Abstract Using the method of equivariant moving frames, we derive the evolution equations for the curvature invariants of arc-length parametrized curves under arc-length preserving geometric flows in two-, three- and four-dimensional Cayley–Klein geometries. In two and three dimensions, we obtain recursion operators, which show that the curvature evolution equations obtained are completely integrable.
We introduce matrix coupled (local and non-local) dispersionless equations, construct GBDT (generalized Bäcklund-Darboux transformation) for these equations, derive wide classes of explicit multipole solutions, give explicit expressions for the corresponding Darboux and wave matrix valued functions and study their asymptotics in some interesting cases. We consider the scalar cases of coupled, complex coupled and non-local dispersionless equations as well.
We use generalized kernel functions to construct explicit solutions by integrals of the non-stationary Schrödinger equation for the Hamiltonian of the elliptic Calogero–Sutherland model (also known as elliptic Knizhnik–Zamolodchikov–Bernard equation). Our solutions provide integral representations of elliptic generalizations of the Jack polynomials.
We prove that the set of closed finite gap curves in hyperbolic 3-space $\mathbb{H}^{3}$ is $W^{2,2}$-dense in the Sobolev space of all closed $W^{2,2}$-curves in $\mathbb{H}^{3}$. We also show that the set of closed finite gap curves in any two-dimensional space form $\mathbb{E}^{2}$ is $W^{2,2}$-dense in the Sobolev space of all closed $W^{2,2}$-curves in $\mathbb{E}^{2}$.
Solutions of Hitchin's self-duality equations corresponds to special real sections in the Deligne-Hitchin moduli space -- twistor lines. A question posed by Simpson in 1997 asks whether all real sections give rise to global solutions of the self-duality equations. An affirmative answer would allow for complex analytic procedure to obtain all solutions of the self-duality equations. The purpose of this paper is to construct counter examples given by certain (branched) Willmore surfaces in $3$-space (with monodromy) via the generalized Whitham flow. Though these higher solutions do not give rise to global solutions of the self-duality equations on the whole Riemann surface $M$, they are solutions on an open dense subset of it. This suggest a deeper connection between Willmore surfaces, i.e., rank $4$ harmonic maps theory, with the rank $2$ self-duality theory.
