UCSD Differential Geometry Seminar (Math 258) 2003-2004
Unless otherwise noted, all seminars are
on Wednesdays 4-5 pm.
|
UCSD Differential Geometry Seminar (Math 258) 2003-2004 Unless otherwise noted, all seminars are
on Wednesdays 4-5 pm. |
|
January 7th, 2004, at APM 7218
Title: Regularity of heat operator on cone
Abstract: In this talk, I will describe some recent work with Paul Loya in which
we studied the differentiability properties of solutions to the heat
equation on a cone in spaces of whole and fractional (H\"{o}lder)
derivatives. The estimates we obtain are good enough to prove short
time existence of solutions to certain semilinear parabolic equations.
The talk will be accessible to general audience including graduate
students of various fields.
January 13th, 2004 (Special date. Joint seminar with UCI)
Title: Local and global groups of diffeomorphisms of CR manifoldsy
Abstract:
Title: The rigidity of asymptotic
hyperbolic spaces
Abstract:
February 20th , 2004 (Special date) at APM 6218
Title: The J-flow and the Mabuchi energy
Abstract:
The J-flow is a parabolic flow on compact Kahler manifolds with two Kahler
metrics. It was discovered by S. Donaldson and X. X. Chen independently.
Donaldson defined it in the setting of moment maps and symplectic
geometry. Chen described the flow as the gradient flow of the
J-functional appearing in his formula for the
Mabuchi energy. The Mabuchi energy is an important functional on the
space of Kahler potentials. Its critical points give constant scalar
curvature metrics, and its lower boundedness is related
to stability in the sense of geometric invariant theory.
I will show that under a condition on the initial data, the J-flow
converges to a critical metric. I will then explain how this implies the
lower boundedness of the Mabuchi energy for an open set of Kahler classes
on manifolds with negative first Chern class.
Title: Mean curvature of some calibrated submanifolds
Abstract:
Title: Yamabe flow on three manifolds
Abstract:
The Ricci flow on surfaces reduces to a scalar evolution equation which
makes perfect sense in higher dimensions as well. It is then called the
Yamabe flow, since it turns out to be the negative gradient flow for the
(normalized) total scalar curvature functional on the space of Riemannian
metrics when restricted to a conformal class. In particular, stationary
solutions of the flow have constant scalar curvature. Therefore the Yamabe
flow can be viewed as a natural geometric deformation of a Riemannian
metric to a conformal metric of constant scalar curvature.
In this talk, we will outline a proof of the general convergence result
for the Yamabe flow on compact three-manifolds of positive conformal
Yamabe invariant: From any initial metric, the evolving metric produced
by the flow converges smoothly to a unique conformal limit metric of
constant scalar curvature as time tends to infinity. Time permitting, We
will comment on some open problems at the end.
March 2nd, 2004 (Special date. Joint seminar with UCI. Held at UCI)
Title: Regularity theory for general curvature flow
Abstract:
Title: Positive Mass Theorem and Stability of Manifolds with Parallel Spinors
Abstract:
Questions: benchow@math.ucsd.edu or lni@math.ucsd.edu