Existence of algebraic minimal surfaces for an arbitrary puncture set

We will show that any punctured Riemann surface can be conformally immersed into a Euclidean -space as a branched complete minimal surface of finite total curvature called an algebraic minimal surface.

     

Maximum principle at infinity for complete minimal surfaces in flat 3-manifolds

The main result of this paper is the following maximum principle at infinity:Theorem.Let M ₁ and M ₂ be two disjoint properly embedded complete minimal surfaces with nonempty boundaries, that are stable in a complete flat 3-manifold. Then dist(M ₁,M ₂)=min(dist(M ₁,M ₂), dist(M ₂,M ₁)).In case one boundary is empty, e.g. M ₁,then dist(M ₁,M ₂)=dist(M ₂,M ₁).If both boundaries are empty, then M ₁ and M ₂ are flat.     

On singly-periodic minimal surfaces with planar ends

The spaces of nondegenerate properly embedded minimal surfaces in quotients of by nontrivial translations or by screw motions with nontrivial rotational part, fixed finite topology and planar type ends, are endowed with natural structures of finite dimensional real analytic manifolds. This nondegeneracy is defined in terms of Jacobi functions. Riemann's minimal examples are characterized as the only nondegenerate surfaces with genus one in their corresponding spaces. We also give natural immersions of these spaces into certain complex Euclidean spaces which turn out to be Lagrangian immersions with respect to the standard symplectic structures.

     

Nonexistence of certain complete minimal surfaces with planar ends

In this paper we prove that two lines bounding an immersed minimal surface in a slab in R ³ homeomorphic to a compact Riemann surface minus two disks and a finite number of points must be parallel. This theorem is extended to a higher dimensional minimal hypersurface. Also it is proved that if the Gauss map of a complete embedded minimal surface of finite total curvature at a planar end has order two, then the intersection of the surface with the plane asymptotic to the planar end cannot admit a one-to-one orthogonal projection onto any line in the plane. Received: November 26, 1998     

Obstacle-type problems for minimal surfaces

We study certain obstacle-type problems involving standard and nonlocal minimal surfaces. We obtain optimal regularity of the solution and a characterization of the free boundary.     

Stability of triply periodic minimal surfaces

In 1992, Ross proved that some classical triply periodic minimal surfaces in three-dimensional Euclidean space (Schwarz P surface, D surface, and Schoen's gyroid) are stable for volume-preserving variations. This paper extends the result to four one-parameter families of triply periodic minimal surfaces, namely, tP family, tD family, rPD family, and H family. We obtain sufficient conditions for volume-preserving stability, and as their numerical applications, we prove that, for each family, every triply periodic minimal surface with Morse index one is volume-preserving stable.     

Stable complete minimal surfaces in hyperkähler manifolds

In this paper we prove that an isometric stable minimal immersion of a complete oriented surface into a hyperkähler 4-manifold is holomorphic with respect to an orthogonal complex structure, if it satisfies a Bernstein-type assumption on the Gauss-lift. This result generalizes a theorem of Micallef for minimal surfaces in the euclidean 4-space. An example found by Atiyah and Hitchin shows that the assumption on the Gauss-lift is necessary.     

The Cauchy problem of Lorentzian minimal surfaces in globally hyperbolic manifolds

In this note a proof is given for global existence and uniqueness of minimal Lorentzian surface maps from a cylinder into a large class of globally hyperbolic Lorentzian manifolds for given initial values up to the first derivatives. The results of this article are part of my PhD thesis written at the Max-Planck institute for Mathematics in the Sciences in Leipzig under the supervision of Prof. Jürgen Jost to whom I want to express my gratitude.     

The spaces of index one minimal surfaces and stable constant mean curvature surfaces embedded in flat three manifolds

It is proved that the spaces of index one minimal surfaces and stable constant mean curvature surfaces with genus greater than one in (non fixed) flat three manifolds are compact in a strong sense: given a sequence of any of the above surfaces we can extract a convergent subsequence of both the surfaces and the ambient manifolds in the topology. These limits preserve the topological type of the surfaces and the affine diffeomorphism class of the ambient manifolds. Some applications to the isoperimetric problem are given.

