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(PhD. Exit Seminar) The Inverse Mean Curvature Flow: Singularities, Dynamical Stability, and Applications to Minimal Surfaces

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Speaker: Brian Harvie, UC Davis
Start time: Fri, Jun 4 2021, 12:30PM

Extrinsic geometric flows are rules for deforming surfaces by their curvature through time that arise in 
several areas of geometry and physics. Key to their applications in these areas is understanding their 
singularities and the dynamical stability of their equilibrium solutions. Specifically, does a surface 
evolving by a particular geometric flow approach some limiting shape that the flow cannot be continued 
from in finite time, or does the flow continue for all time? If the former possibility happens, one would 
like to characterize this limiting shape. If the latter possibility happens, one would like to determine 
if the flow surfaces asymptotically converge to a fixed limit surface such as a round sphere at large 
times, possibly modulo rescalings in time.

This talk will be about my work on the singularities and dynamical stability of the Inverse Mean Curvature 
Flow (IMCF). On the topic of singularities of IMCF, I will show that rotationally symmetric tori evolved 
by IMCF undergo a singularity in which the flow surfaces approach an immersed limit surface. This 
contrasts sharply with the singular behavior of other extrinsic flows. On the topic of dynamical stability, 
I will show that axially symmetric embedded spheres which satisfy a curvature condition do not undergo 
finite-time singularities and asymptotically converge to a round sphere when evolved by IMCF.

I will also discuss how these results on IMCF lead to new insights about physical soap films clinging to 
closed loops of wire. Soap films are modelled by surfaces that minimize area among all surfaces with the 
same boundary curve. In general, area-minimizing surfaces may intersect themselves. However, based on a 
technique that arises from embedded global solutions of IMCF, I demonstrate that an area minimizer cannot 
self-intersect if the boundary curve is confined to certain mean-convex surfaces. This extends previous 
work by Meeks and Yau on the problem of embeddedness of minimal disks in three-manifolds.

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