arXiv research

Discrete maximal surfaces identified from s-embeddings.

This paper answers a question about discrete embeddings to maximal surfaces.

Study discrete analog of zeta-determinant maximization on triangulated surfaces.

In this paper we define and analyze singularities of discrete linear Weingarten surfaces with Weierstrass-type representations in $3$-dimensional Riemannian and Lorentzian spaceforms. In particular, we discuss singularities of discrete surfaces with non-zero constant Gaussian curvature, and parallel surfaces of discret…

We establish what semi-discrete linear Weingarten surfaces with Weierstrass-type representations in $3$-dimensional Riemannian and Lorentzian spaceforms are, confirming their required properties regarding curvatures and parallel surfaces, and then classify them. We then define and analyze their singularities. In partic…

We introduce and study a new class of representations of surface groups into Lie groups of Hermitian type, called {\em weakly maximal} representations. We prove that weakly maximal representations are discrete and injective and we describe the structure of the Zariski closure of their image. Furthermore we prove that t…

Optimizes eigenvalues on surfaces with symmetries.

We introduce and study a new class of representations of surface groups into Lie groups of Hermitian type, called weakly maximal representations. They are defined in terms of invariants in bounded cohomology and extend considerably the scope of maximal representations. We prove that weakly maximal representations are d…

Let G be a connected semisimple Lie group such that the associated symmetric space X is Hermitian and let Gamma be the fundamental group of a compact orientable surface of genus at least 2. We survey the study of maximal representations, that is the subset of Hom(Gamma,G) which is a union of components characterized by…

In this paper we study the moduli space of representations of a surface group (i.e., the fundamental group of a closed oriented surface) in the real symplectic group Sp(2n,R). The moduli space is partitioned by an integer invariant, called the Toledo invariant. This invariant is bounded by a Milnor-Wood type inequality…

We develop the theory of maximal representations of the fundamental group of a compact connected oriented surface with boundary, into a group of Hermitian type. For any such representation we define the Toledo invariant, for which we establish properties such as uniform boundedness on the representation variety, additi…

Arithmetic topology connects surface and $p$-adic field studies, enabling new insights into Galois groups.

Simple constructions of semi-discrete and discrete surfaces using Jacobi elliptic functions.

Consider power utility maximization of terminal wealth in a 1-dimensional continuous-time exponential Levy model with finite time horizon. We discretize the model by restricting portfolio adjustments to an equidistant discrete time grid. Under minimal assumptions we prove convergence of the optimal discrete-time strate…

We define discrete flat surfaces in hyperbolic 3-space from the perspective of discrete integrable systems and prove properties that justify the definition. We show how these surfaces correspond to previously defined discrete constant mean curvature 1 surfaces in hyperbolic 3-space, and we also describe discrete focal …

Solves utility maximization for delayed informed investors.

Study finds Hilbert square of real surfaces can be maximal even when the surface has disconnected real locus.

Paper generalizes discrete CMC surfaces and shows how they can be derived.

In the present paper, we propose a new discrete surface theory on 3-valent embedded graphs in the 3-dimensional Euclidean space which are not necessarily discretization or approximation of smooth surfaces. The Gauss curvature and the mean curvature of discrete surfaces are defined which satisfy properties corresponding…

We obtain a unified theory of discrete minimal surfaces based on discrete holomorphic quadratic differentials via a Weierstrass representation. Our discrete holomorphic quadratic differential are invariant under Möbius transformations. They can be obtained from discrete harmonic functions in the sense of the cotangent …

Permutability of surface transforms yields discrete analogs.

Study on discrete Gaussian curvature for polyhedral surfaces.

We propose a natural discretisation scheme for classical projective minimal surfaces. We follow the classical geometric characterisation and classification of projective minimal surfaces and introduce at each step canonical discrete models of the associated geometric notions and objects. Thus, we introduce discrete ana…

A formula connects discrete harmonic surfaces to holomorphic functions.

New groups discovered with unique properties in a specific space.

Survey on discrete minimal surfaces and their properties.

New representations for discrete surfaces derived from dual transforms.

Study of Ricci flow on discrete surfaces of revolution with constant Gaussian curvature.

We present a definition of discrete channel surfaces in Lie sphere geometry, which reflects several properties for smooth channel surfaces. Various sets of data, defined at vertices, on edges or on faces, are associated with a discrete channel surface that may be used to reconstruct the underlying particular discrete L…

Study on discrete surfaces with constant principal curvature for nanocarbon applications.

MODWST improves classification tasks with wavelet scattering.

We consider a discrete-time financial market model with finite time horizon and give conditions which guarantee the existence of an optimal strategy for the problem of maximizing expected terminal utility. Equivalent martingale measures are constructed using optimal strategies.

This paper explores geometric insights into discrete R-congruences and their envelopes.

This paper completes the classification of discrete conformal structures on surfaces.

Discrete approximation solves Björling's minimal surface problem.

New discrete models for constant mean curvature surfaces and tori.

We translate a classification scheme for periodic CMC surfaces developed by J. Dorfmeister and the author to discrete CMC surfaces in the sense of A. Bobenko and U. Pinkall. The scheme uses the dressing action on discrete CMC surfaces to arrive at a classification for periodic discrete CMC surfaces.

Discrete linear Weingarten surfaces in space forms are characterized as special discrete $Ω$-nets, a discrete analogue of Demoulin's $Ω$-surfaces. It is shown that the Lie-geometric deformation of $Ω$-nets descends to a Lawson transformation for discrete linear Weingarten surfaces, which coincides with the well-known L…

Discretizes special surfaces using Koenigs nets.

Our aim in this paper is to provide a theory of discrete Riemann surfaces based on quadrilateral cellular decompositions of Riemann surfaces together with their complex structure encoded by complex weights. Previous work, in particular of Mercat, mainly focused on real weights corresponding to quadrilateral cells havin…

The paper explores reflection principles for lightlike line segments on maximal surfaces.

The main result of this paper is a discrete Lawson correspondence between discrete CMC surfaces in R^3 and discrete minimal surfaces in S^3. This is a correspondence between two discrete isothermic surfaces. We show that this correspondence is an isometry in the following sense: it preserves the metric coefficients int…

Extends rigidity and existence results for discrete conformal structures on surfaces with boundary.

The paper introduces a new discretization of Gaussian curvature on surfaces.

Paper proves discrete uniformizations converge to continuous for surfaces of genus ≥1.

New Bäcklund transformations for discrete pseudospherical surfaces of revolution are found.

The paper proves a theorem for discretizing Gaussian curvature on surfaces.

The paper studies singularities in discrete indefinite affine minimal surfaces.