# Differential Geometry Seminar Schedule for
Spring 2014

## Fridays 3:00 - 3:50pm, SH 6635

### 4/4 Bogdan Suceava, California State University,
Fullerton
"From a Question of S.-S. Chern to the Geometry of Strongly Minimal
Submanifolds: New Inequalities for Curvature"

Abstract:
In 1968, S.-S. Chern raised to attention the question that there might
be further Riemannian obstructions for a manifold to admit an isometric
minimal immersion into an Euclidean space. This question triggered a
whole new direction of study in the geometry of submanifolds. In the
last two decades, there have been important advances in the study of
new curvature invariants, many of these efforts due to B.-Y. Chen. We
will present a few recent developments in this theory, and we focus our
attention on several classes of inequalities in the geometry of
submanifolds in Riemannian and Kaehlerian context.

### 4/25 Lihan Wang, UCI
"HODGE THEORY ON COMPACT SYMPLECTIC MANIFOLDS WITH BOUNDARY"

Abstract: We study Hodge theory for symplectic Laplacians on compact
symplectic manifolds with boundary. These Laplacians are novel as they
can be associated with symplectic cohomologies and be of fourth-order.
We prove various Hodge decompositions and use them to obtain the
isomorphisms between the cohomologies and the spaces of harmonic fields
with certain prescribed boundary conditions. In order to establish
Hodge theory in the non-vanishing boundary case, we are required to
introduce some new boundary conditions. As an application, our results
can be used to solve boundary value problems of differential forms.
This is a joint work with Li-Sheng Tseng.

### 5/2 Jeffrey Streets, UCI
"On the singularity formation in fourth-order curvature flows"

Abstract: The L2 norm of the Riemannian curvature tensor is a natural
energy to associate to a Riemannian manifold, especially in dimension
4. A natural path for understanding the structure of this
functional
and its minimizers is via its gradient flow, the "L2 flow." This
is a quasi-linear
fourth
order parabolic equation for a Riemannian metric, which one might hope
shares behavior in common with the Yang-Mills flow. We verify
this idea by exhibiting structural results for finite time
singularities of this
flow resembling results on Yang-Mills flow. We also exhibit a new
short-time existence statement for the flow exhibiting a lower bound
for the existence time purely in terms of a measure of the volume
growth of the initial data. As corollaries we establish new compactness
and diffeomorphism finiteness theorems for four-manifolds generalizing
known results to ones with have effectively minimal
hypotheses/dependencies. These results all rely on a new
technique for
controlling the growth of distances along a geometric flow, which is
especially well-suited to the L2 flow.

### 5/9 Robert Ream, UCSB
"Minimal Two-Spheres in Manifolds with Pinched Curvature"

Abstract: When the compact manifold $M$ has a suitably pinched, generic
Riemannian metric, we show that it has many minimal two-spheres of low
energy using Morse inequalities for the $\alpha $-energy of Sacks and

Uhlenbeck. The difficulty is controlling bad behavior of a
sequence of $\alpha $-energy critical points as $\alpha $ approaches
one. The two bad behaviors investigated are convergence toward a
bubble tree or a branched cover of a

minimal sphere of lower energy. We control these difficulties by
making estimates on the index of bubble trees and branched covers. This
proves existence of many minimal two-spheres of low index.

### 5/12-15 Please attend the
distinguished lectures by Prof. Alice
Chang and Paul Yang.

### 5/23 Yuanqi Wang, UCSB
"$C^{2,\alpha}$-estimate for Monge-Ampere equations with
H\"older-continuous right hand side"

Abstract: Given an elliptic complex Monge-Ampere equation
and H\"older exponents $0<\alpha<\alpha^{\prime}$, suppose
$\phi\in C^{2,\alpha^{\prime}}$ is a solution which possesses an
uniform

$C^{1,1}-$bound, we present a new proof to the
$C^{2,\alpha}$-aprori estimate for the solution $\phi$. Our
estimates only depends on the $C^{\alpha^{\prime}}-$norm of
the right hand side of the equation. Our new proof also works
for the real Monge-Ampere equations. Our new
proof has direct applications in the theory of complex Monge-Ampere
equations with conical singularities, including the conical
K\"ahler-Einstein equations, and the conical K\"ahler-Ricci flows.

