Differential Geometry Seminar Schedule for Spring 2006

Fridays 3:30 - 4:30pm, SH 6635


Abstract: This talk will develop part of the foundation needed to develop a partial Morse theory for conformal harmonic maps from a Riemann surface into a Riemannian manifold. Such maps are also called parametrized minimal surfaces. A partial Morse theory for such objects should parallel the well-known Morse theory of smooth closed geodesics. The first step needed is a bumpy metric theorem which states that when a Riemannian manifold has a generic metric, all prime minimal surfaces are free of branch points and lie on nondegenerate critical submanifolds. (A parametrized minimal surface is prime if it does not cover a parametrized minimal surface of lower energy.) We will present such a theorem and describe some applications.

5/19 3pm Jianguo Cao, University of Notre Dame ``A new proof of Gromoll-Grove diameter rigidity theorem"

Abstract: In this lecture, we present a new proof of Gromoll-Grove diameter rigidity theorem. It is well-known that the total Betti number of a compact Riemannian manifold can be estimated by its diameter and lower bound of sectional curvature, by a theorem of Gromov. Along this line, Gromoll and Grove proved that any Riemannian manifold M with sectional curvature >= 1 and diameter >= pi/2 must be either homeomorphic to a twisted sphere or isometric to a compact symmetric space of rank one. The proof of this theorem was previously provided by Gromoll, Grove and Wilking by the careful study of Riemannian submersions from unit sphere to M and loop space of M. We will present an intrinsic proof, which does not use any results on loop spaces. Instead, we introduce a spherical metric on the tangent space of M at a point, together with a Hessian comparison theorem. This lecture is accessible to general audience including graduate students.

5/19 4:15pm Damin Wu, Stanford University ``Asymptotics of Kahler-Einstein metrics on quasi-projective manifolds"

Abstract: For deeper applications of Kahler-Einstein metrics, only knowing existence and uniqueness are not enough. In this talk we will present an iteration method to derive the asymptotic expansion of Kahler-Einstein metrics near the boundary simple normal crossing divisor. Certain isomorphism theorems are derived to finish the proof.

5/26 Vitali Kapovitch, University of Maryland ``Fundamental groups of manifolds with lower Ricci curvature bounds"

Abstract: (joint work with B. Wilking) We prove that the fundamental group of a manifold with an upper diameter and a lower Ricci curvature bound has a presentation with a universally bounded number of generators. We also prove a conjecture of Gromov that a manifold which admits almost nonnegative Ricci curvature has a virtually nilpotent fundamental group.

6/9 Yu Ding, UCSB ``Report on Bohm-Wilking's paper on positive curvature operator"

Differential Geometry Seminar Schedule for Winter 2006

Fridays 3:30 - 4:30pm, SH 6635

1/13 Pengzi Miao, UCSB ``A note on existence and non-existence of minimal surfaces in some asymptotically flat 3-manifolds"

Abstract: We give some observation on existence and non-existence of minimal surfaces in some asymptotically flat and scalar flat manifolds which is topologically $R^3$. The discussion is motivated by problems of existence of horizons in general relativity.

1/20 Neshan Wickramasekera, UCSD ``Structure of singularities of energy minimizing maps"

Abstract: I will describe a Holder continuity result for the singular set of an energy minimizing map from a domain in $R^{n}$ into a compact, real analytic manifold satisfying a certain topological condition. This result is based on a rigidity property for cylindrical tangent maps, which is analogous to rigidity of stable minimal hypercones established recently.

1/27 Jiaping Wang, University of Minnesota, visiting UCI ``Harmonic functions and stable minimal hypersurfaces"

Abstract: We intend to explain how to apply the information of harmonic functions to study the geometry and topology of stable minimal hypersurfaces.

2/3 Xianzhe Dai, UCSB ``Mass under the Ricci flow"

Abstract: Ricci flow is an important geometric evolution equation in Riemannian Geometry. It was introduced by R. Hamilton in 1982 and used extensively by him to prove some outstanding results on 3-manifolds and 4-manifolds. Recently it has been used spectacularly by G.Perelman to study the geometrization conjecture on 3-manifold. The flow has also been very useful in the study of pinching results and metric smoothing process. As a natural geometric tool, Ricci flow should be used to study properties of physically meaningful objects such as mass, entropy, etc. In this talk we would like to discuss the behavior of the ADM mass under the Ricci flow.

