Course overview -- Wintersemester 2011/2012 (Archive)

We start with the classical results by Ecker and Huisken about mean curvature flow for entire graphs. Then we consider more general flows with speed depending on the curvatures. We are interested in understanding which statements hold for speeds that are increasing, homogenous functions of the principal curvatures. In particular if the homogeneity degree is one, we are able to deduce the asymptotic behaviour of entire graphs with linear growth at infinity.

Colding-Minicozzi solved Yau's conjecture that the dimension of the linear space of polynomial growth harmonic functions on Riemannian manifolds with nonnegative Ricci curvature is finite. The technique relies on the volume growth property of the manifold and the mean value inequality of harmonic functions. The Nash-Moser iteration gives the required properties of harmonic functions even in the metric (nonsmooth) setting. Alexandrov spaces are natural metric spaces with sectional curvature bounded below. Recently, the first and second order analysis are well understood in some aspects. Some results of harmonic functions on Alexandrov spaces will be discussed. Then we apply the Alexandrov geometry to develop the analysis on the graph with nonnegative sectional curvature.

Witten genus is the loop space analogue of the Hirzebruch A-hat genus. On a string manifold, the Witten genus is a modular form and is the equivariant index of the Dirac operator on the free loop space. Hohn and Stolz conjectured that existence of a positive Ricci curvature metric on a string manifold implies the vanishing of the Witten genus. In this talk, we will present vanishing results for generalized Witten genus on complete intersections and describe a possible mod 2 extension of the Hohn-Stolz conjecture. The talk is based on the joint work with Qingtao Chen and Weiping Zhang.

Let (M,g) be a riemannian surface and $G$ a g-natural nondegenerate metric on its tangent bundle TM. We compute explicitly the spectrum of some Jacobi operators on (TM , G) and give necessary and sufficient conditions for (TM , G) to be a 4-dimensional Osserman manifold.

Holomorphic curves are the most valuable tools to study the global properties of symplectic manifolds and also play a prominent role in string theory. In order to define algebraic invariants, one has to show that the moduli space of holomorphic curves carries a smooth structure of dimension equal to the Fredholm index of a nonlinear Cauchy-Riemann operator. Assuming that this nonlinear Cauchy-Riemann operator, viewed as a section in a Banach space bundle over a Banach manifold of maps, meets the zero section transversally, the desired result would follow from an infinite-dimensional bundle version of the implicit function theorem. In this talk I will show that multiply-covered holomorphic curves are the reason why transversality does not hold in general. Apart from giving hints at the new infinite-dimensional differential geometry of polyfolds, which were built in order to approach this problem in full generality, I will show how to achieve transversality in interesting special cases and finally illustrate a geometrical application to questions about stable hypersurfaces in symplectic manifolds.