First variation of the fractional $k$-dimensional measure: extending the concept of nonlocal curvature to submanifolds

The fractional $k$-dimensional measure of a submanifold of ${\mathbb{R}}^n$ is a generalization of the fractional perimeter and fractional length appearing in the literature and depends on a parameter $\sigma$ between $0$ and $1$. Here its first variation is computed. The resulting formula is used to define a nonlocal version of the mean-curvature vector for embedded submanifolds. It is shown that in the case where $k=n-1$, this agrees with the nonlocal mean-curvature that has been widely studied.