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

Cornelia Mihaila; Brian Seguin

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arXiv · 2606.24043

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

Cornelia Mihaila·Brian Seguin

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Abstract

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.

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BibTeX RIS

Cornelia Mihaila, Brian Seguin. 2026-06-23. First variation of the fractional $k$-dimensional measure: extending the concept of nonlocal curvature to submanifolds. https://arxiv.org/abs/2606.24043

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