Schwarz H surfaces can be continuously deformed into Meeks surfaces.
problem Classifying and understanding the relationship between Schwarz H surfaces and Meeks surfaces.
method Constructing a 2-parameter family of TPMS of genus three that contains both H and Meeks surfaces.
result 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.
problem Proving the existence of a new family of triply periodic minimal surfaces.
method Proved existence through mathematical proof and analysis of properties.
result New family oΔ does not belong to Meeks' five-dimensional family but shares properties with Schwarz' D. The paper introduces a θ-family of spacelike minimal surfaces in R^4_1.
problem Tackles the construction and properties of spacelike minimal surfaces in R^4_1.
method Introduces a θ-family of surfaces based on holomorphic functions and studies their curvature.
result Proves that the existence of planar points corresponds to solutions of |a_w(w)|^2 =0 and that surfaces cannot be complete.
New theorem connects minimal and maximal surfaces, affecting graphness.
problem Understanding graphness of minimal surfaces in different spaces.
method Introducing a new deformation family and proving Krust-type theorems.
result Graphness of minimal surfaces in isotropic 3-space affects deformed surfaces.
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.
problem Creating embedded doubly periodic minimal surfaces with specific topological properties.
method Constructs a one-parameter family of surfaces with given Weierstrass data and solves the period problem.
result Solves the two dimensional period problem for the constructed surfaces.
The paper calculates Morse indices and nullities for triply periodic minimal surfaces.
problem Computing Morse indices and nullities for triply periodic minimal surfaces.
method Developed an algorithm to compute Morse index and nullity using matrix properties of abelian differentials.
result Explicitly determined key matrices for five families of triply periodic minimal surfaces.
Unified view of integrable systems linking CMC, isothermic, and Willmore surfaces.
problem Understanding the relationships between different types of surfaces and their integrable systems.
method Unified view through families of flat connections and parallel sections.
result Complete description of links between different surface types and their dressing transformations.
The paper constructs isothermic surfaces using Ribaucour transformations.
problem Creating isothermic surfaces with specific properties.
method Applying Ribaucour transformations to the cylinder and obtaining families of complete isothermic surfaces.
result Obtained families of isothermic surfaces with unique properties (e.g., n-bubble surfaces, planar ends, no constant mean curvature).
New families of translation surfaces with multiple oblivious points discovered.
problem Identifying points on translation surfaces without nearby closed geodesics.
method Constructing new families of translation surfaces and proving existence in higher genera.
result Translation surfaces in every genus ≥3 have at least one oblivious point.
Constructs a family to handle unstable fibers on complex surfaces.
problem Handling unstable fibers on complex surfaces.
method Uses Teichmüller theory to construct a degenerating family over the moduli space.
result Any fibered complex surface with unstable fibers can be pulled back from the constructed family.
We use bifurcation theory to determine the existence of infinitely many new examples of triply periodic minimal surfaces in R3. 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.
problem Finding the minimal size of families of curves on surfaces that realize all types of pants decompositions.
method Investigates exponential and superlinear bounds for surfaces without punctures, and provides bounds for surfaces with punctures.
result Provides bounds for the minimal size of families of curves on surfaces with and without punctures.
New insights into minimal surfaces in H^2 x R foliated by arcs and their Jacobi fields.
problem Minimal surfaces in H^2 x R foliated by arcs and their properties.
method Analysis of catenoids, parabolic catenoids, and tall rectangles; study of Jacobi operator and fields.
result Computations of Jacobi fields for deformations to other surfaces in the family.
New minimal surfaces found in ball with boundary constraints.
problem Finding minimal surfaces with boundary conditions.
method Equivariant differential geometry approach.
result A family of free boundary minimal surfaces in the unit ball.
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.
problem Proving the existence of two families of minimal surfaces.
method Extending a technique to non-rectangular branched tori.
result Existence of the tG and rGL 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.
problem Creating minimal surfaces of high genus with specific properties.
method One-parameter family of minimal surfaces constructed in Euclidean 3-space.
result The family includes the Costa-Hoffman-Meeks surfaces with two catenoidal ends and a flat middle end.
Classifies homogeneous affine surfaces and classifies gradient Ricci solitons.
problem Classifying homogeneous affine surfaces and their properties.
method Examined Lie algebra of affine Killing vector fields and classified gradient Ricci solitons.
result Complete classification of homogeneous affine gradient Ricci solitons.
