Martin Kilian, Graham Smith
We adapt Sklyanin's $K$-matrix formalism to the sinh-Gordon equation, and prove that all free boundary constant mean curvature (CMC) annuli in the unit ball in $\mathbb{R}^3$ are of finite type.
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On the elliptic sinh-Gordon equation with integrable boundary conditions II -- Baker-Akhiezer theory
研究带Durham边界条件的椭圆sinh-Gordon方程有限型解，推导边界条件成立的必要有理准则及谱曲线4点附加条件，明确互补边界下准则的充分性，为相关解的构建奠定基础。
We study the space of periodic solutions of the elliptic $\sinh$-Gordon equation by means of spectral data consisting of a Riemann surface $Y$ and a divisor $D$ and prove the existence of certain Darboux coordinates.
This note summarizes results that were obtained by the author in his habilitation thesis (arXiv:1607.08792) concerning the development of a spectral theory for simply periodic, 2-dimensional, complex-valued solutions of the sinh-Gordon equation. Spectral data for such solutions are defined for periodic Cauchy data on a line (following Hitchin and Bobenko) and the space of spectral data is described by an asymptotic characterization. Using methods of asymptotic estimates, the inverse problem for the spectral data of such Cauchy data is answered. Finally a Jacobi variety for the spectral curve i…
We study the elliptic sinh-Gordon and sine-Gordon equations on the real plane and we introduce new families of solutions. We use a Backlund transformation that connects the elliptic versions of sinh-Gordon and sine-Gordon equations. As an application, we construct new harmonic maps between surfaces, when the target is of constant curvature -1.
We study the space of periodic solutions of the elliptic $\sinh$-Gordon equation by means of spectral data consisting of a Riemann surface $Y$ and a divisor $D$. We show that the space $M_g^{\mathbf{p}}$ of real periodic finite type solutions with fixed period $\mathbf{p}$ can be considered as a completely integrable system $(M_g^{\mathbf{p}},Ω,H_2)$ with a symplectic form $Ω$ and a series of commuting Hamiltonians $(H_n)_{n \in \mathbb{N}}$. In particular we relate the gradients of these Hamiltonians to the Jacobi fields $(ω_n)_{n\in \mathbb{N}_0}$ from the Pinkall-Sterling iteration. Moreove…
This preliminary report studies immersed surfaces of constant mean curvature in $H^3$ through their {\it adjusted Gauss maps} (as harmonic maps in $S^2$) and their {\it adjusted frames} in SU(2). Lawson's correspondence between Euclidean CMC surfaces and their hyperbolic cousins is interpreted here under a different perspective: the equivalence of their Weierstrass representations (normalized potentials). This work also presents a construction algorithm for the moving frame, the adjusted frame, their Maurer-Cartan forms, and ultimately the CMC immersion.
Sinh-Gordon equations on finite graphs
本文研究了图上的sinh-Gordon方程，通过引入一致的先验估计来定义拓扑度，逐案计算并证明了在有限、连通且对称图上具有非零规定函数的经典方程总是可解的。
Solutions of the Sinh-Gordon Equation of Spectral Genus Two and constrained Willmore Tori I
该研究探讨谱亏格g<3的椭圆型sinh-Gordon方程解的参数化、流轨道性质及等谱集影响，还利用极限情况构造约束Willmore共形映射并计算Willmore泛函。
Superposition Formulae for the Geometric Bäcklund Transformations of the Hyperbolic and Elliptic Sine-Gordon and Sinh-Gordon Equations
本文提供了三维伪欧几里得空间中六种曲面Bäcklund变换的叠加公式，这些公式在首次积分后能代数地生成无穷多解，并揭示了椭圆方程中解交替的不寻常性质。
In this work a spectral theory for 2-dimensional, simply periodic, complex-valued solutions u of the sinh-Gordon equation is developed. Spectral data for such solutions are defined (following Hitchin and Bobenko) and the space of spectral data is described by an asymptotic characterization. Using methods of asymptotic estimates, the inverse problem for the spectral data is solved along a line, i.e. the solution u is reconstructed on a line from the spectral data. Finally a Jacobi variety and Abel map for the spectral curve is constructed; they are used to describe the change of the spectral da…
New examples of harmonic maps to the hyperbolic plane via Bäcklund transformation
本文利用连接sinh-Gordon与sine-Gordon方程的Bäcklund变换，发展相关机制，构造出从复平面子集到双曲平面的新调和映射。
It is shown that both the sinh--Gordon equation and the elliptic Tzitzeica equation can be interpreted as the Taubes equation for Abelian vortices on a CMC surface embedded in $\R^{2, 1}$, or on a surface conformally related to a hyperbolic affine sphere in $\R^3$. In both cases the Higgs field and the U(1) vortex connection are constructed directly from the Riemannian data of the surface corresponding to the sinh--Gordon or the Tzitzeica equation. Radially symmetric solutions lead to vortices with a topological charge equal to one, and the connection formulae for the resulting third Painlevé …
The spectral curve correspondence for finite-type solutions of the sinh-Gordon equation describes how they arise from and give rise to hyperelliptic curves with a real structure. Constant mean curvature (CMC) 2-tori in $ S ^ 3 $ result when these spectral curves satisfy periodicity conditions. We prove that the spectral curves of CMC tori are dense in the space of smooth spectral curves of finite-type solutions of the sinh-Gordon equation. One consequence of this is the existence of countably many real $ n $-dimensional families of CMC tori in $ S ^ 3 $ for each positive integer $ n $.
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