The struture of stable hypersurfaces with constant mean curvture

Speaker: 

Professor Xu Cheng

Institution: 

Universidade Federal Fluminense, Brazil

Time: 

Tuesday, February 14, 2006 - 4:00pm

Location: 

MSTB 254

We study the global behavior of (weakly) stable constant mean
curvature hypersurfaces in general Riemannian manifolds. We show some
nonexistence of complete and noncompact hypersurfaces with
constant mean curvaure. By using harmonic function theory, we prove
some one-end theorems which are new even for constant mean curvature
hypersurfaces in space forms.

On the Stability of K\"ahler-Einstein Metrics

Speaker: 

Professor Guofang Wei

Institution: 

UCSB

Time: 

Tuesday, January 31, 2006 - 4:00pm

Location: 

MSTB 254

A Einstein metric is stable if the second variation of the total scalar curvature functional is nonpositive in the direction of changes in conformal structures. Using spin^c structure we prove that a compact Einstein metric with nonpositive scalar curvature admits a nonzero parallel spin$^c$ spinor is stable. In particular, all metrics with nonzero parallel spinor (these are Ricci flat with special holonomy such as Calabi-Yau and $G_2$) and Kahler-Einstein metrics with nonpositive scalar curvature are stable. In fact we show that metrics with nonzero parallel spinor are local maxima for the Yamabe invariant and any metric of positive scalar curvature cannot lie too close to them. Similar results also hold for Kahler-Einstein metrics with nonpositive scalar curvature. This is a joint work with Xianzhe Dai and Xiaodong Wang.

The Einstein-scalar field constraint equations on compact manifolds

Speaker: 

Professor Daniel Pollack

Institution: 

University of Washington, Seattle

Time: 

Tuesday, May 9, 2006 - 3:00pm

Location: 

MSTB 254

We introduce the constraint equations for the Einstein-scalar field system on compact manifolds. Using the conformal method we reformulate these equations as a determined system of nonlinear partial differential equations. By introducing a new conformal invariant, which is sensitive to the presence of the initial data for the scalar field, we are able to divide the set of free conformal data into subclasses depending on the possible signs for the coefficients of terms in the resulting Einstein-scalar field Lichnerowicz equation. For most of these subclasses we determine whether or not a solution exists. In contrast to other well studied field theories, there are certain cases, depending on the mean curvature and the potential of the scalar field, for which we are unable to resolve the question of existence of a solution. We consider this system in such generality so as to include the vacuum constraint equations with an arbitrary cosmological constant, the Yamabe equation and even (all cases of) the prescribed scalar curvature problem as special cases.
This is joint work with Yvonne Choquet-Bruhat and Jim Isenberg.

Mean value theorems and local regularity theorem for Ricci flow

Speaker: 

Professor Lei Ni

Institution: 

UCSD

Time: 

Tuesday, March 14, 2006 - 4:00pm

Location: 

MSTB 254

We shall show a general mean value theorem on Riemannian manifold and how it leads to new monotonicity formulae for evloving metrics. As an application we show a local regularity theorem for Ricci flow.

I shall start from the Euclidean mean value theorem and its variations. The first part should be accesible for graduate student.

From Random Matrices to Stochastic Processes, via Integrable Theory

Speaker: 

Professor Pierre van Moerbeke

Institution: 

Brandeis University and UC Berkeley Miller Institute

Time: 

Tuesday, November 15, 2005 - 4:00pm

Location: 

MSTB 254

In a celebrated paper, Dyson shows that the spectrum of a random Hermitian matrix, diffusing according to an Ornstein-Uhlenbeck process, evolves as non-colliding Brownian motions held together by a drift term. The universal edge, bulk and gap scalings for Hermitian random matrices, applied to the Dyson process, lead to novel stochastic processes, Markovian and non-Markovian; among them, the Airy, Sine and Pearcey processes. The integrable theory around the KdV and KP equations provides useful information on these new processes.

Right-veering diffeomorphisms of a compact surface with boundary

Speaker: 

Professor Ko Honda

Institution: 

University of Southern California

Time: 

Tuesday, October 11, 2005 - 4:00pm

Location: 

MSTB 254

We initiate the study of the monoid of right-veering diffeomorphisms on a compact oriented surface with nonempty boundary. The monoid strictly contains the monoid of products of positive Dehn twists. We explain the relationship to tight contact structures and open book decompositions. This is joint work with W. Kazez and G. Mati\'c.

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