# Differential Geometry Seminar Schedule for
Spring 2005

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

### 4/1 Paolo Cascini, UCSB
``K\"ahler-Ricci Flow and the Minimal Model Program for Projective Varieties"

Abstract: One of the most important problems in Algebraic Geometry is the quest
for a Minimal Model for an algebraic variety, that would generalize the
classification of algebraic surfaces in higher dimension.
In case the variety is of general type, then out of a minimal model one
can produce the so called canonical model, a birational model whose
canonical bundle is positive.
On the other hand, around the 1980's, building on the foundational
work of Hamilton in the Riemannian case, H. D. Cao studied the
K\"ahler-Ricci flow for canonical metrics on manifolds with definite
first Chern class, reproving in particular Calabi's conjecture and the
existence of K\"ahler-Einstein metrics in case $c_1 <0$ (the
original proof of Calabi's conjecture is due to Yau, who solved it using
elliptic methods).
We propose to draw a connection between the two theories for projective
varieties of general type, and in fact show that, in complex dimension
two, the K\"ahler-Ricci flow (starting with a suitable metric) produces
the canonical model, generalizing Cao's result.
### 4/8 Farshid Arjomandi, UCSB
``A Survey of the Rozansky-Witten Invariants"

Abstract: It is a rather well-known fact that Riemannian manifolds whose holonomy group is the unitary group U(n) are Kaehler manifolds and vice versa. Compact irreducible Riemannian manifolds whose holonomy group is the symplectic group Sp(n) are called hyper-Kaehler manifolds. While compact Kaehler manifolds carry only one complex structure, compact hyper-Kaehler manifolds possess three complex structures I, J, and K that behave like the quaternions i, j, and k. Remarkably enough, it turns out that every hyper Kaehler manifold can be turned into a holomorphic symplectic manifold and vice versa. The two main example series ofÊcompact hyper Kaehler manifolds includeÊthe Hilbert schemes of points on a K3 surface, and also the generalized Kummer varieties.
ÊA new method for studying a hyper-Kaehler manifold X was invented in 1997 by L. Rozansky and E. Witten via associating an invariant to a pair (X,G), where G is a certain kind of graph known as a Jacobi diagram. Shortly after the invention of these invariants, Kapranov and Kontsevich realized that the Riemannian structure is not really necessary for the construction and they showed how the holomorphic structure of X alone is enough for building up these invariants. In fact, the aforementionedÊquaternionic structures on X not only enableÊone to use algebraic geometry to study these differential-geometric objects, they also lead to a rich geometry on the algebraic side. More interestingly, it turns out that all the Chern invariants of a holomorphic symplectic manifold are Rozansky-Witten (RW) invariants so that in a sense, RW-theory is a generalisation of the Chern-Weil theory of the characteristic classes.
ÊIn this talkÊI plan to talk about the algebraicÊapproach of Kapranov and Kontsevich, starting out by mentioning some definitions and resultsÊon the geometry of compact hyper Kaehler manifolds, followed by a short description of the original 1997 Rozansky-Witten construction. Then I will introduce the Atiyah class, which is the algebraic analog of the Riemann curvature tensor, and the main ingredient in the algebraicÊbuild upÊof the RW-invariants. Next I will define theÊJacobi diagrams, their subspaces, ideals, the IHX relation, and the weight systems. Finally I will show how to construct the RW-invariants, using the machinery thus far developed. If time allows, there will also be a quick presentation of an application of the Wheeling Theorem on relating the L2 norm of the Riemann curvature tensor of a compact hyper-Kaehler manifold to some topological quantity, its Todd genus.
### 4/15 Helena McGahagan, UCSB
``Schr\"odinger Maps"

Abstract: Schr\"odinger maps are solutions of a highly nonlinear PDE with geometric
structure arising from the constraint that the solutions must lie on a
given manifold. PDE techniques that incorporate the geometry allow us to
prove both the existence and uniqueness of Schr\"odinger maps.
Geometric problems that arise in the proof of existence include the
definition of Sobolev spaces in terms of covariant derivatives and the
approximation of maps by smooth maps that lie on the manifold. Also, the
use of parallel transport along geodesics to compare solutions yields an
improved uniqueness result.
### 4/22 no meeting

### 4/29 Xianzhe Dai, UCSB
``Curvature and injectivity radius estimates for Einstein 4-manifolds (apres Cheeger-Tian)"

### 5/6 Xianzhe Dai, UCSB
``Curvature and injectivity radius estimates for Einstein 4-manifolds (apres Cheeger-Tian), continue"

### 5/13 no meeting
``"

### 5/20 Rugang Ye, UCSB
``Curvature and injectivity radius estimates for Einstein 4-manifolds (apres Cheeger-Tian), continue"

### 5/27 Xianzhe Dai, UCSB
``Curvature and injectivity radius estimates for Einstein 4-manifolds (apres Cheeger-Tian), continue"

