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     

Exotic Minimal Surfaces

We prove a general fusion theorem for complete orientable minimal surfaces in ?³ with finite total curvature. As a consequence, complete orientable minimal surfaces of weak finite total curvature with exotic geometry are produced. More specifically, universal surfaces (i.e., surfaces from which all minimal surfaces can be recovered) and space-filling surfaces with arbitrary genus and no symmetries.     

Willmore Surfaces of Revolution with Two Prescribed Boundary Circles

We consider the family of smooth embedded surfaces of revolution in ?³ having two concentric circles contained in two parallel planes of ?³ as boundary. Minimizing the Willmore functional within this class of surfaces we prove the existence of smooth axi-symmetric Willmore surfaces having these circles as boundary. When the radii of the circles tend to zero we prove convergence of these solutions to the round sphere.     

A New Proof of the Existence of Embedded Surfaces with Anosov Geodesic Flow

We give a new proof of the existence of compact surfaces embedded in ?³ with Anosov geodesic flows. This proof starts with a noncompact model surface whose geodesic flow is shown to be Anosov using a uniformly strictly invariant cone condition. Using a sequence of explicit maps based on the standard torus embedding, we produce compact embedded surfaces that can be seen as small perturbations of the Anosov model system and hence are themselves Anosov.     

Finite group actions on bordered surfaces of small genus

This paper considers finite group actions on compact bordered surfaces — quotients of unbordered orientable surfaces under the action of a reflectional symmetry. Classification of such actions (up to topological equivalence) is carried out by means of the theory of non-euclidean crystallographic groups, and determination of normal subgroups of finite index in these groups, up to conjugation within their automorphism group. A result of this investigation is the determination, up to topological equivalence, of all actions of groups of finite order 6 or more on compact (orientable or non-orientable) bordered surfaces of algebraic genus p for 2≤p≤6. We also study actions of groups of order less than 6, or of prime order, on bordered surfaces of arbitrary algebraic genus p≥2.     

The Cauchy problem for the Liouville equation and Bryant surfaces

We give a construction that connects the Cauchy problem for the 2-dimensional elliptic Liouville equation with a certain initial value problem for mean curvature one surfaces in hyperbolic 3-space H³, and solve both of them. We construct the unique mean curvature one surface in H³ that passes through a given curve with a given unit normal along it, and provide diverse applications. In particular, topics such as period problems, symmetries, finite total curvature, planar geodesics, rigidity, etc. are treated for these surfaces.     

Limits on an extension of Carleson’s\bar \partial - Theorem

We study a theorem essentially due to Carleson about solving the $\bar \partial - equation$ on the unit disk. We show that this theorem generalizes to bordered Riemann surfaces with finitely generated fundamental groups. However, our main result is that the constant appearing in the generalized theorem cannot be taken to be independent of the bordered Riemann surface in question. We exhibit a sequence of (topologically equivalent) Riemann surfaces on which the constant tends to ∞. Since Carleson’s $\bar \partial - theorem$ depends on the notion of a Carleson measure, we also discuss Carleson measures at some length in order to define them appropriately on arbitrary Riemann surfaces.     

On the generalized Weyl problem for flat metrics in the hyperbolic 3-space

The paper deals with the study of complete embedded flat surfaces in H³${\mathbb{H}}^3$ with a finite number of isolated singularities. We give a detailed information about its topology, conformal type and metric properties. We show how to solve the generalized Weyl?s problem of realizing isometrically any complete flat metric with Euclidean singularities in H³${\mathbb{H}}^3$ which gives the existence of complete embedded flat surfaces with a finite arbitrary number of isolated singularities.     

Singularities of the affine chord envelope for two surfaces in four-space

We study local singularities of the envelope of the set of straight lines (chords) joining those points of two surfaces embedded in ?⁴ which are not in general position to each other.     

The triply periodic minimal surfaces of Alan Schoen and their constant mean curvature companions

We prove existence of Schoen's and other triply periodic minimal surfaces via conjugate (polygonal) Plateau problems. The simpler of these minimal surfaces can be deformed into constant mean curvature surfaces by solving analogous Plateau problems in S³. The required contours in S³ are obtained by working with the great circle orbits of Hopf S¹-actions in the same way as with families of parallel lines in ³. Annular Plateau problems give new embedded minimal surfaces in S³. For many of the minimal surfaces in ³ global Weierstraß representations are derived.Dedicated to Wilhelm Klingenberg     

Complete flat surfaces with two isolated singularities in hyperbolic 3-space

We construct examples of flat surfaces in H³ which are graphs over a two-punctured horosphere and classify complete embedded flat surfaces in H³ with only one end and at most two isolated singularities.     

On the non existence of some rational 3-folds in ℙ5

In this paper we show that there are no smooth rational 3-folds in ?⁵ (C) which are rational conic bundles, over minimal surfaces, whose generic fibre is embedded as a rational curve of degreeh≥3, (ifh=2 there is a complete classification for these 3-folds as well as for the case of ?¹-bundles). Except for conic bundles, we also give the complete list of rational 3-folds in ?⁵ which are minimal according to Mori’s theory. These are little steps towards the classification of all smooth 3-folds in ?⁵ not of general type.     

