# Differential Geometry Seminar Schedule for
Spring 2006

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

### 4/21 Doug Moore, UCSB ``BUMPY METRICS FOR MINIMAL SURFACES"

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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