### 6/3 Denis Labutin, UCSB
``Maximal function and weak type (1, 1) inequality"

# Differential Geometry Seminar Schedule for
Winter 2005

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

### 1/14 J. Douglas Moore, UCSB
``Towards a Morse Theory for Minimal Surfaces in Riemannian Manifolds"

Abstract: This will be a survey describing some open questions, and recent results that stress the analogies between geodesics and minimal surfaces in Riemannian manifolds. It should be accessible to graduate students who have completed the 240 series.
### 1/21 Per Tomter, visiting UCSB
``Isometric immersions into complex projective space"

Abstract: There is a somewhat vague conjecture, saying that the principal orbit metrics on a homogeneous space G/H which is a principal orbit, are the only homogeneous metrics
that allow an isometric immersion into the Riemannian manifold M on which the compact
Lie group G acts. (This has been proved f.ex. for the action of SO(n) =>SO(n)xSO(n) on
R(2n) (diagonal embedding); the principal orbits are 2.complex Stiefel manifolds of codimension 3). Here we consider the principal orbits of the isotropy action of a complex
projective space CP(n), they are the geodesic 2n-1 spheres. We prove the conjecture for this fundamental case. For n>3, an elaborate use of the Gauss equations will suffice, for
n=3 we also need the Codazzi-Mainardi equations, while the most difficult case n=2 reveals that the fundamental theorem of submanifolds does not generalize to CP(n).
We need to exploit the fact that the complex structure is parallell in CP(n).
We will try to present a survey lecture.
### 1/28 Will Wylie, UCSB
``Long Homotopies and the Tangent Cone at infinity"

Abstract: In 1968 Milnor conjectured that the fundamental group of a
complete manifold with nonnegative Ricci curvature must be finitely
generated. While there are many strong partial results supporting the
conjecture there is still no complete proof or counterexample. Here we
attempt to investigate additional properties of the fundamental group in
cases where the fundamental group has already been shown to be finitely
generated. In particular, we will show in certain cases that if we are
given two homotopic loops in the fundamental group that we can control the
length of the homotopy between them based upon the lengths of the curves.
We will also discuss the relationship between this geometric control on
the fundamental group and special Gromov Hausdorff limits of the manifold
called tangent cones at infinity.
### 2/4 Lei Ni, UCSD
``Ancient solutions to Kahler-Ricci flow"

Abstract: In this talk we discusss the following generalization on a
result of Perelman on Ricci flow, that any
non-flat ancient solution to K\"ahler-Ricci flow with bounded nonnegative
bisectional curvature has asymptotic volume ratio zero.
### 2/11 Xianzhe Dai, UCSB
``On the Stability of K\"ahler-Einstein Metrics"

Abstract: In our previous work, we proved stability of a large class of Ricci flat metrics, namely,
those with special holonomy. A crucial ingredient is the use of parallel spinor, which dictates that the metric will have to be Ricci flat. In order to deal with general Kahler-Einstein metrics, we found that spin^c is good framework and use it to prove the stability of Kahler-Einstein metrics with nonpositive scalar curvature. As with our previous work, we can use it to draw interesting consequences including a local volume comparison. This is joint work with Xiaodong Wang and Guofang Wei.
### 2/18 See the special colloquium by Jeff Cheeger
``Degeneration of Metrics with Special Holonomy"

Abstract:
### 2/25 no meeting

### 3/4 Doug Moore, UCSB
``Generic Properties of Closed Parametrized Minimal Surfaces in Riemannian Manifolds"

Abstract: A basic technique in the theory of manifolds is transversality theory, based upon Sard's theorem, which allows one to prove, for example, that "almost all" maps from a compact surface into a smooth manifold of dimension at least five are imbeddings.
One would like to study some nonlinear partial differential equations by means of calculus on infinite-dimiensional manifolds.Ê Smale developed an extension of Sard's theorem which has proven to be useful in a wide variety of contexts.
In this lecture, we will explore the applications of Smale's extension of Sard's theorem to the theory of minimal surfaces in compact Riemannian manifolds.
### 3/11 Pengzi Miao, UCSB
``On Boundary Effects of Static Asymptotically Flat $3$-manifolds"

Abstract: An asymptotically flat $3$-manifold with boundary is called
static if it arises as a time-symmetric hypersurface in a static
spacetime solution to the Vacuum Einstein Equation. The vacuumness and
the time-independence of such a spacetime suggest there is no
contribution to the ADM mass from either matter distribution or
gravitational radiation. Thus, it has been expected that the only mass
contribution comes from the boundary. In this talk, we give some
uniqueness theorem of static asymptotically flat manifolds with
boundary, established following the above physical spirit. We also talk
about their relation to the Riemannian Penrose Inequality and
Bartnik's quasi-local mass definition.
# Differential Geometry Seminar Schedule for
Fall 2004

