A fractional notion of area in codimension two

Given a set E of finite perimeter in Euclidean space, the squared H^s-seminorm of its characteristic function, suitably rescaled, interpolates between the volume of E (as s goes to 0) and its perimeter (as s goes to 1/2).

In this talk, we introduce an analogous quantity for codimension-two objects: the s-area, defined for codimension-two boundaries in Euclidean space and, more generally, on closed Riemannian manifolds. It is defined as the minimum of the squared H^s-seminorm over circle-valued maps with prescribed singularities along the given codimension-two boundary.

As in the codimension-one case, we show that the s-area, suitably rescaled, approximates the classical codimension-two area as s goes to 1. We also discuss some properties of the s-area for fixed values of s, as well as compactness properties for families of boundaries whose rescaled s-area is uniformly bounded as s goes to 1.

This talk is based on joint work with Michele Caselli and Nicola Picenni.