# 2009 Pacific Northwest Geometry Seminar

## Topic

*Lagrangian Mean Curvature flow for entire Lipschitz graphs*, Albert Chau (University of British Columbia)

**Abstract:** We prove existence of longtime smooth solutions to
mean curvature flow of entire Lipschitz Lagrangian graphs. As an
application of the stimates for the solution, we establish a Bernstein
type result for translating solitons. The results are from joint work
with Jingyi Chen and Weiyong He.

*Collapsing Sequences of Constant Mean Curvature Surfaces in Riemannian Manifolds*, Adrian Butscher (Stanford)

**Abstract:** Since every Riemannian manifold is locally Euclidean
up to second order, small geodesic spheres of radius r have nearly
constant mean curvature of magnitude 2/r. However, it is known that
there are fairly restrictive conditions under which a geodesic sphere
of sufficiently small radius can be perturbed to have exactly constant
mean curvature. I will investigate a related question: whether it is
possible to assemble small geodesic spheres into extended surfaces of
near-constant mean curvature by the gluing technique, and perturbing
these surfaces to have exactly constant mean curvature. It turns out
that the conditions under which this is possible are the result of a
subtle interplay between the surface and the background geometry. A
result of my investigation is a construction of small constant mean
curvature surfaces that locally resemble classical Delaunay surfaces
but exhibit new global properties that are impossible in the classical
setting. In addition, I will relate my investigation to the following
question: given a sequence \Sigma_r of surfaces of mean curvature 2/r
contained within a tubular neighbourhood of size O(r) of a
lower-dimensional variety \Gamma in a Riemannian manifold M, what can
be said about \Gamma?

*The isoperimetric type inequalities and nonlinear PDEs*, Pengfei Guan (McGill)

**Abstract**: We discuss Alexandrov-Fenchel type isoperimetric

inequalities and their relationship with elliptic and parabolic

type partial differential equations. We give a proof of the

isoperimetric inequality for quermassintegrals of non-convex

starshaped domains, using an expanding geometric curvature flow. We also prove a similar type of inequalities for functions

on $\mathbb S^n$ using elliptic Hessian equation.

*The rate of change of width under flows*, Bill Minicozzi (Johns Hopkins)

**Abstract**: I will discuss a geometric invariant, that we call
the width, of a manifold and first show how it can be realized as the
sum of areas of minimal 2-spheres. When $M$ is a homotopy 3-sphere,
the width is loosely speaking the area of the smallest 2-sphere needed
to ``pull over'' $M$. Second, we will estimate the rate of change of
width under various geometric Flows, including flows by mean curvature
and Ricci curvature, to prove sharp estimates for extinction times.
This is joint work with Toby Colding.

*Smooth Metric Measure Spaces*, Guofang Wei (UCSB)

**Abstract**: Smooth metric measure spaces are Riemannian manifolds
with a conformal change of the Riemannian measure and occur naturally
as measured Gromov-Hausdorff limit of Riemannian manifolds. The
important curvature quantity here is the Bakry-Emery Ricci tensor,
which corresponds to the (synthetic) Ricci curvature lower bound for
(nonsmooth) metric measure spaces. What geometric and topological
results for Ricci curvature can be extended to Bakry-Emery Ricci
tensor? Recently there are many developments. We will discuss
comparison geometry and rigidity in this direction.

*A general regularity theory for stable codimension 1 integral varifolds*, Neshan Wickramasekera (Cambridge)

**Abstract:** The focus of this talk will be a new regularity
theorem for the class of singular stable minimal hypersurfaces
(stable codimension 1 integral varifolds) of an open all. Making no
assumption a priori on the size of the singular set, the theorem gives
a natural, geometric structural condition or a hypersurface in this
class to be smooth and embedded in he interior up to a lower
dimensional, generally unavoidable, singular set. Precisely, suppose
that a hypersurface in this class has the property that no singular
point has a neighborhood in which the hypersurface is a union of
$C^{1, \alpha}$ (for some arbitrarily chosen \alpha \in (0, 1)$)
hypersurfaces-with-boundary meeting

(only) along their common boundary. Then it is smooth and embedded away from the boundary of the ball and away from a

possible interior singular set of codimension at least 7 (which is
empty if the dimension of the hypersurface is $\leq 6$). The

work generalizes the regularity theory of R. Schoen and L. Simon. Some
applications as well as what can be said in the absence of the above
structural condition will also be discussed.

## Details

* Oregon State University

* Portland State University

* Stanford University

* University of British Columbia

* University of Oregon

* University of Utah

* University of Washington

The meetings are supported by the National Science Foundation (NSF), the Pacific Institute for the Mathematical Sciences (PIMS), and the host institutions.

## Additional Information

For more information, please visit

http://www.math.washington.edu/~lee/PNGS/2009-spring/

or download the webarchive attached above.

Albert Chau (UBC) | |

Guofang Wei (UCSB) | |

N. Wickramasekera (Cambridge) |

**Scientific, Conference**

**May 2â€“3, 2009**

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