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

### 9/24 organizational meeting

### 10/1 Pengzi Miao, UCSB
``On Small Data Solution to Static Metric Extension Problem"

Abstract: Motivated by R. Bartnik's quasi-local mass definition in
general relativity, we study the small data solution to the static
metric extension conjecture. We first recall the basic definition and
property of static metrics. Then we state the static metric extension
problem with emphasis on explaining both the interior equation and the
boundary condition. In the end we present an existence result on small
data solution. Open questions on relation among static metrics, Einstein
metrics and constant scalar curvature metrics may also be discussed.
### 10/8 Pengzi Miao, UCSB
``On Static Metric Extension in General Relativity II"

Abstract: We introduced the static metric extension problem proposed by
R. Bartnik in the previous talk. In this talk we will give it rigorous
mathematical analysis. First, we will analyze static metrics from a
scalar curvature deformation point of view and
state J. Corvino's theorem on local scalar curvature deformation. Then
we will reformulate our variational problem to prove that a mass
minimizer, if exists, must statisfy the system of static equations and
the expected boundary condition. Then we will focus on small data
solutions for compact domains in R^3 and prove that there always exists
solution to the extension problem if the metric deformation of standard
balls satisfies a symmetry condition.
### 10/15 Xiaofeng Sun, Harvard Univ.
``Canonical metrics on the moduli space of Riemann surfaces"

Abstract: We study two new complete Kahler metrics, the Ricci metric and
perturbed Ricci metric on the moduli space of closed Riemann surfaces and
show that these metrics have nice curvature properties. Both of these
metrics have bounded geometry and have Poincare growth near the boundary
of the moduli space. Furthermore, the holomorphic sectional curvature and
the Ricci curvature of the perturbed Ricci metric are pinched between
negative constants.
By using these new metrics we showed that all known complete metrics on
the Teichmuller space and moduli space are equivalent in the sense that
they are quasi-isometric. This proved the conjecture of Yau about the
equivalence between the Teichmuller metric and the Cheng-Yau-Mok
Kahler-Einstein metric and the conjecture of Bers about the equivalence
between the Kobayashi metric and the Bergman metric.
We then investigate the behavior of the Kahler class of the
Kahler-Einstein metric and derive algebro-geometric properties of the
moduli space. As a direct consequence, we showed that the
moduli space is of log general type. More importantly, the logarithm
contangent bundle of the moduli space is Mumford stable. Also, following
Yau's estimates, we show that the Kahler-Einstein metric has bounded
geometry. This is a joint work with Prof. Kefeng Liu
and Prof. Yau.
### 10/22 (cancelled) Doug Moore, UCSB ``Bumpy metrics
for minimal surfaces"

Abstract: Morse theory on finite-dimensional manifolds establishes
inequalities between the number of critical points of generic
function on a compact manifold M and the Betti numbers of M.
In the Morse theory of closed geodesics in a compact manifold M, one
cannot modify the action function, so instead one chooses a generic
metric on M. Abraham proved that for a generic choice of metric,
nonconstant closed geodesics lie on nondegenerate critical
submanifolds of dimension one. Using this theorem one can establish
Morse inequalities between the number of critical submanifolds and
Betti numbers of the free loop space of M.
This lecture explores the properties of parametrized minimal surfaces
in compact manifolds with generic metrics. It turns out that for
generic metrics, prime minimal two-spheres lie on nondegenerate
critical submanifolds of dimension six, prime minimal tori lie on
such submanifolds of dimension two, while minimal surfaces of higher
genus are nondegenerate in the usual sense.
Some simple applications of this result will be described.

### 10/29 Xiaoling Wang, UCSB
``Modular Invariance and Witten's Anomaly Cancellation Formula"

Abstract: To research super string theory, in 1983, the physicists
Alvarez-Gaume and Witten found a so-called "miraculous cancellation"
which reveals a beautiful relation between the Hirzebruch L-form and
the Hirzebruch-Â-form of a 12-dimensional Riemannian manifold.
Kerfeng Liu established higher dimensional "miraculous cancellation"
fornulas for (8k+4)-dimensional smooth manifolds by developing modular
invariance properties of characteristic forms.
For each (8k+2)-dimensional closed Riemannian manifolds with a
rank two real oriented vector bundle over it, Weiping Zhang and Fei Han
established a cancellation formulas on the level of Characteristic classes.
In fact, it actually holds on the level of differential forms, and I will
give a direct proof by using the modular invariance method developed by
Liu. We then will obtain more cancellation formulas for even dimensional
manifolds with a complex line bundle involved.

### 11/5 Jingyi Chen, UBC, visiting Stanford University
``Mean curvature flow of calibrated submanifolds"

### 11/12, Pengzi Miao, UCSB
``Remark on static, asymptotically flat and scalar flat 3-manifolds
with horizon boundary"

Abstract: Given a static asymptotically flat and scalar flat 3-manifold
M with non-empty boundary, a classic result of Bunting and Masood-ul
Alam states that if the static potential function vanishes on the
boundary, then M is isometric to the half Schwarzschild manifold.
Motivated by Bartnik's minimal mass extenstion conjecture, we discuss
how the boundary assumption on the static potential function can be
replaced by geometric boundary conditions of the manifold.
### 11/19 no meeting

### 11/26 Thanksgiving

### 12/3 no meeting

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