Differential Geometry Seminar Schedule for
Spring 2004
Fridays 3:30 - 4:30pm, SH 6617
4/2 Xianzhe Dai, UCSB
``Local Index Theorem and the Asymptotic Expansion of Bergman Kernel"
Abstract : Localization is an important principle in mathematics and local index
theorem is an example of such localization. We will discuss the application of
the local index Theorem technique to the question of asymptotic expansion of
Bergman kernel, which has played an essential role in Donaldson's work on
extremal metrics. This is joint work with Kefeng Liu, Xiaonan Ma and
Xiaowei Wang.
4/9 Guofang Wei, UCSB
``Model high curvature region by $\kapa$-solutions, 12.1 of Perelman"
4/16 no meeting
4/23 Krishnan Shankar, University of Oklahama
``Spherical Rank Rigidity and Blashcke Manifolds"
Abstract: Let $M^n$ be a complete Riemannian manifold whose sectional
curvature is bounded above by 1. We say that $M$ has positive spherical
rank if along every geodesic there is a conjugate point at $t = \pi$
(geodesics are assumed to be parametrized by arc-length). In case $M$
has positive spherical rank, the equality discussion of the Rauch
Comparison theorem implies that along every geodesic of $M$ we have a
spherical Jacobi field i.e., a Jacobi field of the form $J(t) = \sin(t)
E(t)$, where is $E(t)$ is a parallel vector field along the geodesic.
This notion of rank is analogous to the notion of geometric rank for
upper curvature bound 0 or upper curvature bound -1 studied by Ballmann,
Burns, Spatzier, Eberlein, Hamenstaedt etc. In the case of spherical
rank we show: Let $M^n$ be a complete, simply connected Riemannian
manifold with upper curvature bound 1 and positive spherical rank. Then
$M^n$ is isometric to a compact rank one symmetric space i.e., isometric
to a sphere or projective space. This is joint work with Ralf Spatzier and
Burkhard Wilking.
4/30 Guofang Wei, UCSB
``Model high curvature region by $\kapa$-solutions, 12.1 of Perelman, continue"
5/7 Pengzi Miao, MSRI,
``Ricci Curvature Rigidity of Asymptotically Hyperbolic Manifolds"
Abstract: Let $n$ be a dimension for which the classical Positive
Mass Theorem holds in general relativity. Let $(X^{n+1}, g)$ be a
conformally compact manifold whose Ricci curvature is bounded below
by $-n$. We show that $(X^{n+1}, g)$ is isometric to the standard
hyperbolic space provided the conformal infinity of $(X^{n+1}, g)$
is the standard sphere and $Ric(g) + n$ decays faster than $r^2$
near infinity, where $r$ is any boundary defining function. The proof
is based on a quasi-local mass characterization of Euclidean balls,
which is essentially a local version of the Positive
Mass Theorem. This is a joint work with J. Qing.
5/14 Rugang Ye, UCSB
``An introduction to the logarithmic Sobolev inequality"
5/21 Xianzhe Dai, UCSB
``Einstein 4-manifolds and Hitchin-Thorpe inequality"
5/28 William C Wylie, UCSB
``An Application of the Cheeger-Gromoll splitting theorem (work of Christina
Sormani)"
Abstract. The Cheeger-Gromoll splitting theorem states that a complete
Riemannian manifold with nonnegative Ricci curvature is isometric to the
product of some Euclidean space and a manifold N where N has no lines, a line
being defined as a smooth curve that minimizes distance in both directions. We
will prove that any complete Riemannian manifold which does not have a property
called the geodesic loops to infinity property has a line in its universal
cover. We will then use this theorem along with the splitting theorem to
discuss the relationship between the loops to infinity property and the
structure of manifolds with nonnegative Ricci curvature. This is work of
Christina Sormani.
