Geometric curvature energies: facts, trends, and open problems [chapter]

Paweł Strzelecki, Heiko von der Mosel, Philipp Reiter, Simon Blatt, Armin Schikorra
2017 New Directions in Geometric and Applied Knot Theory  
This survey focuses on geometric curvature functionals, that is, geometrically defined self-avoidance energies for curves, surfaces, or more general kdimensional sets in R d . Previous investigations of the authors and collaborators concentrated on the regularising effects of such energies, with a priori estimates in the regime above scale-invariance that allowed for compactness and variational applications for knotted curves and surfaces under topological restrictions. We briefly describe the
more » ... mpact of geometric curvature energies on geometric knot theory. Currently, various attempts are being made to obtain a deeper understanding of the energy landscape of these highly singular and nonlinear nonlocal interaction energies. Moreover, a regularity theory for critical points is being developed in the setting of fractional Sobolev spaces. We describe some of these current trends and present a list of open problems. Energies. Geometric curvature functionals are characterised as geometrically defined energies on a priorily non-smooth k-dimensional subsets Σ k of R d , and these functionals are designed to penalise self-intersections. In addition, there are regularising effects: finite energy implies some higher degree of smoothness of Σ. One of the first examples is that of the Möbius energy on rectifiable curves ⊂ R d , introduced by J. O'Hara [82] and investigated analytically by M. H. Freedman et al. [41, 50] , where d (x, y) denotes the intrinsic distance between the points x, y on the curve , and H 1 stands for the one-dimensional Hausdorff-measure. The first summand in the integrand resembles a Coulomb-type repulsive potential suitably regularised by the second term so as to obtain finite energy for smooth embedded curves. Another exam-
doi:10.1515/9783110571493-002 fatcat:suxf4lbq6rembov3en7wp4vxlq