arXiv research

New solutions found for elliptic sinh-Gordon and sine-Gordon equations.

Formulae for Bäcklund transformations of hyperbolic and elliptic sine-Gordon/sinh-Gordon equations.

Study finite-type solutions of elliptic sinh-Gordon equation with Durham boundary conditions.

Study proves all free boundary CMC annuli are of finite type.

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.

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 conn…

In this article it is shown that the study of harmonic diffeomorphisms, with nonvanishing Hopf differential, reduces to the study of the Beltrami equation of a certain type: the imaginary part of the logarithm of the Beltrami function coincides with the imaginary part of the logarithm of the Hopf differential, therefor…

Vortices on conical surfaces embedded in hyperbolic space.

We investigate solutions of the elliptic sinh-Gordon equation of spectral genus g<3. These solutions are parametrized by complex matrix-valued polynomials called potentials. On the space of these potentials there act two commuting flows. The orbits of these flows are called Polynomial Killing fields and are double peri…

The sinh-Gordon equation is solved on finite, symmetric graphs.

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 s…

New harmonic maps to hyperbolic plane via Bäcklund transformation.

Considering the kinematics of the moving frame associated with a constant mean curvature surface immersed in S^3 we derive a linear problem with the spectral parameter corresponding to elliptic sinh-Gordon equation. The spectral parameter is related to the radius R of the sphere S^3. The application of the Sym formula …

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…

Study on surfaces in Heisenberg group with constant mean curvature.

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 spectra…

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 …

The paper classifies hypersurfaces in a product of two spheres with constant curvature.

Solves constant pre-factor problem for tt*-Toda equations using asymptotic data and symplectic structures.

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 pe…

New method for analyzing elliptic and parabolic equations.

Constructs universal local deformations for curves and differential forms.

Two classification theorems for Willmore surfaces in S² × S².

Proves C^2,alpha estimates for elliptic equations on hyperkähler manifolds.

The paper examines ellipticity of specific equations on vector bundles.

Note on advancements in nonlinear elliptic equations' regularity theory.

Study constant mean curvature tori in R^3 using spectral data and Whitham deformations.

Proves solutions to elliptic equations on Hermitian manifolds with optimal conditions.

Paper proves a Liouville theorem for a generalized elliptic equation on H-type groups.

Derives estimates for geometric elliptic equations on complex manifolds.

Study shows solutions to certain equations form smooth manifolds.

We give a review of the systematic construction of hierarchies of soliton flows and integrable elliptic equations associated to a complex semi-simple Lie algebra and finite order automorphisms. For example, the non-linear Schrödinger equation, the n-wave equation, and the sigma-model are soliton flows; and the equation…

Study fully nonlinear elliptic equations on complex manifolds.

Deep neural nets solve complex insurance math equations.

Solves open problems for fully nonlinear elliptic equations on manifolds.

We establish a general theorem improving regularity of solutions of elliptic pseudodifferential equations. It allows to resolve in a unified way the regularity issue for a broad class of nonlinear elliptic equations and systems appearing in different areas of geometry and analysis.

Study improves understanding of solutions to complex equations in geometry.

Study elliptic equations on hyperhermitian manifolds with flat hyperkähler metric.

Study fully nonlinear equations on Hermitian manifolds to find metrics with specific curvature.

Researchers solve an inverse problem for a semilinear elliptic equation on complex manifolds.

We prove a priori interior $C^{2,α}$ estimates for solutions of fully nonlinear elliptic equations of twisted type. For example, our estimates apply to equations of the type convex + concave. These results are particularly well suited to equations arising from elliptic regularization. As application, we obtain a new pr…

The paper concerns singular solutions of nonlinear elliptic equations.

Solves a specific Dirichlet problem on Riemannian manifolds.

New method solves elliptic equations on manifolds without grids.

Mathematical formulas for elliptic curve integrals solve anomaly equations.

Uniqueness found for elliptic equations with drift on manifolds.

Proves regularity for quasilinear elliptic equations in metric spaces.

We prove the existence of non-smooth solutions to fully nonlinear uniformly elliptic equations.