### 5/30 Lee Kennard, UCSB
"What rational homotopy theory tells us about the Wilhelm conjecture"

Abstract: A conjecture of Fred Wilhelm is the following: Given a
Riemannian manifold M with positive sectional curvature, if M admits a
Riemannian submersion to another manifold B, then the dimension of M is
less than twice the dimension of B. We discuss some examples supporting
this conjecture and some evidence for it. I will then discuss new
evidence for the conjecture obtained using topological methods. This is
joint work with Manuel Amann.

# Differential Geometry Seminar Schedule for
Winter 2014

## Fridays 3:00 - 3:50pm, SH 6635

### 1/31 Guofang Wei, UCSB
"Monotonicity Formulas for Ricci Curvature"

Abstract: Bishop-Gromov's volume comparison has been a very useful tool
for studying manifolds with Ricci curvature lower bound. Its essential
information is contained in the monotonicity of a volume ratio.
Perelman obtained several monotonicity formulas along the Ricci flow
which are key tools in his work on the geometrization conjecture.
Recently Colding and Minicozzi-Colding introduced some monotonicity
formulas associated with harmonic functions. Joint with B. Song and G.
Wu we extend and generalize these to Bakry-Emery Ricci curvature.

### 2/7 Changliang Wang, UCSB
"Linear and dynamical stability of Einstein metrics"

Abstract: There are two kinds of stability problems for Einstein
metrics, i.e. linear stability and dynamical stability. Natasa
Sesum proved that in Ricci flat case dynamical stability implies linear
stability and linear stability together with an integrability
assumption imply dynamical
stability. Recently, Robert Haslhofer and Reto Muller improved Sesum's
result by getting rid of the integrability assumption. Then more
recently, Klaus Kroncker generalized Haslhofer and Muller's result to
general
Einstein metrics.

### 2/14 Yuanqi Wang, UCSB
"Liouville theorem for complex Monge-Ampere equations with conic

singularities"

Abstract: Following Calabi, Pogorelov, Evans-Krylov-Safanov, and
Trudinger's pioneer work on interior regularities and liouville
theorems for Monge-Ampere equations, we prove the Liouville
theorem for conic K\"ahler-Ricci flat metrics. We

also discuss various applications of this Liouville theorem to conic
K\"ahler geometry.

### 2/21 Peng Wu, Cornell University
"Einstein four-manifolds of three-positive curvature operator"

Abstract:
We will show that Einstein four-manifolds of three-positive

curvature operator are isometric to $(S^4, g_0)$, $(\mathbb{R}P^4,
g_0)$, or

$(\mathbb{C}P^2, g_{FS})$. This is joint work with Xiaodong Cao.

### 2/28 Yuanqi Wang, UCSB
"Liouville theorem for complex Monge-Ampere equations with conic

singularities II"

Abstract: Following Calabi, Pogorelov, Evans-Krylov-Safanov, and
Trudinger's pioneer work on interior regularities and liouville
theorems for Monge-Ampere equations, we prove the Liouville
theorem for conic K\"ahler-Ricci flat metrics. We

also discuss various applications of this Liouville theorem to conic
K\"ahler geometry.

### 3/7 Doug Moore, UCSB
"Morse Theory for Minimal Tori"

Abstract:
Morse theory might have developed in three stages. The first
stage would have described the relationship between critical points on
a finite dimensional manifold and the topology of the manifold.
The second might have been "calculus of variations in the large" for
ODE's including the Morse theory of geodesics. Palais and Smale
formulated this in terms of infinite-dimensional manifolds, and Smale
expressed the hope that this approach might lead to a similar theory
for minimal surfaces, and related PDE's. Sacks and Uhlenbeck described
how to perturb the theory of minimal surfaces and develop a Morse
theory for the perturbed problem, initiating the third stage.

In this talk I will describe what happens when the perturbation is
turned off. One gets a partial Morse theory in the limit.
Applications to the existence of minimal tori in Riemannian manifolds
will be presented.

### 3/14 Tiancong Chen, UCSB
"Prescribing Curvature to Graphs"

Abstract:
We review various theorems on the existence of graphs whose curvature
is prescribed. Then we present a new theorem. The is a joint work with
Q. Han.

