On closed finite gap curves in spaceforms II

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 $\mathbb{E}^{2}$.

Properly embedded minimal annuli in $\mathbb{S}^2 \times \mathbb{R}$

We prove that every properly embedded minimal annulus in $\mathbb{S}^2\times\mathbb{R}$ is foliated by circles. We show that such minimal annuli are given by periodic harmonic maps $\mathbb{C} \to \mathbb{S}^2$ of finite type. Such harmonic maps are parameterized by spectral data, and we show that continuous deformations of the spectral data preserve the embeddedness of the corresponding annuli. A curvature estimate of Meeks and Rosenberg is used to show that each connected component of spectral data of embedded minimal annuli contains a maximum of the flux of the third coordinate. A classification of these maxima allows us to identify the spectral data of properly embedded minimal annuli with the spectral data of minimal annuli foliated by circles.

Exact solutions by integrals of the non-stationary elliptic Calogero–Sutherland equation

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.

Geometric curve flows in low dimensional Cayley–Klein geometries

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.

GBDT version of the Darboux transformation for the matrix coupled dispersionless equations (local and non-local cases)

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.

Higher solutions of Hitchin’s self-duality equations

Solutions of Hitchin’s self-duality equations correspond to special real sections of 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 in principle allow for complex analytic procedures to obtain all solutions of the self-duality equations. The purpose of this article is to construct counter examples given by certain (branched) Willmore surfaces in three-space (with monodromy) via the generalized Whitham flow. Though these sections do not give rise to global solutions of the self-duality equations on the whole Riemann surface M, they induce solutions on an open and dense subset of it. This suggest a connection between Willmore surfaces, i.e., rank 4 harmonic maps theory, with the rank 2 self-duality theory.

Voros coefficients for the hypergeometric differential equations and Eynard–Orantin’s topological recursion: Part II: For confluent family of hypergeometric equations

We show that each member of the confluent family of the Gauss hypergeometric equations is realized as quantum curves for appropriate spectral curves. As an application, relations between the Voros coefficients of those equations and the free energy of their classical limit computed by the topological recursion are established. We will also find explicit expressions of the free energy and the Voros coefficients in terms of the Bernoulli numbers and Bernoulli polynomials. Communicated by: Youjin Zhang

Set-theoretical solutions to the reflection equation associated to the quantum affine algebra of type $\boldsymbol{A^{(1)}_{n-1}}$

A trick to obtain a solution to the set-theoretical reflection equation from a known one to the Yang–Baxter equation is applied to crystals and geometric crystals associated to the quantum affine algebra of type $A^{(1)}_{n-1}$.

Affine screening operators, affine Laumon spaces and conjectures concerning non-stationary Ruijsenaars functions

Based on the screened vertex operators associated with the affine screening operators, we introduce the formal power series $f^{\widehat{\mathfrak gl}_N}(x,p|s,\kappa|q,t)$ which we call the non-stationary Ruijsenaars function. We identify it with the generating function for the Euler characteristics of the affine Laumon spaces. When the parameters $s$ and $\kappa$ are suitably chosen, the limit $t\rightarrow q$ of $f^{\widehat{\mathfrak gl}_N}(x,p|s,\kappa|q,q/t)$ gives us the dominant integrable characters of $\widehat{\mathfrak sl}_N$ multiplied by $1/(p^N;p^N)_\infty$ (i.e. the $\widehat{\mathfrak gl}_1$ character). Several conjectures are presented for $f^{\widehat{\mathfrak gl}_N}(x,p|s,\kappa|q,t)$, including the bispectral and the Poincaré dualities, and the evaluation formula. The main conjecture asserts that (i) one can normalize $f^{\widehat{\mathfrak gl}_N}(x,p|s,\kappa|q,t)$ in such a way that the limit $\kappa\rightarrow 1$ exists, and (ii) the limit $f^{{\rm st.}\,\widehat{\mathfrak gl}_N}(x,p|s|q,t)$ gives us the eigenfunction of the elliptic Ruijsenaars operator. The non-stationary affine $q$-difference Toda operator ${\mathcal T}^{\widehat{\mathfrak gl}_N}(\kappa)$ is introduced, which comes as an outcome of the study of the Poincaré duality conjecture in the affine Toda limit $t\rightarrow 0$. The main conjecture is examined also in the limiting cases of the affine $q$-difference Toda ($t\rightarrow 0$), and the elliptic Calogero–Sutherland ($q,t\rightarrow 1$) equations.