PDE Seminar Abstracts

The stability of constant mean curvature hypersurfaces is an important topic in mathematics and influences problems such as the dynamics of the volume preserving mean curvature flow. We consider the case where the hypersurfaces have free boundary contained in parallel planes, this results in the periodic Delaunay hypersurfaces. They include spheres (which are always stable), cylinders (which are stable at large radii), and nodoids (which are unstable). The final case of unduloids is more complex. All unduloids of dimension two through seven are unstable, but there exists stable unduloids of dimension nine and greater. In this talk, we will consider the missing case of whether there exists stable unduloids of dimension eight. We will also work towards finding a criterion to determine if a specific unduloid is stable or unstable.

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