arXiv research

DPW method reconstructs minimal and symmetric CMC surfaces in 3-sphere.

Paper constructs Lawson surfaces using Fuchsian DPW potentials.

New proof of high genus Lawson surfaces with area estimates.

Global harmonic maps into SU(1,1) constructed from Smyth potentials using DPW method.

Python tool for constructing CMC surfaces using DPW method.

We consider constant mean curvature 1 surfaces in $\mathbb{R}^3$ arising via the DPW method from a holomorphic perturbation of the standard Delaunay potential on the punctured disk. Kilian, Rossman and Schmitt have proven that such a surface is asymptotic to a Delaunay surface. We consider families of such potentials p…

We use the DPW method to obtain the associate family of Delaunay surfaces and derive a formula for the neck size of the surface in terms of the entries of the holomorphic potential.

Survey of Willmore surfaces in spheres using DPW method.

New method constructs translationally equivariant hyperbolic affine spheres.

Solutions of tt*-equation from SU(2)_k fusion algebra.

Paper develops a new method for harmonic maps into symmetric spaces.

We construct constant mean curvature surfaces in euclidean space with genus zero and n ends asymptotic to Delaunay surfaces using the DPW method.

We construct constant mean curvature surfaces in euclidean space by gluing n half Delaunay surfaces to a non-degenerate minimal n-noid, using the DPW method.

We combine the DPW method and Opening Nodes to construct embedded surfaces of positive constant mean curvature with Delaunay ends in euclidean space, with no limitation to the genus or number of ends.

The paper describes a new method for Willmore surfaces in spheres.

Applying the DPW version of the theory developed by Burstall and Guest for harmonic maps of finite uniton type, we derive a coarse classification of Willmore two-spheres in $S^{n+2}$ in terms of the normalized potential of their (harmonic) conformal Gauss maps. Moreover, for the case of $S^6$, some geometric properties…

We consider compact minimal surfaces $f\colon M\to S^3$ of genus 2 which are homotopic to an embedding. We assume that the associated holomorphic bundle is stable. We prove that these surfaces can be constructed from a globally defined family of meromorphic connections by the DPW method. The poles of the meromorphic co…

We investigate the Lawson genus $2$ surface by methods from integrable system theory. We prove that the associated family of flat connections comes from a family of flat connections on a $4-$punctured sphere. We describe the symmetries of the holonomy and show that it is already determined by the holonomy around one of…

New method constructs surfaces with constant mean curvature.

The paper constructs families of high genus CMC surfaces in the 3-sphere.

In this paper we describe how the operation of adding a uniton arises via the DPW method of obtaining harmonic maps into compact Riemannian symmetric spaces out of certain holomorphic one forms. We exploit this point of view to investigate which unitons preserve finite type property of harmonic maps. In particular, we …

Study Lorentz harmonic maps and spacelike surfaces in anti-de Sitter space.

In this paper we numerically construct CMC deformations of the Lawson minimal surfaces $ξ_{g,1}$ using a spectral curve and a DPW approach to CMC surfaces in spaceforms.

Minimal Lagrangian surfaces in complex hyperbolic quadric via loop group method.

Constructs special surfaces in hyperbolic space with constant mean curvature.

Let $G$ be a complex Lie group and $ΛG$ denote the group of maps from the unit circle ${\mathbb S}^1$ into $G$, of a suitable class. A differentiable map $F$ from a manifold $M$ into $ΛG$, is said to be of \emph{connection order $(_a^b)$} if the Fourier expansion in the loop parameter $λ$ of the ${\mathbb S}^1$-family …

The quantum cohomology of CP^1 provides a distinguished solution of the third Painleve equation. S. Cecotti and C. Vafa discovered this from a physical viewpoint. We derive it from a differential geometric viewpoint, using the theory of harmonic maps and in particular the generalized Weierstrass representation (DPW rep…

This paper concerns rigidity results to Serrin's overdetermined problem in an epigraph $\{\begin{aligned} &Δu+ f(u)=0,\ \ \ {in}\ Ω=\{(x^\prime,x_n): x_n>\varphi (x^\prime)\},\\ &u>0,\ \ \ {in}\ Ω,\\ &u=0,\ \ \ {on}\ \partialΩ,\\ &|\nabla u|=const. {on} \partialΩ. \end{aligned}.$ We prove that up to isometry the ep…

Develops a new method for minimal Lagrangian surfaces in complex quadrics.

We study two aspects of the loop group formulation for isometric immersions with flat normal bundle of space forms. The first aspect is to examine the loop group maps along different ranges of the loop parameter. This leads to various equivalences between global isometric immersion problems among different space forms …

Skew parallelogram nets factorize, encompassing discrete differential geometry.

In this article we obtain total masses of solutions to the Toda system associated to a general simple Lie algebra with singular sources at the origin. The determination of such total masses is one of the important steps towards establishing the a priori bound for solutions to the mean field type of Toda system on compa…

In this paper we deal with the global properties of Willmore surfaces in spheres via the harmonic conformal Gauss map using loop groups. We first derive a global description of those harmonic maps which can be realized as conformal Gauss maps of some Willmore surfaces (Theorem 3.4, Theorem 3.11 and Theorem 3.18). Then …

Develops explicit formulas for minimal immersions in 5D space.

In this article we discuss the distribution of asset price movements by the market potential function. From the principle of free energy minimization we analyze two different kinds of market potentials. We obtain a U-shaped potential when market reversion (i.e. contrarian investors) is dominant. On the other hand, if t…

Develops potential theory for WZW equation in Kähler potentials space.

The paper examines stability of harmonic and symphonic maps with forms and potentials.

Extracts interpretable potential energy from Hamiltonian systems.

The paper examines stability of subelliptic harmonic maps with potential.

The paper describes flat Hessian metrics on surfaces and their potentials.

A hyperKähler potential is a function rho that is a Kähler potential for each complex structure compatible with the hyperKähler structure. Nilpotent orbits in a complex simple Lie algebra are known to carry hyperKähler metrics admitting such potentials. In this paper, we explicitly calculate the hyperKähler potential w…

Investigates how adding a scalar potential affects Dirac-harmonic maps.

Article provides Bernstein gradient estimates for heat equations with potential terms.

In this paper we study potential function of gradient steady Ricci solitons. We prove that infimum of potential function decays linearly; in particular, potential function of rectifiable gradient steady Ricci solitons decays linearly. As a consequence, we show that a gradient steady Ricci soliton with bounded potential…

We consider the geodesic equation for the generalized Kahler potential with only mixed second derivatives bounded. We show that given such two generalized Kahler potentials, there is a unique geodesic segment such that for each point on the geodesic, the generalized Kahler potential has uniformly bounded mixed second d…

We show two results about the Conway potential function which is known as the normalized multivariable Alexander polynomial. We first show that the Conway potential function introduced by Kauffman in "Formal Knot Theory" is indeed a link invariant. Next we show that Kauffman's potential function equals Hartley's potent…

The paper characterizes potential functions whose level sets are orbits in mechanical systems.

The paper studies $m$-quasi Einstein manifolds with convex potential and finds constant scalar curvature.