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

**Special Events**

Speaker: | Brian Harvie, UC Davis |

Location: | |

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.

Join Zoom Meeting https://ucdavis.zoom.us/j/93761795880?pwd=b0ZSVmRsSEJqNkpwK203ZWV3bEdQUT09 Meeting ID: 937 6179 5880 Passcode: exit