restrict to the setting of a Hermitian vector
bundle over a compact, complex, Hermitian (but not necessarily Kaehler)
surface and the initial connection has curvature of type (1,1), then the
Yang-Mills gradient flow exists for all time, though bubble
singularities may (and in certain cases must) occur in the limit as time
tends to infinity. The Lojasiewicz-Simon gradient inequality plays a crucial role in our approach and we develop two versions of that inequality for the
Yang-Mills energy functional.

Pursuing an idea motivated by a question of S.-S. Chern from 1968 on the existence of
intrinsic Riemannian obstructions to minimality [Chern, S.-S.: Minimal submanifolds
in a Riemannian manifold (1968)], an important study of the very idea of curvature
was deepened after 1993 by B.-Y. Chen, then by other authors. In the last two decades,
B.-Y. Chen’s fundamental inequalities have been investigated by many authors in the
context of various geometric structures. In this talk, we start by presenting B.-Y. Chen’s
fundamental inequality for Kähler submanifolds in complex space forms, and we recall
why the case of Kähler surfaces in C^3 satisfying scal(p) = 4 inf sec(π^r ) appears
naturally and is important. Then we provide several characterizations of strongly minimal
complex surfaces in the complex three dimensional space. We focus our study on the question
of finding further examples of strongly minimal Kähler surfaces, as the question of a
complete classification of these geometric objects is still open.

We present a joint work with Richard Bamler.
We consider Ricci flows that satisfy certain scalar curvature bounds. It is found that the time derivative for the solution of the heat equation and the curvature tensor have better than expected bounds. Based on these, we derive a number results. They are: bounds on distance distortion at different times and Gaussian bounds for the heat kernel, backward pseudolocality, L^2-curvature bounds in
dimension 4.

In this talk I will describe a cohomological formula for a higher
index pairing between invariant elliptic differential operators and
differentiable group cohomology classes. This index theorem generalizes the
Connes-Moscovici L^2-index theorem and its variants. This is joint work with
Markus Pflaum and Hessel Posthuma.