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We define and discuss the notion of pseudospherical surfaces in asymptotic coordinates on time scales. Two special cases, namely dicrete pseudospherical surfaces and smooth pseudosperical surfaces are consistent with this description. In…
This survey article is about discrete constant mean curvature surfaces defined by an approach related to integrable systems techniques. We introduce the notion of discrete constant mean curvature surfaces by first introducing properties of…
We make a systematic study of the focal surface of a congruence of lines in the projective space. Using differential techniques together with techniques from intersection theory, we reobtain in particular all the invariants of the focal…
The maximum principle forms an important qualitative property of second order elliptic equations, therefore its discrete analogues, the so-called discrete maximum principles (DMPs) have drawn much attention. In this paper DMPs are…
We introduce a novel type of approximation spaces for functions with values in a nonlinear manifold. The discrete functions are constructed by piecewise polynomial interpolation in a Euclidean embedding space, and then projecting pointwise…
In this paper we prove that a complete, embedded minimal surface $M$ in $\mathbb{R}^3$ with finite topology and compact boundary (possibly empty) is conformally a compact Riemann surface $\overline{M}$ with boundary punctured in a finite…
We introduce a class of maps from an affine flat into a Riemannian manifold that solve an elliptic system defined by the natural second order elliptic operator of the affine structure and the nonlinear Riemann geometry of the target. These…
Finite metric spaces arise in many different contexts. Enormous bodies of data, scientific, commercial and others can often be viewed as large metric spaces. It turns out that the metric of graphs reveals a lot of interesting information.…
We describe and study geometric properties of discrete circular and spherical means of directional derivatives of functions, as well as discrete approximations of higher order differential operators. For an arbitrary dimension we present a…
We study compact polyhedral surfaces as Riemann surfaces and their discrete counterparts obtained through quadrilateral cellular decompositions and a linear discretization of the Cauchy-Riemann equation. By ensuring uniformly bounded…
We study invariant sets and measures generated by iterated function systems defined on countable discrete spaces that are uniform grids of a finite dimension. The discrete spaces of this type can be considered as models of spaces in which…
In this article we address a number of features of the moduli space of spherical metrics on connected, compact, orientable surfaces with conical singularities of assigned angles, such as its non-emptiness and connectedness. We also consider…
In this paper, we introduce a definition of discrete conformality for triangulated surfaces with flat cone metrics and describe an algorithm for solving the problem of prescribing curvature, that is to deform the metric discrete conformally…
We prove a sharp Schwarz type inequality for the Weierstrass- Enneper representation of the minimal surfaces. It states the following. If $F:\mathbf{D}\to \Sigma$ is a conformal harmonic parameterization of a minimal disk $\Sigma$, where…
A branched affine structure on a compact topological surface with marked points is a complex affine structure outside the marked points. We give a proof of an unpublished foundational theorem of Veech, stating that any branched affine…
We define an abstract framework called {\it discrete finite differences embedding} which can be used to obtain discrete analogue of formal functional relations in the spirit of category theory. For ordinary differential equations we exhibit…
We introduce a characterization for affine equivalence of two surfaces of translation defined by either rational or meromorphic generators. In turn, this induces a similar characterization for minimal surfaces. In the rational case, our…
In this paper, we introduce restricted products for families of locally convex spaces and formulate criteria ensuring that mappings into such products are continuous or smooth. As a special case, can define restricted products of weighted…
Demoulin surfaces in real projective $3$-space are investigated. Our result enable us to establish a generalized Weierstrass type representation for definite Demoulin surfaces by virtue of primitive maps into a certain semi-Riemannian…
We construct new examples of immersed minimal surfaces with catenoid ends and finite total curvature, of both genus zero and higher genus. In the genus zero case, we classify all such surfaces with at most $2n+1$ ends, and with symmetry…