On the second variation of integral Menger curvature
- Über die zweite Variation der integralen Mengerkrümmung
Knappmann, Jan; von der Mosel, Heiko (Thesis advisor); Wagner, Alfred (Thesis advisor)
Dissertation / PhD Thesis
Dissertation, RWTH Aachen University, 2020
In this thesis, we show the existence of a second variation of the integral Menger curvature functional M_p in the situation of p>3, compute its explicit formula, and use it to prove the common conjecture that the round (meaning undeformed) circle is an isolated local minimizer for the knot energy among all space curves in R^3. The existence of a first variation of the integral Menger curvature was already proven by Hermes in 2014 who could also use this formula to show that the round circle is a critical point of M_p. With some adaptations and further developements applied to Hermes' methods, we can show the existence and compute the formulas of both the second and third variation of M_p and derive a conjecture for the n-th variation from that. After we discuss integral Menger curvature in the context of geometric knot energies, we consider a scale invariant version of M_p denoted by E_p and insert the round circle c in its second variation D^2E_p(c)[ , ]. In order to show that the round circle is an isolated local minimizer of E_p, we express the perturbation Phi by Fourier series in each component and convert the term D^2E_p(c)[Phi,Phi] into a series expression consisting only of the Fourier coefficients of Phi. A close investigation of this series assures positive semi-definiteness for D^2E_p(c)[Phi,Phi] and a Taylor expansion of E_p(c+t Phi) up to the second degree allows to prove that for a sufficiently small t the energy values of E_p(c+t Phi) is strictly greater than E_p(c) for all Phi satisfying D^2E_p(c)[Phi,Phi]>0. A deeper analysis of the null space N_E of the quadratic form D^2E_p(c)[ , ] enables us to transfer this result to all possible perturbations Phi which is the claimed local minimizing property for the round circle c. We also apply the described strategies to the closely related tangent point energy TP_q, more accurately its scale invariant version F_q, for which it is already known that the round circle is the unique global minimizer. Although the proof of minimality can not be simply reproduced, the applied methods also provide a series expression for D^2F_q(c)[Phi,Phi] and new insights on the corresponding null space of D^2F_q(c)[ , ].