Mathematics Colloquia and Seminars

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Maximum principles are powerful tools for determining the profiles of solutions to elliptic and parabolic PDE. One may adapt the classical maximum principle for solutions to linear elliptic PDE to a maximum principle for surfaces in Euclidean Space through what is known as the method of moving planes. Using a maximum principle for quasi-linear elliptic PDE, one can establish that two graphs with the same mean curvature which touch at a point actually touch everywhere. In the context of the method of moving planes, one utilizes this fact to show that the only solution to the Isoperimetric Problem in any dimension is a round sphere.

Parabolic PDE also arise in differential geometry in the form of geometric flows, rules for deforming a surface over time according to its curvature at each point. One then naturally asks if there is a parabolic version of the method of moving planes to apply to these flows, and what it might tell us about the ultimate fate of the evolving hypersurface. In this talk I will begin with a review of basic elliptic maximum principles, then introduce the elliptic method of moving planes. After explaining the basics about geometric flows, I will conclude by presenting some original work about applying the parabolic version of the method of moving planes to the Inverse Mean Curvature Flow to determine when solutions exist globally.