Related papers: Dynamics of Induced Surfaces in Four-Dimensional E…
A marginally trapped surface in the four-dimensional Minkowski space is a spacelike surface whose mean curvature vector is lightlike at each point. We associate a geometrically determined moving frame field to such a surface and using the…
In this paper we continue the study of the superconformal index of four-dimensional $\mathcal{N}=2$ theories of class $\mathcal{S}$ in the presence of surface defects. Our main result is the construction of an algebra of difference…
We consider harmonic maps on simply connected Riemann surfaces into the group $\mathrm{U}(n)$ of unitary matrices of order $n$. It is known that a harmonic map with an associated algebraic extended solution can be deformed into a new…
We give a local representation for the pseudoholomorphic surfaces in Euclidean spheres in terms of holomorphic data. Similar to the case of the generalized Weierstrass representation of Hoffman and Osserman, we assign such a surface in…
Dilation surfaces, or twisted quadratic differentials, are variants of translation surfaces. In this paper, we study the question of what elements or subgroups of the mapping class group can be realized as affine automorphisms of dilation…
We show that the discrete operator stemming from the time and space discretization of evolutionary partial differential equations can be represented in terms of a single Sylvester matrix equation. A novel solution strategy that combines…
We present a fully general derivation of the Laplace--Young formula and discuss the interplay between the intrinsic surface geometry and the extrinsic one ensuing from the immersion of the surface in the ordinary euclidean three-dimensional…
The discrete Laplacian on Euclidean triangulated surfaces is a well-established notion. We introduce discrete Laplacians on spherical and hyperbolic triangulated surfaces. On the one hand, our definitions are close to the Euclidean one in…
A classification of 2-dimensional surfaces imbedded in spacetime is presented, according to the algebraic properties of their shape tensor. The classification has five levels, and provides among other things a refinement of the concepts of…
This paper develops a discrete theory of real Riemann surfaces based on quadrilateral cellular decompositions (quad-graphs) and a linear discretization of the Cauchy-Riemann equations. We construct a discrete analogue of an antiholomorphic…
In this paper, we consider both differential and algebraic properties of surfaces associated with sigma models. It is shown that surfaces defined by the generalized Weierstrass formula for immersion for solutions of the CP^{N-1} sigma model…
Modeling arbitrarily large deformations of surfaces smoothly embedded in three-dimensional space is challenging. The difficulties come from two aspects: the existing geometry processing or forward simulation methods penalize the difference…
Using the Schwarzian derivative we construct a sequence $\left(P_{\ell}\right)_{\ell \geqslant 2}$ of meromorphic differentials on every non-flat oriented minimal surface in Euclidean $3$-space. The differentials…
We study surfaces with parallel normalized mean curvature vector field in Euclidean or Minkowski 4-space. On any such surface we introduce special isothermal parameters (canonical parameters) and describe these surfaces in terms of three…
We study the problem posed by F. Burstall of developing a theory of isothermic Euclidean submanifolds of dimension greater than or equal to three. As a natural extension of the definition in the surface case, we call a Euclidean submanifold…
In this paper we develop an abstract theory for the Codazzi equation on surfaces, and use it as an analytic tool to derive new global results for surfaces in the space forms ${\bb R}^3$, ${\bb S}^3$ and ${\bb H}^3$. We give essentially…
We introduce the Loop Weierstrass Representation for minimal surfaces in Euclidean space and constant mean curvature 1 surfaces in hyperbolic space by applying integral system methods to the Weierstrass and Bryant representations. We unify…
We derive the vorticity equation for an incompressible fluid on a 2-dimensional surface with arbitrary topology embedded in 3-dimensional Euclidean space by using a tailored Clebsch parametrization of the flow. In the inviscid limit, we…
In this paper we prove two results. The first shows that the Dirichlet-Neumann map of the operator $\Delta_g+q$ on a Riemannian surface can determine its topological, differential, and metric structure. Earlier work of this type assumes a…
In this paper, we study the inverse surfaces in 3-dimensional Euclidean space $\mathbb{E}^{3}$. We obtain some results relating Christoffel symbols, the normal curvatures, the shape operators and the third fundamental forms of the inverse…