[Geometry, Topology and Dynamics Seminar] On the nonlocal curvature of surfaces and curves by Dr. Brian Seguin (Loyola University Chicago)

Abstract: Motivated by generalizations of the Ginsburg-Landau energy and the diffusion equation in which derivatives are replaced by fractional derivatives, Caffarelli, Roquejoffre, and Savin studied the minimizers of a fractional perimeter functional on sets involving a parameter between 0 and 1.  Such minimizers have to satisfy a pointwise condition on their boundary, which can be used to define a notion of nonlocal mean-curvature.  This definition only holds for surfaces which are the boundary of a set.  I will describe how to define a nonlocal notion of mean-curvature for any surface by introducing a fractional area functional and considering its minimizers.  Moreover, I will describe how these ideas can be extended to curves by defining a fractional length and an associated nonlocal curvature for a curve.