Constructs continuous families of minimal surfaces and holomorphic immersions.
problem Creating continuous families of minimal surfaces and holomorphic immersions.
method Continuous family of complex structures and conformal minimal immersions.
result Continuous families of proper Jb-conformal minimal immersions and holomorphic null immersions. The paper studies vector bundles over surfaces, focusing on singularity formation.
problem Understanding singularity formation in rank two holomorphic vector bundles over surfaces.
method Defining fertile families bearing bubbles and using elementary modifications to prove their existence.
result Existence of fertile families bearing bubbles for certain types of vector bundles.
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.
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…
Rational configurations in K3 surfaces and simply-connected pg=1 surfaces for K2=1,2,3,4,5,6,7,8,9math.AG The study finds families of surfaces with specific properties and connects them to rational curve configurations in K3 surfaces.
problem Existence and properties of surfaces with specific canonical and geometric genus conditions.
method Study of rational curve configurations and use of Q-Gorenstein smoothings. result Existence of (20−2K2)-dimensional families of simply-connected surfaces with pg=1 and K2=1,2,3,4,5,6,7,8,9. 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.
problem Determining which state surfaces are fibered.
method Algebraic characterization of fibers from state graphs, decomposing graphs into planar components.
result Characterizes fibering for many families of state surfaces.
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.
problem Minimizing crossings on congruence surfaces.
method Examining systoles and their intersections.
result Modular systoles have minimal crossing numbers.
The paper studies surfaces with spherical curvature lines and their generation by constrained elastic curves.
problem Understanding surfaces with spherical curvature lines and their generation mechanisms.
method The approach involves Lie sphere transformations, Legendre curves, and polynomial conserved quantities of connections.
result Lie applicable surfaces with exactly one family of spherical curvature lines are generated by the lift of constrained elastic curves.
Study local topological types of binary differential equations for surface families.
problem Classify bifurcations of asymptotic curves at parabolic and umbilical points.
method Compare projective classification of Monge forms with general BDE classification.
result Classify generic bifurcations of parabolic curves and new flecnodal curves.
Constructs solutions for gravitational instantons from minimal surfaces.
problem Finding solutions for gravitational instantons.
method Using correspondence between minimal surfaces and gravitational instantons, derived explicit maximal surface solutions.
result Explicit solutions for maximal surface with zero mean curvature.
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.
problem Computing monodromy groups of surface families over smooth quartic curves.
method Analyzing cyclic branched covers of P2 over smooth quartic curves, computing monodromy groups for del Pezzo and K3 surfaces. result Obtained monodromy groups for del Pezzo and K3 surfaces, including Weyl group $W\left(E_{7}
ight)$ and arithmetic lattice $U\left(h_{L_{-}}
ight)$.
The study finds infinite non-embedded surfaces with constant length second fundamental forms.
problem Characterizing surfaces with constant length second fundamental forms.
method Analyzing complete rotational surfaces in R3. result Only round spheres, circular cylinders, and a specific family of surfaces have constant length second fundamental forms.
Infinite-genus surfaces have many isospectral hyperbolic structures.
problem Finding many isospectral hyperbolic structures on infinite-genus surfaces.
method Constructing families of isospectral hyperbolic structures on infinite-type surfaces without planar ends.
result Uncountable families of isospectral and quasiconformally distinct hyperbolic structures on infinite-genus surfaces with self-similar end spaces.
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.
problem Understanding the ambiguity in discrete R-congruences and their envelopes.
method Analyzes discrete R-congruences that are enveloped by specific types of surfaces and maps.
result Discovers a 2-parameter family of discrete enveloping surfaces for discrete R-congruences.
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 R3. 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.
problem Existence of convex translating surfaces under flow by α-th power of Gauss curvature. method Constructing a family of translating surfaces using Jacobi fields and analyzing their effective growth rates.
result The family of translating surfaces is a topological manifold with quantitative convergence rates.
Curved flats linked to pairs of Lie applicable surfaces.
problem Understanding curved flats in Lie sphere geometry.
method One-to-one correspondence with pairs of Demoulin families of Lie applicable surfaces via Darboux transformation.
result Curved flats correspond to specific Lie applicable surface pairs.
In this work, we consider spacelike surfaces in Minkowski space E that satisfy a linear Weingarten condition of type κ1=mκ2+n, where m and n are constant and κ1 and κ2 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 τr,m minimally immersed in spheres to a three-parametric family Ta,b,c of tori and Klein bottles minimally immersed in spheres. It was remarked that this family inclu…