Weierstrass representation of Lagrangian surfaces in four-dimensional space using spinors and quaternions

We derive a Weierstrass-type formula for conformal Lagrangian immersions in Euclidean 4-space, and show that the data satisfies an equation similar to Dirac equation with complex potential. Alternatively this representation has a simple formulation using quaternions. We apply it to the Hamiltonian stationary case and construct all possible tori, thus obtaining a first approach to a moduli space in terms of a simple algebraic-geometric problem on the plane. We also classify Hamiltonian stationary Klein bottles and show they self-intersect. Received: January 25, 2000.     

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.     

MINIMAL SURFACES IN 3—DIMENSIONAL SOLVABLE LIE GROUPS

The author studies minimal surfaces in 3-dimensional solvable Lie groups with left invariant Riemannian metrics. A Weierstraβ type integral representation formula for minimal surfaces is obtained.     

Some Picard theorems for minimal surfaces

This paper deals with the study of those closed subsets for which the following statement holds:

If is a properly immersed minimal surface in of finite topology that is eventually disjoint from then has finite total curvature.

The same question is also considered when the conclusion is finite type or parabolicity.

     

Dense solutions to the Cauchy problem for minimal surfaces

We show a general way to produce in explicit coordinates complete minimal surfaces in ³ that lie densely in the whole space. This construction relies on solving the Björling problem for adequate initial data.1 Partially supported by MCYT-FEDER, Grant no. BFM2001-3318.2 Partially supported byMCYT-FEDER, Grant no. BFM2001-2871 and CARM Programa Séneca, Grant no PI-3/00854/FS/01.     

Ruled minimal surfaces in the Berger sphere

We show that any ruled minimal surface in the Berger sphere is a helicoid whose axis is a Hopf fiber by solving the ruled minimal surface equation in the parametric form.     

Laguerre minimal surfaces in ℝ3

Laguerre geometry of surfaces in R^3 is given in the book of Blaschke, and has been studied by Musso and Nicolodi, Palmer, Li and Wang and other authors. In this paper we study Laguerre minimal surface in 3-dimensional Euclidean space R^3. We show that any Laguerre minimal surface in R^3 can be constructed by using at most two holomorphic functions. We show also that any Laguerre minimal surface in R^3 with constant Laguerre curvature is Laguerre equivalent to a surface with vanishing mean curvature in the 3-dimensional degenerate space R0^3.     

Computer-oriented, n-body modelling of minimal surfaces

In this paper a new approach to the modelling of minimal surfaces is described and applied. Rather than use a continuous model, we develop a discrete, _n-body model with basic tensile interactions derived from classical molecular force formulae. Computer results are described and discussed.     

Centroaffine minimal surfaces whose centroaffine curvature and Pick function are constants

We classify centroaffine minimal surfaces whose centroaffine curvature and Pick function are constants locally, which also gives classification of centroaffine minimal surfaces whose centroaffine curvature and generalized Pick function are constants locally.     

Totally real flat minimal surfaces in the hyperquadric

In this paper, we study geometry of totally real minimal surfaces in the complex hyperquadric $Q_{N-2}$, and obtain some characterizations of the harmonic sequence generated by these minimal immersions. For totally real flat surfaces that are minimal immersed in both $Q_{N-2}$ and $\mathbb{C}{P}^{N-1}$, we determine them for $N=4,5,6$, and give a classification theorem when they are Clifford solutions.     

The minimal genus problem in rational surfaces CP~2#nCP~2

There are three key ingredients in the study of the minimal genus problem for rational surfaces CP2#nCP2: the generalized adjunction formula, the action of the orthogonal group of the Lorentz space and the geometric construction. In this paper, we prove the uniqueness of the standard form (see Definition 1.1 and Theorem 1.1) of a 2-dimensional homology class under the action of the subgroup of the Lorentz orthogonal group that is realized by the diffeomorphisms of CP2#nCP2.Using the geometric construction, we determine the minimal genera of some classes (see Theorem 1.2).     

Generalized Weierstrass representation for surfaces in Heisenberg spaces

We establish a spinorial representation for surfaces immersed with prescribed mean curvature in Heisenberg space. This permits to obtain minimal immersions starting with a harmonic Gauss map whose target is either the Poincaré disc or a hemisphere of the round sphere.     

A Weierstrass representation for linear Weingarten spacelike surfaces of maximal type in the Lorentz-Minkowski space

In this work we extend the Weierstrass representation for maximal spacelike surfaces in the 3-dimensional Lorentz-Minkowski space to spacelike surfaces whose mean curvature is proportional to its Gaussian curvature (linear Weingarten surfaces of maximal type). We use this representation in order to study the Gaussian curvature and the Gauss map of such surfaces when the immersion is complete, proving that the surface is a plane or the supremum of its Gaussian curvature is a negative constant and its Gauss map is a diffeomorphism onto the hyperbolic plane. Finally, we classify the rotation linear Weingarten surfaces of maximal type.     

The closed embedded minimal surfaces in an almost positively curved three manifold

The compactness theorem of the closed embedded minimal surfaces of fixed genus in a 3-dimensional closed Riemannian manifoldN is proved, providedN is simply connected and the nonpositive value set of Ricci curvature is sufficiently concentrated within finite balls and the minimal surfaces are uniformly away from these balls.     

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.     

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.     

On the Surjectivity of the Exponential Function of Solvable Lie Groups

For a solvable Lie group G the surjectivity of the exponential function exp_G is equivalent to the connectedness of the near-Cartan subgroups and to the connectedness of the centralizers in a Cartan subgroup of all nilpotent elements in its Lie algebra g. Furthermore, these conditions are satisfied if and only if for all elements g ? G there is an x ? g with g = exp_G x in which exp_G is regular.