# Differential Geometry Seminar Schedule for
Fall 2013

## Fridays 3:00 - 3:50pm, SH 6635

### 10/4 Yuanqi Wang, UCSB
"On the long time behavior of the Conical K\"ahler-Ricci flow"

Abstract: We show that the conical K\"abler-Ricci flow actually exists
for all time. This long time existence results gives

convergence Theorems when the twisted C_{1,beta}(M) is nonpositive,
thus completely settled down existence problem

of K\"ahler-Einstein metrics log Calabi-Yau pairs. We also discuss some
related topics.

### 10/11 Xin Zhou, MSRI and MIT
"On the min-max hypersurface in manifold of positive Ricci curvature"

Abstract: In this talk, we will discuss the shape of the min-max
minimal hypersurface produced by Almgren-Pitts corresponding to the
fundamental class of a Riemannian manifold (M^{n+1},g) of positive
Ricci curvature with 2 ≤ n ≤ 6. We characterize the Morse index,
volume and multiplicity of this min-max hypersurface.

### 10/17 please attend the RTG
Seminar by Greg Galloway

### 10/18 please attend the RTG
Seminar by Greg Galloway

### 10/25 Maree Jaramillo, UCSB
"Fundamental Groups of Spaces with Bakry-Emery Ricci Tensor Bounded
Below"

Abstract: The Bakry-Emery Ricci tensor is a natural extension of Ricci
curvature on smooth metric measure spaces. Since topological and
geometric information can be obtained for manifolds with Ricci
curvature bounded
from below, it is natural to ask if the same information holds true for
smooth metric measure spaces with Bakry-Emery Ricci tensor bounded from
below. Using Guofang Wei and Will Wylie’s
comparison theorems and an
extension of Kevin Brighton’s gradient estimate on smooth metric
measure spaces, we extend the Almost Splitting Theorem of
Cheeger-Colding to the smooth metric space setting. Using this
Almost Splitting theorem, we

show that the fundamental group of the smooth metric measure space with
a
lower bound on volume has almost abelian fundamental group. We also
show
that the number of generators of the fundamental group of a smooth
metric
measure space with Bakry-Emery Ricci tensor bounded from below is
uniformly bounded. The results on the fundamental group are extensions
of theorems which
hold for Riemannian manifolds with Ricci curvature
bounded from below.
### 11/8 Lee Kennard, UCSB
"On Chern's question with symmetry"

Abstract: In 1965, Chern asked the following: If M is a compact
manifold with positive sectional curvature, is every abelian subgroup
of the fundamental group cyclic? Shankar proved in 1998 that the answer
is no. I wil discuss conditions under which the answer is yes. The
proof involves three obstructions to free actions on positively curved
rational homology spheres.

### 11/15 Paul Lee, The Chinese University of Hong Kong, visiting MSRI
"Ricci curvature type lower bounds for sub-Riemannian structures on
Sasakian manifolds"

Abstract: In this talk, we introduce a type of Ricci curvature lower
bound
for a natural sub-Riemannian structure on Sasakian manifolds. We will
show
that analogues of measure contraction properties, Bishop volume
comparison
theorem, and Laplacian comparison theorem hold under this lower bound.

### 11/22 Xianzhe Dai, UCSB
"Analytic torsion and degeneration via Eguchi-Hanson instanton"

Abstract: Analytic torsion is a combination of determinants of the
Laplacians on differential forms
introduced by Ray-Singer as an analytic counterpart of the topological
invariant, the Reidemeister
torsion. It is trivial in even dimensions, but there is a complex
version defined for complex manifolds.
Motivated by questions from modular forms and mirror symmetry, we study
the degeneration of
the analytic torsion when the underlying manifolds develop orbifold
singularities (via Eguchi-Hanson
instanton). This is joint work with Ken-Ichi Yoshikawa.

### 12/6 Hao Fang, University of Iowa
"A spectral approach to singularity theory"

Abstract: In a joint work in progress with Fuijun Fan, motivated by LG
mirror symmetry theory in physics, we study a modified d-bar operator
over complete Kahler manifolds. We study its spectral properties
and establish a corresponding index theorem. We construct new torsion
type invariants and point out possible applications to singularity
theory.

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