Ribaucour transformations for flat surfaces in the hyperbolic 3-space

We consider Ribaucour transformations for flat surfaces in the hyperbolic 3-space, H³${\mathbb{H}}^3$. We show that such transformations produce complete, embedded ends of horosphere type and curves of singularities which generically are cuspidal edges. Moreover, we prove that these ends and curves of singularities do not intersect. We apply Ribaucour transformations to rotational flat surfaces in H³${\mathbb{H}}^3$ providing new families of explicitly given flat surfaces H³${\mathbb{H}}^3$ which are determined by several parameters. For special choices of the parameters, we get surfaces that are periodic in one variable and surfaces with any even number or an infinite number of embedded ends of horosphere type.     

A characterization of Weingarten surfaces in hyperbolic 3-space

We study 2-dimensional submanifolds of the space \({\mathbb{L}}({\mathbb{H}}^{3})\) of oriented geodesics of hyperbolic 3-space, endowed with the canonical neutral Kähler structure. Such a surface is Lagrangian iff there exists a surface in ?³ orthogonal to the geodesics of Σ.We prove that the induced metric on a Lagrangian surface in \({\mathbb{L}}({\mathbb{H}}^{3})\) has zero Gauss curvature iff the orthogonal surfaces in ?³ are Weingarten: the eigenvalues of the second fundamental form are functionally related. We then classify the totally null surfaces in \({\mathbb{L}}({\mathbb{H}}^{3})\) and recover the well-known holomorphic constructions of flat and CMC 1 surfaces in ?³.     

A General Existence Theorem for Embedded Minimal Surfaces with Free Boundary

In this paper, we prove a general existence theorem for properly embedded minimal surfaces with free boundary in any compact Riemannian 3‐manifold M with boundary ?M. These minimal surfaces are either disjoint from ?M or meet ?M orthogonally. The main feature of our result is that there is no assumptions on the curvature of M or convexity of ?M. We prove the boundary regularity of the minimal surfaces at their free boundaries. Furthermore, we define a topological invariant, the filling genus, for compact 3‐manifolds with boundary and show that we can bound the genus of the minimal surface constructed above in terms of the filling genus of the ambient manifold M. Our proof employs a variant of the min‐max construction used by Colding and De Lellis on closed embedded minimal surfaces, which were first developed by Almgren and Pitts.© 2014 Wiley Periodicals, Inc.     

Laguerre isothermic surfaces in ℝ3 and their Darboux transformation

In this paper,we study Laguerre isothermic surfaces in R3.We show that the Darboux transformation of a Laguerre isothermic surface x produces a new Laguerre isothermic surface x and their respective Laguerre Gauss maps form a Darboux pair of each other at the corresponding point.We also classify the surfaces which are both Laguerre isothermic and Laguerre minimal and show that they must be Laguerre equivalent to surfaces with vanishing mean curvature in R3,R13 or R03.     

Some remarks on complete simply connected minimal surfaces meeting the planes x3 = Constant Transversally

The helicoid and plane are the only known complete embedded minimal surfaces inR ³ that are simply connected. We prove the helicoid and plane are the only surfaces of this type that have bounded curvature and meet each plane x₃ = constant in (at most) one smooth connected curve.     

Some remarks on embedded hypersurfaces in hyperbolic space of constant curvature and spherical boundary

We consider embedded hypersurfacesM in hyperbolic space with compact boundaryC and somer ^th mean curvature functionH _r a positive constant. We investigate when symmetries ofC are symmetries ofM. We prove that if 0H _r1 andC is a sphere thenM is a part of an equidistant sphere. Forr=1 (H ₁ is the mean curvature) we obtain results whenC is convex.     

Moduli of (1, 7)–Polarized Abelian Surfaces via Syzygies

We prove that the moduli space X(1,7) of (1,7)–polarized abelian surfaces with canonical level–structure is birational to the Fano 3–fold V₂₂ of polar hexagons of the Klein quartic (7). In particular X(1,7) is rational and the birational map to ℙ³ is defined over ℚ. As a byproduct we obtain explicitely the equations of the (1,7)–very–ample–polarized abelian surfaces embedded in ℙ⁶.     

Symmetries of PDE systems and correspondences between jet spaces

Let M be a manifold. A PDE system can be prolonged to another one R^?⊆T^?M (Jiménez et al. (2005) [10]). In analogy with the higher-order symmetries, symmetries of R^? will be called higher-dimensional symmetries of R. For a broad class of PDE systems we prove that every (infinitesimal or finite) symmetry of R comes from another one of R^?. We show that R^? does not have internal (infinitesimal) symmetries (modulo trivial symmetries). This fact allows us, in the infinitesimal case, to compute the internal symmetries of R as external symmetries of R^?. We also give an algorithmic method to obtain solutions of R invariant by a given internal symmetry.