2/10 Yu Zheng, Eastern China Normal University, visiting UCSD ``One application of Moser's weak maximal principal on the $\epsilon$ -regularity estimation "

Abstract: In this paper one flow for ASD connection in $\Omega^{2+}(Ad P)$ for the principal bundle $P$ over four dimensional compact Riemannian manifold $(M, g)$ was studied. The local existence and the analysis of the phenomenon of blowing up including the local small energy regularity for this flow are given based on the application of Moser's weak maximal principal.

Special time 2pm 2/17 Detang Zhou, Universidade Federal Fluminense, visiting UCI ``Global properties of constant mean curvature hypersurfaces"

Abstract: We consider constant mean curvature hypersurfaces in a Riemannian manifold. Instead of using strong stability which is natural for minimal hypersurfaces, we use weak stability for constant mean curvature hypersurfaces which comes naturally from soap bubble problems and isoperimetric problems. Some structure theorems and nonexistences of complete noncompact weakly stable constant mean curvature hypersurfaces are proved.

2/24 no meeting

3/3 Reiner Schatzle, University of Tubingen ``Estimates for the Willmore flow"

3/10 no meeting

3/17 (cancelled) Pengzi Miao, UCSB ``On an inequality between the ADM mass and the boundary capacity"

Abstract: In his proof of the Riemannian Penrose Inequality, Bray found an interesting inequality between the ADM mass and the boundary capacity for asymptotically flat 3-manifolds with minimal surfaces boundary. In this talk, we give a similar inequality for asymptotically flat 3-manifolds whose boundary does not necessarily have vanishing mean curvature. As a byproduct, we obtain new upper estimates of electrostatic capacity of surfaces in the Euclidean space. Our main tool is the weak inverse mean curvature flow technique developed by Huisken and Ilmanen. This is a joint result with H. Bray.

Differential Geometry Seminar Schedule for Fall 2005

Fridays 3:30 - 4:30pm, SH 6635

9/23 Rafe Mazzeo, Stanford University, visiting KITP ``Some linear and nonlinear problems in higher rank geometry"

9/30 Richard Melrose, MIT, visiting KITP ``Compactification of moduli spaces of monopoles"

10/7 Gary Horowitz, UCSB ``An Informal Lecture: Black Holes in Higher Dimensions"

10/14 Feng Luo, Rutgers University ``Volume and constant curvature metrics on triangulated 3-manifolds"

Abstract: We introduce an finite dimensional variational approach to find constant curvature metrics on triangulated closed 3-manifolds. The approach is based on the Schlaefli formula for volume of tetrahedra. Schlaefli formula suggests that the volume is best expressed in terms of dihedral angles than in edge lengths. Based on this observation, we defined the concept of an angle structure on a tetrahedron and on a triangulated closed 3-manifold, and defined their volume. These are natural generalizations of constant sectional curvature metrics and their volume. It is shown that the volume functional can be extended continuously to the compact closure of the moduli space of angle structure, verifying a conjecture of John Milnor. The main result shows that for a 1-vertex triangulation of a closed 3-manifold if the volume function on the moduli space of all angle structures has a local maximum point, then either the manifold admits a constant curvature Riemannian metric, or the manifold is reducible.

10/21 Dan Freed, University of Texas, Austin, visiting KITP ``Invariants of Dirac operators in odd dimensions and applications to M-theory"

10/28 Rugang Ye, UCSB ``Curvature Estimates for the Ricci Flow"

11/4 Pengzi Miao, UCSB ``Black hole topology in higher dimensions (after Galloway and Schoen)"

11/11, Veterans' day

11/18 Hisham Sati, visiting KITP ``The partition functions of M-theory and type II string theory, and generalized cohomology"

ABSTRACT: The degree-four field of M-theory can be described in terms of an E8 gauge theory in eleven dimensions, and hence the partition function can be calculated using integral cohomology. In type IIA string theory, the Ramond-Ramond fields are taken as elements of K-theory of spacetime, and hence the partition function can be calculated using K-theory. Diaconescu, Moore and Witten (DMW) have demonstrated the match between the two partition functions. We report on recent work with V. Mathai and with I. Kriz on the corresponding situation obtained by including the Neveu-Schwarz fields and by cancelling the DMW anomaly. This leads to a refinement of the K-theoretic description to twisted K-theory and to elliptic cohomology, respectively. We also briefly discuss a proposal for a generalized cohomology theory describing the non-gravitational field(s) of M-theory.

11/25 Thanksgiving

12/2 Yu Ding, UCSB ``Introduction to conformally compact Einstein manifolds: a brief survey on the recent work of Anderson"

12/9 Yu Ding, UCSB ``Introduction to conformally compact Einstein manifolds: a brief survey on the recent work of Anderson, continue"

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