Gradient Flows of Curvature Energies

What is the nicest shape of a knot? What means "nice" in this context? Or more precisely: is this clew of thread here knotted of not? And: Is there a natural way to transform a given knot into its optimal shape? To answer these questions, mathematicians coined the terms of curvature energy and knot energy and studied those energies. This is an attempt to measure the niceness of a knot. The smaller the energy the nicer the knot. These energies thus give an answer or actually many different answers, as we have many different knot energies to the first question we raised. There is a simple physical idea behind the most well-known energy of this kind: One wants to punish different threads of the curve that get close to each other. The inventor of this energy, the Japanese mathematician Jun O`Hara, used the self-repelling effects of electric charge. He distributed a quantum of electrical charge on the knot and calculated the potential energy of this charge distribution. The further different threads of the knot are, the smaller is this potential energy. He had to use the coulomb energy in four dimensional space though, in order to actually get the desired self-repelling effects. This research project deals with the last of questions mentioned above: Is there a natural way to transform a given knot into its optimal shape? Mathematicians therefore like to think of the energy as a mountain range and as ambitious mountaineers try to follow the "direttissima" -- the way of steepest ascend or in our case the way of steepest descent. For knot energies this approach has hardly been investigated before. The reason for that is certainly that the equations that appear in this context have a new structure they are so-called quasilinear, fractional parabolic partial differential equations that are in addition highly non-local; and this structure is deeply hidden within these equations. Completely new techniques have to be found in order to deal with these equations. There are various motivations for this project: On the one hand it is simply thrilling new mathematics at the border between mathematical disciplinces as entirely different as analysis, geometry, and topology. On the other hand there are interesting interconnections between some of these energies and the modeling of proteins and DNA or the modeling of the energies themselves are based on physical ideas. There are connections to other topics in current research like the modeling of membranes, especially the Willmore energy, and some deep topological questions that may be solved with these techniques. Last but not least fractional equations are a hot topic in the analysis of partial differential equations.

In this project, we addressed the question of what the most beautiful form of a geometric object is and whether there is a natural way of transforming a given geometric form into this most beautiful shape? In doing so, we dealt with research directions that go back to the last century, but also looked at approaches that are new and have only emerged in the last three decades. We contributed significantly to further research into the theory of elastica, which goes back to the work of Bernoulli and Euler, as well as to the theory of so-called knot energies, which were developed to determine the "most beautiful shape" of a geometric knot. We showed that a very natural way to find the optimal shape of a geometric object or knot can be put into practice. Motivated by the behavior of very ambitious climbers who follow the "direttissima" and always ascend or descend in the steepest direction in order to reach the summit or return to the valley as quickly as possible, mathematicians interpret the energy as a kind of mountain landscape and walk along the direction of the maximum gradient. We were able to prove that this tactic shows the way to an optimal shape, which was unknown especially for the knot energies mentioned but also for the p-elastic energies studied. By using advanced and technically difficult mathematical techniques from areas such as harmonic analysis and fractional operator theory, we were able to answer this question. We proved that critical points of numerous nodal energies are indeed analytic. This result is one of the first of its kind for solutions of nonlocal partial differential equations. In a tour de force, we managed to extend this result to a kind of "black box" for this type of nonlinear nonlocal equations. This allowed us to solve a variant of Hilbert's nineteenth problem for integro-differential equations and, in particular, to prove the analyticity of so-called fractional minimal surfaces - another previously unsolved problem. This is just a small selection of the exciting new discoveries that we have made as part of this research project in the conext of curvature energies. The numerous prizes won as part of the project (Stegbuchner Prize, Marie-Andreßner Prize, Marshall Fund Scholarship, Price of Excellence) underline the importance and appreciation of the research carried out by the participating research team.