Special Session on Ricci Flow/Riemannian Geometry
Spring Western Section Meeting of the AMS
UC Santa Barbara, April 16-17, 2005.
Organizers:
Guofang Wei
(wei@math.ucsb.edu) University of California, Santa Barbara
Rugang Ye (yer@math.ucsb.edu)
University of California, Santa Barbara
Information:
Saturday Program:
Room 1119, Girvetz Hall
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8:00am Zhuang-dan Daniel Guan , UC Riverside, "On modified Ricci flow and modified Calabi flow"
Abstract: In this talk I will explain recent approach of
finding K\"ahler-Einstein type metric on compact
K\"ahler manifolds by fourth order curvature flows.
First, I will explain the quasi-second-order fourth order
modified Ricci flow for extremal-soliton metrics and
will try to prove the convergence of the flow on compact
almost homogeneous manifolds with two ends. I will explain
the weakness of this flow. This was done in 1993
(see a related paper appeared in International Journal of Mathematics 1995).
Then, I will treat the
fourth order modified Calabi flow and prove the $C^{\infty}$
convergence on this manifolds. It shows that the
Calabi-Robinson-Trautment flow is more natural.
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9:00am Natasa Sesum, Courant Institute, New York University, "Linear and dynamical stability of Ricci flat "
Abstract: We can talk about two kinds of stability of the Ricci flow at Ricci
flat metrics. One of them is a linear stability, defined with respect
to Perelman's functional $\mathcal{F}$. The other one is a dynamical
stability and it refers to a convergence of a Ricci flow starting at
any metric in a neighbourhood of a considered Ricci flat metric. We
show that dynamical stability implies linear stability. We also show
that a linear stability together with the integrability assumption
imply dynamical stability. As a corollary we get a stability result
for $K3$ surfaces part of which has been done by Guenther, Isenberg and Knopf.
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10:00am John Lott , Michigan University, "Wasserstein space and Ricci flow"
Abstract: Given a compact metric space X, its Wasserstein space is the space
of probability measures on X, equipped with a certain metric.
It has arisen recently in synthetic notions of Ricci curvature.
The talk will be about some relations between the Wasserstein space
of a Riemannian manifold and Perelman's modified Ricci flow.
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3:00pm Lei Ni , UC San Diego, "A parabolic relative volume comparison theorem"
Abstract: We explore further the implication of Perelman's new ideas in the simpler case
without Ricci flow. A little surprised, this also leads to some new results,
which are also colsely related to the previous works of Cheeger-Yau and Li-Yau.
If time allows we also would like to discuss an interpolation between the
entropy formula of Perelman and a new matrix Li-Yau-Hamilton inequality for
Ricci flow.
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4:00pm Dan Knopf , University of Texas at Austin, "Positivity of Ricci curvature under the
K\"ahler--Ricci flow"
Abstract: In all complex dimensions $n\geq2$, we construct complete K\"ahler manifolds of
bounded curvature and non-negative Ricci curvature whose K\"ahler--Ricci
evolutions immediately acquire Ricci curvature of mixed sign.
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5:00pm Peng Lu, University of Oregon, Eugene "Ricci flow on homogeneous 4-manifolds"
Abstract:
The Ricci flow on homogeneous 3-manifolds are studied by J. Isenberg and M.
Jackson. In later work Knopf and McLeod classify the quasi-equivalence class of
such flow. In this talk
we will discuss the Ricci flow on homogeneous 4-manifolds.
We first list the classification of such manifolds,
for each family of initial metrics there are subfamily
of the initial metrics such that we can diagonalize them and
the Ricci flow will preserve the diagonalization,
finally we will analysis long time behavior of the subfamily.
In particular we find that if such a solution exists for
all the time then it is type III singularities in the sense of
Hamilton.
Sunday Program:
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8:00am Doug Moore , UCSB, "Generic Properties of Closed Parametrized Minimal
Surfaces in Riemannian Manifolds"
Abstract: This talk will describe the foundations needed to develop a partial Morse
theory for conformal harmonic maps $f:\Sigma \rightarrow M$, where $\Sigma $ is
a compact Riemann surface and $M$ is a compact Riemannian manifold. This
theory should parallel the well-known Morse theory of smooth closed
geodesics.\par
The first step in developing such a theory consists of providing an analog of
Abraham's bumpy metric theorem in the context of parametrized minimal surfaces.
We have developed such a bumpy metric theorem, which states in part that when
the ambient manifold $M$ is given a generic metric, all prime closed
parametrized minimal surfaces are as Morse nondegenerate as the group of
conformal transformations of $\Sigma $ allows. They are Morse nondegenerate in
the usual sense if $\Sigma $ has genus at least two, lie on two-dimensional
nondegenerate critical submanifolds if $\Sigma $ has genus one, and on
six-dimensional nondegenerate critical submanifolds if $\Sigma $ has genus
zero.\par
We have also proven some results on the properties of minimal surfaces in a
Riemannian manifold with generic metric, including the fact that such minimal
surfaces are free of branch points. Additional generic properties will be
described, as well as potential applications.
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9:00am Marcus Khuri , Stanford University, "Qualitative Properties of Solutions to the Yamabe"
Abstract: The Yamabe problem asks if every compact Riemannian manifold is conformally
equivalent to one of constant scalar curvature. As is well-known, an
affirmative answer was provided through the works of Yamabe, Trudinger, Aubin,
and Schoen. However, their methods only give limited information about the
solution(s). We will review the current state of knowledge, and introduce some
new results.
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10:00am Yu Ding , Princeton University, "Orbifolds and collapsing"
Abstract: We study Riemannian orbifolds with bounded curvature that is
sufficiently collapsed, in particular, the existence of a nilpotent
Killing structure, similar to those defined on manifolds by Cheeger, Gromov and
Fukaya. We also analyze the corresponding F-structure, with applications to
certain gap phenomena as in
the manifold case.
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10:30am William Wylie , UCSB, "Far 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
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.
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3:00pm Weiqing Gu , Harvey Mudd College, "Calibrated Exceptional Geometry"
Abstract: In this talk, first we will quickly review the background of
exceptional geometry and the method of calibrations as an effective
tool for identifying volume-minimizing cycles in Riemannian manifolds
such as Grassmann manifolds,
Calabi-Yau 4-folds and exceptional holonomy manifolds. Then we will
determine several families of so-called associative,
coassociative, and Cayley cycles
in $G_2$ and $Spin_7$ manifolds. Those cycles are
supersymmetric cycles in string theory.
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4:00pm Chadwick Sprouse, California State University at Northridge, " Convergence of manifolds with lower Ricci and $L^1$ sectional curvature bounds"
Abstract: We consider a sequence of manifolds with $\rm{Ric} \geq -k^2$ converging to a
Gromov-Hausdorff limit $X$. We use the Cheeger-Colding segment inequality to
show that if the amount of sectional curvature below $K$ approaches $0$ in an
$L^1$ sense then the $X$ is an Alexandrov space of curvature $\geq K$. Several
examples and applications are presented.