On the Nonlocal Curvatures of Surfaces with or without Boundary

TL;DR

This paper introduces new nonlocal curvature concepts for surfaces, including those with boundaries, using a novel fractional area functional, advancing the mathematical understanding of surface geometry.

Contribution

It develops alternative nonlocal curvature notions applicable to surfaces with boundary, based on a new fractional area functional, expanding previous work on boundaryless surfaces.

Findings

New nonlocal curvature definitions for surfaces with boundary

A fractional area functional for compact surfaces

Enhanced understanding of surface geometry with nonlocal tools

Abstract

For surfaces without boundary, nonlocal notions of directional and mean curvatures have been recently given. Here, we develop alternative notions, special cases of which apply to surfaces with boundary. Our main tool is a new fractional or nonlocal area functional for compact surfaces.