Embedded minimal surfaces: Forces,topology and symmetries

We prove topological uniqueness theorems for embedded minimal surfaces in ³ under the assumption that certain forces associated to these surfaces are vertical. We give applications to minimal surfaces with symmetries and with free boundary.Partially supported by DGICYT grant PB94-0796     

The bridge principle for stable minimal surfaces

The author was partially funded by NSF grants DMS85-53231(PYI), DMS-92-07704, and by the IHES.     

Non-existence of regular algebraic minimal surfaces of spheres of degree 3

In this paper we prove that if is a closed minimal surface, then, , for any homogeneous polynomial f of degree 3 with 0 a regular value of the function .     

On the thread problem for minimal surfaces

For a given one-dimensional fixed boundary $\Gamma$ in and a given constant we consider any one-dimensional free boundary $F$ in subject to the conditions that the length of is equal to , that and form a closed boundary, and that the minimal surface of dimension two being bounded by and minimizes the area among all comparison surfaces being bounded by and some with length equal to . This variational problem is known as the thread problem for minimal surfaces and stems from soap film experiments, in which the fixed boundary parts are pieces of wires and the free boundary parts are threads. The new result of this article will be that has no singular points in , provided the admissible surfaces and boundary parts are supposed to be rectifiable flat chains modulo two. Received February 16, 1995 / Accepted October 20, 1995     

Incompressibility and least-area surfaces

We show that if F is a smooth, closed, orientable surface embedded in a closed, orientable 3-manifold M such that for each Riemannian metric g on M, F   is isotopic to a least-area surface F(g)$F(g)$, then F is incompressible.     

Eigenvalue estimates for minimal hypersurfaces in hyperbolic space

Recently Candel [A. Candel, Eigenvalue estimates for minimal surfaces in hyperbolic space, Trans. Amer. Math. Soc. 359 (2007) 3567-3575] proved that if M is a simply-connected stable minimal surface isometrically immersed in H³, then the first eigenvalue of M satisfies 1/4?λ(M)?4/3 and he asked whether the bound is sharp and gave an example such that the lower bound is attained. In this note, we prove that the upper bound can never be attained. Also we extend the result by proving that if M is compact stable minimal hypersurface isometrically immersed in Hn+1 where n?3 such that its smooth Yamabe invariant is negative, then (n−1)/4?λ(M)?n²(n−2)/(7n−6).     

Algebraic curves and the Gauss map of algebraic minimal surfaces

In this paper, we first establish a second main theorem for algebraic curves into the n-dimensional projective space. We then use it to study the ramified values for the Gauss map of the complete (regular) minimal surfaces in Rm with finite total curvature, as well as the uniqueness problem.     

A method to exclude branch points of minimal surfaces

The central problem of this paper is to exclude boundary branch points of minimal surfaces. The method consists in showing that the third derivative of the Dirichlet energy is negative along well-chosen paths in admissible Jacobi field directions, if a “Schüffler condition” is satisfied. Received July 21, 1997 / Accepted October 3, 1997     

The spiral minimal surfaces and their Legendre and Weierstrass representations

A class of spiral minimal surfaces in E³ is constructed using a symmetry reduction. The reduction leads to a cubic-nonlinear ODE whose phase portrait is described using an auxiliary Riccati's equation and the Warzewski topological principle for its solutions. The new surfaces are invariant with respect to the composition of rotation and dilation. The solutions are obtained in parametric form through the Legendre and the Weierstrass representations, and also their asymptotic behaviour is described.     

Minimal surfaces,Hopf differentials and the Ricci condition

We deal with minimal surfaces in a sphere and investigate certain invariants of geometric significance, the Hopf differentials, which are defined in terms of the complex structure and the higher fundamental forms. We discuss the holomorphicity of Hopf differentials and provide a geometric interpretation for it in terms of the higher curvature ellipses. This motivates the study of a class of minimal surfaces, which we call exceptional. We show that exceptional minimal surfaces are related to Lawson’s conjecture regarding the Ricci condition. Indeed, we prove that, under certain conditions, compact minimal surfaces in spheres which satisfy the Ricci condition are exceptional. Thus, under these conditions, the proof of Lawson’s conjecture is reduced to its confirmation for exceptional minimal surfaces. In fact, we provide an affirmative answer to Lawson’s conjecture for exceptional minimal surfaces in odd dimensional spheres or in S ^4m .     

Triply periodic minimal surfaces bounded by vertical symmetry planes

We give a uniform and elementary treatment of many classical and new triply periodic minimal surfaces in Euclidean space, based on a Schwarz–Christoffel formula for periodic polygons in the plane. Our surfaces share the property that vertical symmetry planes cut them into simply connected pieces. S. Fujimori was partially supported by JSPS Grant-in-Aid for Young Scientists (Start-up) 19840035. M. Weber’s material is based upon work for the NSF under Award No. DMS-0139476.     

A characterization of constant mean curvature surfaces in homogeneous 3-manifolds

It has been recently shown by Abresch and Rosenberg that a certain Hopf differential is holomorphic on every constant mean curvature surface in a Riemannian homogeneous 3-manifold with isometry group of dimension 4. In this paper we describe all the surfaces with holomorphic Hopf differential in the homogeneous 3-manifolds isometric to H²×R or having isometry group isomorphic either to the one of the universal cover of PSL(2,R), or to the one of a certain class of Berger spheres. It turns out that, except for the case of these Berger spheres, there exist some exceptional surfaces with holomorphic Hopf differential and non-constant mean curvature.     

A notion of branching rank for semilattices with descending chain condition

Given a maximal subchainC of a semilatticeS, there are some natural leaves ofS attached to it. These are subsemilattices ofS which may have a simpler structure thanS itself. We look atS as build up fromC together with its leaves. Starting with one-point subsemilattices, the (branching) rank ofS is defined to be the least number of steps needed to recoverS. For technical reasons, only semilattices with no infinite descending chains are considered. The main result states that ifR is a subsemilattice ofS and rankS is defined, then rankRrankS. On the other hand, rank does not behave well with respect to epimorphisms. Several examples are presented as well as various results concerning finite semilattices and trees.This work was supported, in part, by NSERC Grant A4044.     

A characterization of the Clifford torus

We provide a characterization of the Clifford torus via a Ricci type condition among minimal surfaces in S⁴. More precisely, we prove that a compact minimal surface in S⁴, with induced metric ds² and Gaussian curvature K, for which the metric is flat away from points where K = 1, is the Clifford torus, provided that m is an integer with m > 2.Received: 8 September 2004