arXiv research

Schwarz H surfaces can be continuously deformed into Meeks surfaces.

New 2-parameter family of triply periodic minimal surfaces, sharing properties with Schwarz' D but distinct.

The paper introduces a θ-family of spacelike minimal surfaces in R^4_1.

New theorem connects minimal and maximal surfaces, affecting graphness.

We classify Riemannian surfaces admitting associated families in three dimensional homogeneous spaces with four-dimensional isometry groups and in a wide family of (semi-Riemannian) warped products, with an extra natural condition (namely, rotating structure vector field). We prove that, provided the surface is not tot…

It is well-known that in any codimension a simply connected Euclidean minimal surface has an associated one-parameter family of minimal isometric deformations. In this paper, we show that this is just a special case of the associated family to any simply connected elliptic surface for which all curvature ellipses of a …

We study the one parameter family of genus 2 Riemann surfaces defined by the orbit of the L-shaped translation surface tiled by three squares under the Teichmüller geodesic flow. These surfaces are real algebraic curves with three real components. We are interested in describing these surfaces by their period matrices.…

Constructs a family of genus three minimal surfaces with parallel ends.

The paper calculates Morse indices and nullities for triply periodic minimal surfaces.

Unified view of integrable systems linking CMC, isothermic, and Willmore surfaces.

The paper constructs isothermic surfaces using Ribaucour transformations.

New families of translation surfaces with multiple oblivious points discovered.

Constructs a family to handle unstable fibers on complex surfaces.

We use bifurcation theory to determine the existence of infinitely many new examples of triply periodic minimal surfaces in $\mathbb R^3$. These new examples form branches issuing from the H-family, the rPD-family, the tP-family, and the tD-family, that converge to some degenerate embedding of the families. As to nonde…

The paper studies families of curves on surfaces that realize all types of pants decompositions.

New insights into minimal surfaces in H^2 x R foliated by arcs and their Jacobi fields.

New minimal surfaces found in ball with boundary constraints.

We introduce a new technique to solve period problems on minimal surfaces called limit-method. If a family of surfaces has Weierstrass-data converging to the data of a known example, and this presents a transversal solution of periods, then the original family contains a sub-family with closed periods.

Existence proof for two families of triply periodic minimal surfaces.

In the paper we formulate and derive the family blowup formula of family Seiberg-Witten invariants. The formula has been used in the enumerative application of counting singular curves on algebraic surfaces. We first give a topological derivation of the formula by using family index theorem. Then we define the algebrai…

The paper constructs surfaces of high genus with three ends.

Constructs continuous families of minimal surfaces and holomorphic immersions.

The paper studies vector bundles over surfaces, focusing on singularity formation.

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.

The homogeneous affine surfaces have been classified by Opozda. They may be grouped into 3 families, which are not disjoint. The connections which arise as the Levi-Civita connection of a surface with a metric of constant Gauss curvature form one family; there are, however, two other families. For a surface in one of t…

In this study, we define a family of ruled surfaces in the Euclidean 3-space E^3 and called similar ruled surfaces. We obtain some properties of these special surfaces and we show that developable ruled surfaces form a family of similar ruled surfaces if and only if the striction curves of the surfaces are similar curv…

A Laguerre minimal surface is an immersed surface in the Euclidean space being an extremal of the functional \int (H^2/K - 1) dA. In the present paper, we prove that the only ruled Laguerre minimal surfaces are up to isometry the surfaces R(u,v) = (Au, Bu, Cu + D cos 2u) + v (sin u, cos u, 0), where A, B, C, D are fixe…

The study finds families of surfaces with specific properties and connects them to rational curve configurations in K3 surfaces.

All complete, axially symmetric surfaces of constant mean curvature in R^3 lie in the one-parameter family D_tau of Delaunay surfaces. The elements of this family which are embedded are called unduloids; all other elements, which correspond to parameter value tau element in R^-, are immersed and are called nodoids. The…

Classifies fibering of state surfaces for various knot families.

We analyzed the problem of finding a surfaces family through an asymptotic curve with Cartan frame. We obtain the parametric representation for surfaces family whose members have the same as an asymptotic curve. By using the Cartan frame of the given null curve, we present the surface as a linear combination of this fr…

Optimal curves minimize crossings on surfaces.

The paper studies surfaces with spherical curvature lines and their generation by constrained elastic curves.

Study local topological types of binary differential equations for surface families.

Constructs solutions for gravitational instantons from minimal surfaces.

We exhibit several transformations of surfaces in R^4. First, one that takes a flat surface and gets a surface with flat normal bundle; then, one that takes a surface with flat normal bundle and gets a flat surface; finally, a one-parameter family of transformations on a flat surface with flat normal bundle and gives a…

Researchers compute monodromy groups of surface families over quartic curves.

The study finds infinite non-embedded surfaces with constant length second fundamental forms.

Infinite-genus surfaces have many isospectral hyperbolic structures.

We prove the existence of a family of embedded doubly periodic minimal surfaces of (quotient) genus $g$ with orthogonal ends that generalizes the classical doubly periodic surface of Scherk and the genus-one Scherk surface of Karcher. The proof of the family of immersed surfaces is by induction on genus, while the proo…

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

In this work we give a method for constructing a one-parameter family of complete CMC-1 (i.e. constant mean curvature 1) surfaces in hyperbolic 3-space that correspond to a given complete minimal surface with finite total curvature in Euclidean 3-space. We show that this one-parameter family of surfaces with the same s…

We study the Weil-Petersson geometry for holomorphic families of Riemann Surfaces equipped with the unique conical metric of constant curvature -1.

We investigate minimal surfaces passing a given curve in $R^{3}$. Using the Frenet frame of a given curve and isothermal parameter, we derive the necessary and sufficient condition for minimal surface. Also we derive the parametric representation of two minimal surface families passing a circle and a helix as examples.

The paper proves the existence of a continuous family of translating surfaces under a specific curvature flow.

Curved flats linked to pairs of Lie applicable surfaces.

In this work, we consider spacelike surfaces in Minkowski space $\hbox{\bf E}$ that satisfy a linear Weingarten condition of type $κ_{1}=mκ_{2}+n$, where $m$ and $n$ are constant and $κ_{1}$ and $κ_{2}$ denote the principal curvatures at each point of the surface. We study the family of surfaces foliated by a …

Recently Penskoi [J. Geom. Anal. 25 (2015), 2645-2666, arXiv:1308.1628] generalized the well known two-parametric family of Lawson tau-surfaces $τ_{r,m}$ minimally immersed in spheres to a three-parametric family $T_{a,b,c}$ of tori and Klein bottles minimally immersed in spheres. It was remarked that this family inclu…