Mathematics Department - Geometric Analysis Seminar - Spring 2017

Geometric Analysis Seminar - Spring 2017



Organizer(s)

Paul Feehan, Manousos Maridakis, Natasa Sesum

Archive



Upcoming Talks


Tuesday, March 7th

Siao-Hao Guo , Rutgers University

"Analysis of Velázquez's solution to the mean curvature flow with a type II singularity"

Time: 3:00 PM
Location: Hill 705
Abstract: J.J.L. Velázquez in 1994 used the degree theory to show that there is a perturbation of Simons' cone, starting from which the mean curvature flow develops a type II singularity at the origin. He also showed that under a proper time-dependent rescaling of the solution around the origin, the rescaled flow converges in the C^0 sense to a minimal hypersurface which is tangent to Simons' cone at infi nity. In this talk, we will present that the rescaled flow actually converges locally smoothly to the minimal hypersurface, which appears to be the singularity model of the type II singularity. In addition, we will show that the mean curvature of the solution blows up near the origin at a rate which is smaller than that of the second fundamental form.

This is a joint work with N. Sesum.





Past Talks


Tuesday, February 21st

Bing Wang , University of Wisconsin

"The extension problem of the mean curvature flow (I)"

Time: 3:00 PM
Location: Hill 705
Abstract: We show that the mean curvature blows up at the first finite singular time for a closed smooth embedded mean curvature flow in R^3.

This is a joint work with H.Z. Li.


Tuesday, February 14th

Kyeongsu Choi , Columbia University

"Free boundary problems on the Gauss curvature flow"

Time: 3:00 PM
Location: Hill 705
Abstract: We will discuss about the optimal $C^{1,frac{1}{n-1}}$ regularity of the Gauss curvature flow with a flat side and the $C^{infty}$ regularity of its free boundary, namely the boundary of the flat side. We establish estimates by applying the maximum principle for chart independent geometric quantities.


Tuesday, February 7th

Xiaoliu Wang , Southeast University in China and Rutgers University

"Nonlocal Curvature Flows"

Time: 3:00 PM
Location: Hill 705
Abstract: In this talk, some nonlocal planar cuvature flows will be introduced, including the area-preserving flow and the length-preserving one, etc. The locally convex initial curve could be embedded or immersed. Some sharp conditions on immersed initial curves are established to yield the singularity of the flow at a finte time. In addition, the symmetry of curves are found to play an important role in the flow's evolution.


Tuesday, January 31st

Siao-Hao Guo, Rutgers University

"Analysis of Velázquez's solution to the mean curvature flow with a type II singularity"

Time: 3:00 PM
Location: Hill 705
Abstract: J.J.L. Velázquez in 1994 used the degree theory to show that there is a perturbation of Simons' cone, starting from which the mean curvature flow develops a type II singularity at the origin. He also showed that under a proper time-dependent rescaling of the solution around the origin, the rescaled flow converges in the C^0 sense to a minimal hypersurface which is tangent to Simons' cone at infi nity. In this talk, we will present that the rescaled flow actually converges locally smoothly to the minimal hypersurface, which appears to be the singularity model of the type II singularity. In addition, we will show that the mean curvature of the solution blows up near the origin at a rate which is smaller than that of the second fundamental form.

This is a joint work with N. Sesum.


Tuesday, January 24th

Special Geometric Analysis Seminar

Professor Lu Wang, University of Wisconsin

" Surfaces of Low Entropy"

Time: 3:00 PM
Location: Hill 705
Abstract: The entropy of a (hyper)-surface is given by the supremum over all Gaussian integrals with varying centers and scales. It is a geometric invariant that measures complexity of the surface. This quantity is introduced by Colding and Minicozzi to understand the singularity formation of mean curvature flow of surfaces. In this talk, I will discuss several results which show that closed surfaces that have small entropy are simple in various senses.

This is joint with Jacob Bernstein.


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