Geometric curvature energies
Zusammenfassung der Projektergebnisse
This project is focused on the study of geometric curvature energies, that is, geometrically defined energies that impose self-avoidance along with a very weak control of curvature of the underlying non-smooth curves, or surfaces, or of more general subsets of Euclidean space. The quantities to be maximized or averaged over are built from simple elementary geometric functions, such as radii of circles or spheres that intersect or touch the sets in several distinct points. Alternatively, such discrete curvatures can be defined as multi-point functions on higher-dimensional simplices whose vertices are all on the set. The most prominent examples on curves are the ropelength functional as a non-smooth maximization over triples of curve points as a sort of steric constraint on the one end, and the (pure) integral Menger curvature, averaging the same three-point function over all point triples of the curve, on the other end. The results of this project regarding one-dimensional curves and sets are quite complete: we have sharp geometric Morrey-Sobolev embedding results for embedded curves of finite energy for all of these energies, providing insight into the topological and regularizing effects of such energies in the range of integrability exponents above the respective scale-invariant case. We have proven several valuable applications in geometric knot theory, such as bounding the number of isotopy types, bounds of the stick number and of the average crossing number solely in terms of these energies, and also some isotopy results in terms of energy and Hausdorff distance. We can minimize each of these energies in given knot classes. The topic can also be communicated well to the general public as our account on sphere-filling ropes shows: it appeared in the American Mathematical Monthly and was awarded the Merten Hasse Prize meant to honor “inspiring and dedicated teachers” and for “a noteworthy expository paper appearing in an Association publication.” Young analysts who have started their scientific career partly funded by this project have now proceeded on their own, such as S. Blatt who was able to characterize finite energy curves for most of these energies in terms of fractional Sobolev spaces, and who is the first one to treat analytically a particular family of repulsive energies with a gradient flow. New techniques in the flavor of pseudodifferential calculus and fractional Sobolev spaces were developed by him in cooperation with two other of my former Ph.D. students, P. Reiter and A. Schikorra, to attack the notoriously difficult question of higher regularity for minimizers or critical points of such singular energy functionals. We have also successfully generalized these energy concepts to higher dimension and co-dimension. We can now minimize these energies on surfaces of given genus, or in given isotopy classes. We also have all the tools to prove finiteness theorems in the class of non-smooth surfaces, resembling corresponding famous results of Cheeger, Anderson and others in the smooth category of Riemannian geometry. This opens up new questions regarding topological quantities for higher-dimensional isotopy classes possibly bounding higherdimensional knot-invariants that can be controlled by our energies, as well as the challenging questions of optimal regularity (not to speak of the shape) of critical points of such higher-dimensional geometric curvature functionals. The corresponding gradient flow problem remains completely open. Some of our results find interesting parallels in harmonic analysis where one considers similar quantities in the regime below scale-invariance. Ongoing work, e.g., the dissertation of M. Meurer on higher-dimensional integral Menger curvatures on measurable sets of arbitrary dimension might connect well to this other community. There are so many threads to follow now, in particular inspired by the recent research activities in cooperation with the community of geometric knot theory, as well as by some challenging problems in some subfields of theoretical physics, that we are planning a joint application for new funds in the near future supporting this line of research.
Projektbezogene Publikationen (Auswahl)
- A geometric curvature double integral of Menger type for space curves. Ann. Acad. Sci. Fenn. Math., 34(1):195–214, 2009
Paweł Strzelecki, Marta Szumanska, and Heiko von der Mosel
- Regularity theory for the Möbius energy. Commun. Pure Appl. Anal., 9(5):1463– 1471, 2010
Philipp Reiter
- Integral Menger curvature for surfaces. Adv. Math., 226:2233–2304, 2011
Paweł Strzelecki and Heiko von der Mosel
- On sphere-filling ropes. Amer. Math. Monthly, 118(10):863–876, 2011
Henryk Gerlach and Heiko von der Mosel
- Tangent-point repulsive potentials for a class of nonsmooth m-dimensional sets in Rn . Part I: Smoothing and self-avoidance effects, 2011. arXiv:1102.3642; J. Geom. Anal.
Paweł Strzelecki and Heiko von der Mosel
(Siehe online unter https://doi.org/10.1007/s12220-011-9275-z) - What are the longest ropes on the unit sphere? Arch. Ration. Mech. Anal., 201(1):303–342, 2011
Henryk Gerlach and Heiko von der Mosel
- A characterisation of inner product spaces by the maximal circumradius of spheres, 2012
Sebastian Scholtes
- On some knot energies involving Menger curvature, 2012
Marta Szumanska, Paweł Strzelecki, and Heiko von der Mosel
(Siehe online unter https://doi.org/10.1016/j.topol.2013.05.022) - Repulsive knot energies and pseudodifferential calculus for O’Hara’s knot energy family E(α) , α ∈ [2, 3). Math. Nachr., 285(7):889–913, 2012
Philipp Reiter
- Sławomir Kolasinski, Paweł Strzelecki, and Heiko von der Mosel. Characterizing W2,p -submanifolds n by p-integrability of global curvatures, 2012; Geometric and Functional Analysis
Sławomir Kolasinski, Paweł Strzelecki, and Heiko von der Mosel
- Tangency properties of sets with finite geometric curvature energies. Fund. Math., 218(2):165–191, 2012
Sebastian Scholtes
- Tangent-point self-avoidance energies for curves. J. Knot Theory Ramifications, 21(5):28 pages, 2012
Paweł Strzelecki and Heiko von der Mosel
- How averaged Menger curvatures control regularity and topology of curves and surfaces, 2013. Preprint 60, Inst. f. Mathematik, RWTH Aachen University
Paweł Strzelecki and Heiko von der Mosel
(Siehe online unter https://doi.org/10.1088/1742-6596/544/1/012018) - Menger curvature as a knot energy. Physics Reports, 2013. Preprint 59, Inst. f. Mathematik, RWTH Aachen University
Paweł Strzelecki and Heiko von der Mosel
(Siehe online unter https://doi.org/10.1016/j.physrep.2013.05.003) - On hypersurfaces of positive reach, alternating Steiner formulae and Hadwiger’s problem, 2013
Sebastian Scholtes
