Differential Geometry Seminar Schedule for
Spring 2010
Fridays 3:00 - 3:50pm, SH 6617
4/30 Jeffrey Case, UCSB
``A new perspective on smooth metric measure spaces"
Abstract: The concept of a smooth metric measure space has recently arisen
as a useful object within Riemannian geometry, for example in Perelman's
formulation of Ricci flow as a gradient flow. There have been two
different methods proposed to study them. The first is by introducing the
Bakry-Emery tensor as a generalization of the Ricci curvature, which
allows one to recover many geometric properties (e.g. Bishop-Gromov
comparison theorem), and has recently been used by Lott, Villani, and
Sturm to define the notion of Ricci curvature lower bounds on a
(nonsmooth) metric measure space. The second is by introducing a
conformally invariant generalization of the Ricci curvature, which allows
one to recover many analytic properties (e.g. Sobolev inequalities), as
introduced by Chang, Gursky, and Yang. In both formulations, one
introduces a ``hidden dimension'' which, when it is infinite, recovers
many of the objects introduced by Perelman. One can also talk about
``quasi-Einstein'' metrics in this setting, which include conformally
Einstein metrics, gradient Ricci solitons, and warped product Einstein
metrics.
In this talk, I will introduce what I call ``conformally warped
manifolds'' as a way to unite these two points of view. I will discuss
the proper generalization of Ricci and scalar curvature in this setting,
and introduce the notion of quasi-Einstein metrics on these manifolds. I
will then discuss how these metrics arise as solutions of the natural
variational problem, and use this to discuss the natural analog of the
Yamabe problem to this setting. Throughout this talk, I will stress the
connections between the usual Riemannian setting and the $m=\infty$
setting, as it appears in Perelman's work. In particular, I will discuss
how Perelman's $F$ and $W$ functionals and the consequences of their study
naturally arise.
5/7 Yat-Sun Poon, UCR
``Deformation on Holomorphic Symplectic Spaces"
Abstract: In the realm of generalized complex geometry, a complex manifold could deform to a symplectic manifold. This deformation is part of the extended deformation theory of Kontsevich. The extended parameter spaces inherit Manin's weak Frobenius structure. We investigate the stability of such structures when the underlying complex manifold is a holomophic symplectic nilmanifold, and the deformtion is induced by an associated holomorphic Poisson vector field.
5/14 Qingtao Chen, USC
``Modular Forms, string^c structure, mod 2 Witten genus and generalized Hohn-Stolz Conjecture"
Abstract:The Witten genus is the loop space analogue of the Hirzebruch A-hat genus. On a string manifold, the Witten genus is a level 1 modular form over SL(2,Z). The homotopy refinement of the Witten genus leads to the theory of topological modular forms. In this talk, I will discuss two extensions of the original Witten genus. We construct generalized Witten genera on a kind of spin^c manifolds, which we call string^c manifolds and the generalized Witten genera become a level 1 modular form over SL(2,Z) on a string^c manifold. The other one is the mod 2 extension. Hohn and Stolz conjectured that existence of a positive Ricci curvature metric on a string manifold implies the vanishing of the Witten genus. We will present vanishing results for these generalized Witten genera on complete intersections and describe a possible mod 2 extension of the Hohn-Stolz conjecture. The talk is based on the joint work with Fei Han and Weiping Zhang.
6/8, 11am-12noon, Jiayu Li, Chinese Academy of Sciences
``Two ways to find holomorphic curves in K\"ahler
surfaces"
Abstract: Let $M$ be a K\"ahler surface and $\Sigma$ be a closed real
surface smoothly immersed in $M$. Let $\alpha$ be the K\"ahler
angle of $\Sigma$ in $M$. If $\cos\alpha>0$, we say $\Sigma$ is a
symplectic surface. We study the problem "{\it whether there is a
holomorphic curve in the homotopy class of a symplectic surface}".
In the talk we will present two approaches to the problem, one is
the mean curvature flow method, another one is the variational
approach.
