A Berstein type theorem of entire Willmore graphs in R^3
Dienstag, 25.10.11, 16:15-17:15, Raum 127, Eckerstr. 1
Finite volume schemes for balance laws on time dependent Riemannian manifolds: An error estimate
Mittwoch, 26.10.11, 16:15-17:15, Hörsaal II, Albertstr. 23b
Singular spin structures and Witten spinors
Mittwoch, 2.11.11, 16:15-17:15, Hörsaal II, Albertstr. 23b
Unlike a 3-dimensional manifold, a higher dimensional manifold need not be spin. On an oriented Riemannian manifold the obstruction to having a spin structure is given by the second Stiefel-Whitney class. I will show that even when this obstruction does not vanish, it is still possible to define a notion of singular spin structure and associated singular Dirac operator. Then, modeling on Witten's proof of the Positive Mass Theorem, I will define the notion of Witten spinor on an asymptotically flat nonspin manifold, show their existence and describe their properties.
Entire graphs moving by curvature flows: mean curvature flow and beyond
Dienstag, 8.11.11, 16:15-17:15, Raum 127, Eckerstr. 1
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.\n
Some Remarks on Huisken's Monotonicity
Dienstag, 15.11.11, 16:15-17:15, Raum 127, Eckerstr. 1
Existence of immersed spheres minimizing curvature functionals in compact 3-manifolds
Dienstag, 22.11.11, 16:15-17:15, Raum 127, Eckerstr. 1
Polynomial Growth Harmonic Functions in Metric Spaces
Dienstag, 29.11.11, 16:15-17:15, Raum 127, Eckerstr. 1
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.
Generalized Witten Genus and Complete Intersections
Dienstag, 6.12.11, 16:15-17:15, Raum 127, Eckerstr. 1
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.
Osserman natural tangent bundles on surfaces
Dienstag, 13.12.11, 16:15-17:15, Raum 127, Eckerstr. 1
Let (M,g) be a riemannian surface and \(G\) a\ng-natural nondegenerate\nmetric on its tangent bundle TM. We compute explicitly the\nspectrum of some Jacobi operators on (TM , G) and give\nnecessary and sufficient conditions for\n(TM , G) to be a 4-dimensional Osserman manifold.\n\n
On the transversality problem for the Cauchy-Riemann operator in symplectic geometry
Dienstag, 20.12.11, 16:15-17:15, Raum 127, Eckerstr. 1
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.
Regularity for the partitioning problem: Exclusion of false branch points using a Douglas condition
Dienstag, 10.1.12, 16:15-17:15, Raum 127, Eckerstr. 1
Volume and area renormalizations for Conformally compact Einstein metrics.
Dienstag, 17.1.12, 16:15-17:15, Raum 127, Eckerstr. 1
Volume and area renormalizations for Conformally compact Einstein metrics II
Dienstag, 31.1.12, 16:15-17:15, Raum 127, Eckerstr. 1
Gauge Theory on G2-manifolds
Mittwoch, 1.2.12, 16:15-17:15, Hörsaal II, Albertstr. 23b
G2-manifolds are a special kind of Ricci-flat 7-manifolds. On a G2-manifold one can study a special class of Yang-Mills connections, called G2-instantons. The theory of G2-instantons is governed by an index zero supercritical elliptic equation.\n\nConjecturally, to each bundle over a G2-manifold one can associate a "G2 Casson invariant" obtained by "counting" the moduli space of G2-instantons. Such an invariant could prove to be very useful in understanding the landscape of G2-manifolds. Unfortunately, there are quite a large number of technical problems in the theory of G2-instantons. I will briefly discuss one of the major issues having to do with non-compactness phenomena.\n\n An important class of G2-manifolds are those arising from Joyce's generalised Kummer construction. I will briefly review this construction, and explain a program to study G2-instantons on these G2-manifolds. In particular, I will give an existence result obtained via a gluing method as well as a partial compactness theorem. Time permitting, I will describe a remaining analytic problem whose solution would make the "G2 Casson invariant" rigorous and, in fact, computable on certain bundles over generalised Kummer constructions.\n
Volume and area renormalizations for Conformally compact Einstein metrics III
Dienstag, 7.2.12, 16:15-17:15, Raum 127, Eckerstr. 1
Volume and area renormalizations for Conformally compact Einstein metrics IV
Dienstag, 14.2.12, 16:15-17:15, Raum 127, Eckerstr. 1
Lagrangian mean curvature flow and Gauss maps
Mittwoch, 15.2.12, 16:15-17:15, Hörsaal II, Albertstr. 23b
In the first part of the talk we will give an overview\nof the Lagrangian mean curvature flow. In the second part we develop a connection between the Gauss maps of certain hypersurfaces and the Lagrangian mean curvature flow of their Gauss maps into Grassmannians and present some convergence results in the two-positive case.\n\n
Regularity for Fractional Harmonic Maps in the critical dimension
Mittwoch, 22.2.12, 16:15-17:15, Hörsaal II, Albertstr. 23b
The theory of fractional harmonic maps can be seen as an extension of Riviere's celebrated result for critical points of conformally invariant variational functionals in two dimensions.\nI will present some arguments necessary for the regularity results, as well as applications of these arguments for related energies.\n