2022/12/29 by Mucci, Domenico, Saracco, Alberto · 1 citation
#49J45 #53A04 #74K10 #Differential Geometry (math.DG) #FOS: Mathematics
paper · doi:10.48550/arxiv.2212.14380
A weak notion of elastic energy for (not necessarily regular) rectifiable curves in any space dimension is proposed. Our p-energy is defined through a relaxation process, where a suitable p-rotation of inscribed polygonals is adopted. The discrete p-rotation we choose has a geometric flavor: a polygonal is viewed as an approximation to a smooth curve and hence its discrete curvature is spread out into a smooth density. For any exponent p greater than one, the p-energy is finite if and only if the arc-length parameterization of the curve has a second order summability with the same growth exponent. In that case, moreover, the energy agrees with the natural extension of the integral of the p-th power of the scalar curvature. Finally, a comparison with other definitions of discrete curvatures is discussed.