Differential Geometry Seminar Schedule for
Winter 2010
Fridays 3:00 - 3:50pm, SH 6617
1/8 Yi-Jun Yao, Penn State
``Hopf cyclic cohomology of proper actions (joint work with X. Tang and W. Zhang)"
Abstract: We introduce a Hopf algebroid associated to a locally compact Lie
group proper action on a smooth manifold. We prove that the cyclic
cohomology of this Hopf algebroid is equal to the cohomology of
invariant differential forms. When the action is cocompact, with
Hodge theory, we show that every cyclic cohomology class of the Hopf
algebroid is represented by a generalized harmonic form. This
implies that the space of cyclic cohomology of the Hopf algebroid is
finitely dimensional.
1/22 Ilesanmi Adeboye, UCSB
``Rank One Symmetric Spaces and Volume"
Abstract: The group of isometries of hyperbolic n-space is a semisimple
Lie group. We show how an upper bound on the sectional curvature of this
group leads to an explicit lower bound on the volume of a hyperbolic
n-orbifold. We then discuss progress in extending this result to the
remaining non compact rank one symmetric spaces, i.e., complex hyperbolic
space, quaternionic hyperbolic space and the octonionic hyperbolic plane.
This is joint work with Guofang Wei.
1/29 Brian Clarke, Stanford University
``Metric Structures on the Manifold of Riemannian Metrics"
Abstract: The manifold $\mathcal{M}$ of all smooth Riemannian metrics over a closed, finite-dimensional base manifold carries itself several natural Riemannian metrics. We discuss the metric geometry of a particular one of these, the $L^2$ metric, chosen for its applications in Teichm\"uller theory and in previous investigations of the geometry and topology of $\mathcal{M}$. The main result is a description of the completion of $\mathcal{M}$ with respect to the $L^2$ metric.
At the end of the talk, we discuss some directions for further study, including applications to Teichm\"uller theory and the moduli space of Riemannian metrics. We also discuss how the metric geometry of the $L^2$ metric relates to that of a newly discovered metric structure on $\mathcal{M}$.
2/12 Boris Vertman, Stanford University
``Heat Kernel and Analytic Torsion on Manifolds with Edge Singularities"
Abstract: We discuss the construction of the heat kernel for the Friedrichs
extension of the Hodge Laplacian on Riemannian manifolds with non-iterated
incomplete edge singularities. We derive the heat trace asymptotics and
define the analytic torsion in this singular setup. We discuss variation
of the analytic torsion under metric changes. Under additional conditions
on the dimension and metric structures the analytic torsion is shown to
form a topological invariant, independent of the particular choice of an
edge metric.
2/19 Kevin Brighton, UCSB
``A Liouville-type theorem for smooth metric measure spaces"
Abstract: In 1976 Yau proved that any bounded harmonic function on a complete Riemannian manifold with nonnegative Ricci curvature must be constant. In this talk I'll review the classical case and then prove an extension of the result to smooth metric measure spaces with nonnegative Bakry-Emery-Ricci tensor.
Differential Geometry Seminar Schedule for
Fall 2009
Fridays 3:00 - 3:50pm, SH 6617
10/9 Rugang Ye, UCSB
``G2 geometry and the Laplacian flow I"
10/16 Rugang Ye, UCSB
``G2 geometry and the Laplacian flow II"
10/30 Guihua Gong, University of Puerto Rico
``Positive scalar curvature and
non-commutative geometry"
11/6, Jeffrey Case, UCSB
``On the nonexistence of quasi-Einstein metrics"
Abstract: We study Riemannian manifolds satisfying the equation $Ric+Hess(f)-\frac{1}{m}df^2=0$ by studying the associated PDE $\Delta_f f + ke^{2f/m}=0$ for $k\leq 0$. We develop a gradient estimate for such functions, and show that the only solutions on a complete manifold are constant functions. As an application, we show that there are no nontrivial Ricci flat warped products whose fibers are Einstein with nonpositive scalar curvature. We also show that one can take the limit $m\to\infty$ and get that for nontrivial steady gradient Ricci solitons $R+|df|^2$ is a positive constant.
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