Related papers: Genus three embedded doubly periodic minimal surfa…
We construct three kinds of complete embedded minimal surfaces in $\Bbb H^2\times \Bbb R$. The first is a simply connected, singly periodic, infinite total curvature surface. The second is an annular finite total curvature surface. These…
We construct 1-parameter families of non-periodic embedded minimal surfaces of infinite genus in $T \times \mathbb{R}$, where $T$ denotes a flat 2-tori. Each of our families converges to a foliation of $T \times \mathbb{R}$ by $T$. These…
We prove: a properly embedded, genus-one minimal surface that is asymptotic to a helicoid and that contains two straight lines must intersect that helicoid precisely in those two lines. In particular, the two lines divide the surface into…
We solve a certain case of the minimal genus problem for embedded surfaces in elliptic 4-manifolds. The proofs involve a restricted transitivity property of the action of the orientation preserving diffeomorphism group on the second…
Entangled embedded periodic nets and crystal frameworks are defined, along with their dimension type, homogeneity type, adjacency depth and periodic isotopy type. We obtain periodic isotopy classifications for various families of embedded…
Minimal surfaces of general type in Euclidean 4-space are characterized with the conditions that the ellipse of curvature at any point is centered at this point and has two different principal axes. Any minimal surface of general type…
This paper develops new tools for understanding surfaces with more than one end (and usually, of infinite topology) which properly minimally embed into Euclidean three-space. On such a surface, the set of ends forms a compact Hausdorff…
We prove the existence of complete minimal surfaces of genus g>1 which minimize the total curvature for their genus. Our method is first to identify this (Weierstrass high dimensional period) problem with the problem of finding a particular…
We provide an existence proof for two 1-parameter families of embedded triply periodic minimal surfaces of genus three, namely the tG family with tetragonal symmetry that contains the gyroid, and the rGL family with rhombohedral symmetry…
In the present paper which a sequel to dg-ga/9511005 and dg-ga//9610013 a global Weierstrass representation of an arbitrary closed oriented surface of genus $\geq 1$ in the the three-space is constructed. The Weierstrass spectrum of a torus…
A triangulated piecewise-linear minimal surface in Euclidean 3-space defined using a variational characterization is critical for area amongst all continuous piecewise-linear variations with compact support that preserve the simplicial…
We construct and study a family of double-periodic almost entire solutions of the maximal surface equation. The solutions are parameterized by a submanifold of $3\times 3$-matrices (the so-called generating matrices). We show that the…
We present the novel method for generation of periodic surfaces based on the simple Landau-Ginzburg model of microemulsion. We test the method on four minimal surfaces (P,D,G, and I-WP), find two new surfaces of cubic symmetry, show how to…
We describe the family of minimal graphs on strips with boundary values $\pm\infty$ disposed alternately on edges of length one, and whose conjugate graphs are contained in horizontal slabs of width one in $\mathbb{R}^3$. We can obtain as…
We give a positive answer to M. Traizet's open question about the existence of complete embedded minimal surfaces with Scherk-ends without planar geodesics. In the singly periodic case, these examples get close to an extension of Traizet's…
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…
The isometry group of the classical Lawson embedded minimal surface $\xi_{2,1}\subset \mathbb{S}^3$ of genus 2 is isomorphic to the product $S_3\times D_4$ of the permutation group of three elements and the dihedral group of order 8…
In this paper, we show that a complete embedded minimal surface in $\Real^3$ with finite topology and one end is conformal to a once-punctured compact Riemann surface. Moreover, using the conformality and embeddedness, we examine the…
We show that the number of genus $g$ embedded minimal surfaces in $\mathbb{S}^3$ tends to infinity as $g\rightarrow\infty$. The surfaces we construct resemble doublings of the Clifford torus with curvature blowing up along torus knots as…
We discuss in detail Alan Schoen's I-WP surface, an embedded triply periodic minimal surface of genus 4 with cubical symmetries. We exhibit various geometric realizations of this surface with the same conformal structure and use them to…