arXiv research

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

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

Survey of Willmore surfaces in spheres using DPW method.

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.

Paper constructs Lawson surfaces using Fuchsian DPW potentials.

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

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

New method constructs surfaces with constant mean curvature.

Starting at a saddle tower surface, we give a new existence proof of the Lawson surfaces $ξ_{m,k}$ of high genus by deforming the corresponding DPW potential. As a byproduct, we obtain for fixed $m$ estimates on the area of $ξ_{m,k}$ in terms of their genus $g=m k \gg1$.

The paper builds a DPW approach of Willmore surfaces via conformal Gauss maps. As applications, we provide descriptions of minimal surfaces in $\mathbb R^{n+2}$, isotropic surfaces in $S^4$ and homogeneous Willmore tori via the loop group method. A new example of a Willmore two-sphere in $S^6$ without dual surfaces is …

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.

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…

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…

Using the DPW method, we construct genus zero Alexandrov-embedded constant mean curvature (greater than one) surfaces with any number of Delaunay ends in hyperbolic 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.

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.

New method constructs translationally equivariant hyperbolic affine spheres.

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…

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 …

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

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

Skew parallelogram nets factorize, encompassing discrete differential geometry.

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

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 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…

Develops explicit formulas for minimal immersions in 5D space.

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…

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 …

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 …

New representation theory for closed geodesic subflows.

Proves EGF representations in specific geometric contexts.

The paper establishes isomorphisms and constructs colored versions of Lawrence representations.

Study k-positive surface group representations and their degenerations.

Develops theory of Anosov representations for Fuchsian groups, showing stability and analytical properties.

New representations defined for groups and graphs, with applications to stable representations.

Collar lemma proven for certain surface group representations.

Study subgroup actions on mapping class groups using Heisenberg representations.

Researchers describe unitary representations of mixed braid groups.

This paper addresses law invariant coherent risk measures and their Kusuoka representations. By elaborating the existence of a minimal representation we show that every Kusuoka representation can be reduced to its minimal representation. Uniqueness -- in a sense specified in the paper -- of the risk measure's Kusuoka r…

A very popular problem on braid groups has recently been solved by Bigelow and Krammer, namely, they have found a faithful linear representation for the braid group B_n. In their papers, Bigelow and Krammer suggested that their representation is the monodromy representation of a certain fibration. Our goal in this pape…

Let S be a closed orientable surface of genus at least 2 and let G be a semisimple real algebraic group of non-compact type. We consider a class of representations from the fundamental group of S to G called positively ratioed representations. These are Anosov representations with the additional condition that certain …

In this paper, we introduce a study of prolongations of representations of Lie groups. We obtain a faithful (one-to-one) representation of TG where G is a finite-dimensional Lie group and TG is the tangent bundle of G, by using (not necessarily faithful) representations of G. We show that tangent functions of Lie group…

Polynomial representations found in surface braid and mapping class groups.

Paper addresses the disparity between sampled and mean representations in disentangled learning.