Differential Geometry Seminar Schedule for
Winter 2004
Fridays 3:30 - 4:30pm, SH 6617
1/9 Xianzhe Dai, UCSB
``On the Stability of Riemannian Manifold with Parallel Spinors"
Abstract: Inspired by the recent work of Hertog-Horowitz-Maeda, we prove two
stability results for compact Riemannian manifolds with nonzero
parallel spinors. Our first result says that Ricci flat metrics
which also admits nonzero parallel spinors are stable (in the
direction of changes in conformal structures) as the critical
points of the total scalar curvature functional. In fact, we show
that the Lichnerowicz Laplacian, which governs the second
variation, is the square of a twisted Dirac operator. Our second
result, which is a local version of the first one, shows that any
metrics of positive scalar curvature cannot lie too close to a
metric with nonzero parallel spinor. As an interesting
application, we prove that scalar flat deformations of Calabi-Yau
metric must be Calabi-Yau. We also explore the connection with a
positive mass theorem of X. Dai. This is a joint work with Xiaodong Wang and
Guofang Wei.
1/16 Vitali Kapovitch, UCSB
``Pinching estimates for negatively curved manifolds with
nilpotent fundamental groups"
Abstract: We compute optimal pinching constants for negatively curved
manifolds with almost nilpotent fundamental groups. Namely, we show
that if a manifold $M$ with almost nilpotent fundamental group
admits a metric with $-a^2\le sec(M)\le -1$, then the infimum of
$a^2$ over all such metrics is equal to square of the nilpotency
class of the fundamental group of $M$.
1/23 Mark Haskins, IHES
``Isolated conical singularities of special Lagrangian varieties"
Abstract: we describe recent progress in understanding isolated cone-like
singularities of special Lagrangian varieties in dimension 3.
1/30 Rugang Ye, UCSB
``Uniqueness of 2-D k-solutions of the Ricci flow according to Perelman"
2/6 see Colloquium by Prof. G. Tian
2/13 see Colloquium by Prof. B. Chow
2/20 see math/physics talk by Prof. Freddy Cachazo
2/27 see math/physics talk by Prof. Sergei Gukov
3/5 Burkhard Wilking, Munster, Germany
``A duality theorem for singular Riemannian foliations in nonnegative
curvature"
Abstract: A singular foliation of a Riemannian manifold
is a subdivision into submanifolds that are locally equidistant.
A piecewise smooth curve on such a manifold is then called
horizontal if its tangentfield is everywhere normal to
the corresponding leaf. Given such a foliation
one can try to assign a new singualr foliation by
defining the leaf of a point as the set of all point that
can be connected to that point by a horizontal curve.
We will show that if the ambient manifold has nonnegtive curvature
and the orginal foliation is nonsingular, then
the dual foliation defines a singular Riemannian foliation.
As a consequence we show that the Sharafutdinov retraction is of
class $C^\infty$.
3/12 Is Singer, UCSB
``The projective Dirac operator and its rational index"
Differential Geometry Seminar Schedule for
Fall 2003
Fridays 3:30 - 4:30pm, SH 6617
9/26 organization meeting
10/3 Rugang Ye, UCSB
``Ricci flow and geometrization of 3-manifolds, introduction"
Abstract: Thurston's geometrization program conjectures that closed 3-manifolds
can be cut into peices in a suitable way such that each piece has a homogenous
geometric structure. This includes the Poincare conjecture. Hamilton has
developed the approach of Ricci flow to carry out this program. Recently,
Perelman annouced that he carried out the Ricci flow approach and hence proved
Thurston's conjecture and the Poincare conjecture. We will give an introduction
to Perelman's papers.
10/10, Rugang Ye, UCSB
``Ricci flow and geometrization of 3-manifolds, introduction, continue"
10/17 Anda Degeratu, MSRI
``Geometrical McKay Correspondence"
Abstract: A Calabi-Yau orbifold is locally modeled on C^n/G,
where G is a finite subgroup of SL(n, C). One way to handle
this type of orbifolds is to resolve them using a crepant resolution of
singularities. We use analytical techniques to give a description of the
topology of the crepant resolution in terms of the finite group G. This
gives a generalization of the McKay Correspondence.