     

The concept of hyperellipticity on surfaces with nodes

A surface with nodes X is hyperelliptic if there exists an involution such that the genus of X/〈h〉 is 0. We prove that this definition is equivalent, as in the category of surfaces without nodes, to the existence of a degree 2 morphism satisfying an additional condition where the genus of Y is 0. Other question is if the hyperelliptic involution is unique or not. We shall prove that the hyperelliptic involution is unique in the case of stable Riemann surfaces but is not unique in the case of Klein surfaces with nodes. Finally, we shall prove that a complex double of a hyperelliptic Klein surface with nodes could not be hyperelliptic.     

Sur les metriques admettant les plans comme surfaces minimales

We establish the system of partial differential equations satisfied by the riemannian metrics on open subsets of which admit planes as minimal surfaces. This is a nonlinear system of 10 partial differential equations, with the euclidian metric as a particular solution. In a previous work, we solved this system for axially symmetrical metrics. In this paper we linearize the system at the euclidian metric and solve the linear system. We obtain a 20-dimensional space of solutions.

     

On a free boundary problem and minimal surfaces

From minimal surfaces such as Simons' cone and catenoids, using refined Lyapunov–Schmidt reduction method, we construct new solutions for a free boundary problem whose free boundary has two components. In dimension 8, using variational arguments, we also obtain solutions which are global minimizers of the corresponding energy functional. This shows that the theorem of Valdinoci et al. [41], [42] is optimal.     

Topologically minimal surfaces versus self-amalgamated Heegaard surfaces

Let V ∪SW be a Heegaard splitting of M,such that αM = α-W = F1 ∪ F2 and g(S) = 2g(F1)= 2g(F2). Let V * ∪S*W * be the self-amalgamation of V ∪SW. We show if d(S) 3 then S* is not a topologically minimal surface.     

Designing approximation minimal parametric surfaces with geodesics

Geodesic is an important curve in practical application, especially in shoe design and garment design. In practical applications, we not only hope the shoe and garment surfaces possess characteristic curves, but also we hope minimal cost of material to build surfaces. In this paper, we combine the geodesic and minimal surface. We study the approximation minimal surface with geodesics by using Dirichlet function. The extremal of such a function can be easily computed as the solutions of linear systems, which avoid the high nonlinearity of the area function. They are not extremal of the area function but they are a fine approximation in some cases.     

Meshing skin surfaces with certified topology

Skin surfaces are used for the visualization of molecules. They form a class of tangent continuous surfaces defined in terms of a set of balls (the atoms of the molecule) and a shrink factor. More recently, skin surfaces have been used for approximation purposes.

We present an algorithm that approximates a skin surface with a topologically correct mesh. The complexity of the mesh is linear in the size of the Delaunay triangulation of the balls, which is worst case optimal.

We also adapt two existing refinement algorithms to improve the quality of the mesh and show that the same algorithm can be used for meshing a union of balls.     

The singly periodic genus-one helicoid

We prove the existence of a complete, embedded, singly periodic minimal surface, whose quotient by vertical translations has genus one and two ends. The existence of this surface was announced in our paper in Bulletin of the AMS, 29(1):77-84, 1993. Its ends in the quotient are asymptotic to one full turn of the helicoid, and, like the helicoid, it contains a vertical line. Modulo vertical translations, it has two parallel horizontal lines crossing the vertical axis. The nontrivial symmetries of the surface, modulo vertical translations, consist of: -rotation about the vertical line; rotation about the horizontal lines (the same symmetry); and their composition. Received: May 1996; revised October 1996.     

Characterization of order 3 algebraic immersed minimal surfaces of <Emphasis Type="Italic">S</Emphasis><Superscript>3</Superscript>

In this paper we prove that if is a minimal immersion of a compact surface and , for some homogeneous polynomial f of degree 3 on R ⁴, then, M is a torus and is one of the examples given by Lawson (1970, Complete minimal surfaces in S ³. Ann. Math. 92(2), 335–374).