10/24 (Special time 3-4pm) Jaeup So, visiting UCSB ``On
G-invariant Minimal Hypersurfaces with Constant Scalar Curvatures in S^5"
Abstract: Let M^n be a closed minimally immersed hypersurface in the unit sphere
S^{n+1}, and h its second fundamental form. Denote by R and S its scalar
curvature and the square norm of h respectively. It is well known that
S=n(n-1)-R from the structure equations of both M^n and S^{n+1}. In particular,
S is constant if and only if M has constant scalar curvature.
In 1968, J. Simons observed that if S <,= n everywhere and S is constant,
then S =0 or n. Clearly, M^n is an equatorial sphere if S=0. And when S=n, M^n
is indeed a product of spheres, due to the works of Chern, do Carmo, and
Kobayashi.
We are concerned about the following conjecture posed by Chern,
Chern Conjecture: For any n>2, the set R_n of the real numbers each of which
can be realized as the constant scalar curvature of a closed minimally immersed
hypersurface in S^{n+1} is discrete.
Theorem [Chang, 1993] A closed minimally immersed hypersurface with constant
scalar curvature in S^4 is either an equatorial 3-sphere, a product of spheres,
or a Cartan's minimal hypersurface. In particular, R_n= {0, 3, 6 }.
Theorem [Yang and Cheng, 1998] Let M^n be a closed minimally immersed
hypersurface with constant scalar curvature in S^{n+1}. If S>n, then S>,= n+n/3.
Let G = O(k)xO(k)xO(q), k=2 or 3 and set 2k+q=n+2. Then W. Y. Hsiang [4]
investigated G-invariant, minimal hypersurfaces, M^n in S^{n+1} and showed that
there exit infinitely many closed minimal hypersurfaces in S^{n+1} for all n>1.
Theorem [Hsiang, 1987] For each dimension n>1, there exist infinitely many,
mutually noncongruent closed G-invariant minimal hypersurfaces in S^{n+1},
where G = O(k)xO(k)x(q) and k=2 or 3.
We studied G-invariant minimal hypersurfaces with constant scalar curvatures in
S^{5}. We prove the following theorem:
Theorem: A closed G-invariant minimal hypersurface with constant scalar
curvature in S^5 is a product of spheres, where G=O(k)xO(k)xO(q). In
particular, S=4.
10/31 Rugang Ye, UCSB
``Asymptotic Soliton of Ricci Flow"
11/7, Rugang Ye, UCSB
``Asymptotic Soliton of Ricci Flow, continue"
11/14, Guofang Wei, UCSB
``Laplace spectrum, Length Spectrum and Covering Spectrum"
Abstract:
One of the most important subfields of Riemannian Geometry is the study of
the Laplace spectrum of a compact Riemannian manifold. Another spectrum
defined in an entirely different manner is the length spectrum of a manifold:
the set of lengths of smoothly closed geodesics. We define a new spectrum for
compact length spaces and Riemannian manifolds called the ``covering
spectrum" which roughly measures the size of the one dimensional
holes in the space. We investigate the relationship between these spectrums.
We analyze the behavior of the covering spectrum under Gromov-Hausdorff
convergence and study its gap phenomenon. This is a joint work with Christina
Sormani.
11/21 Denis Labutin, UCSB
``Polar sets for scalar flat metrics"
11/28 Thanksgiving
12/5 Vitali Kapovitch, UCSB
``Classification of negatively pinched manifolds with almost
nilpotent fundamental groups"
Abstract:
We prove that an open manifold with virtually nilpotent fundamental group
admits a complete metric of pinched negative curvature if and only if it is
diffeomorphic to a product of a line and the total space of a flat
vector bundle over a closed infranil manifold.
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