Related papers: Embedded, Doubly Periodic Minimal Surfaces

For any m > 0, we construct properly embedded minimal surfaces in H^2 x R with genus zero, infinitely many vertical planar ends and m limit ends. We also provide examples with an infinite countable number of limit ends. All these examples…

This paper is devoted to topological phenomena in normal metals with rather complicated Fermi surface. The results of the article are based on the deep topological theorems concerning the geometry of non-compact plane sections of level…

We introduce on any smooth oriented minimal surface in Euclidean $3$-space a meromorphic quadratic differential, $P$, which we call the entropy differential. This differential arises naturally in a number of different contexts. Of…

This article explains a program to study complete and properly embedded minimal surfaces in $\mathbb{R}^3$ developed jointly with W.H. Meeks and A. Ros in the last three decades. It follows closely the structure of my invited ICM talk with…

Using min-max theory, we show that in any closed Riemannian manifold of dimension at least 3 and at most 7, there exist infinitely many smoothly embedded closed minimal hypersurfaces. It proves a conjecture of S.-T. Yau. This paper builds…

In this paper we develop the theory of properly immersed minimal surfaces in the quotient space $\mathbb H^2\times\mathbb R/G,$ where $G$ is a subgroup of isometries generated by a vertical translation and a horizontal isometry in $\mathbb…

Consider a convex domain B of space. We prove that there exist complete minimal surfaces which are properly immersed in B. We also demonstrate that if D and D' are convex domains with D bounded and the closure of D contained in D' then any…

A maximal surface $\sb$ with isolated singularities in a complete flat Lorentzian 3-manifold $\N$ is said to be entire if it lifts to a (periodic) entire multigraph $\tilde{\sb}$ in $\l^3.$ In addition, $\sb$ is called of finite type if it…

In this note we propose a min-max theory for embedded hypersurfaces with a fixed boundary and apply it to prove several theorems about the existence of embedded minimal hypersurfaces with a given boundary. A simpler variant of these…

In this work we prove the existence of embedded closed minimal hypersurfaces in non-compact manifolds containing a bounded open subset with smooth and strictly mean-concave boundary and a natural behavior on the geometry at infinity. For…

For 3 $\leq$ n $\leq$ 7, we prove that a bumpy closed Riemannian n-manifold contains a sequence of connected embedded closed minimal surfaces with unbounded area.

We study the structure of complex points on real surfaces, embedded into complex Elliptic surfaces. We show, for example, that any compact surface has a totally real embedding into a blow-up of a K3 surface. We also exhibit smooth disc…

In this paper, we construct a one-parameter family of minimal surfaces in the Euclidean $3$-space of arbitrarily high genus and with three ends. Each member of this family is immersed, complete and with finite total curvature. Another…

Utilizing the Weierstrass representation for embedded doubly periodic minimal surfaces with parallel ends, we construct entire singly periodic graphs of spacelike maximal surfaces with isolated cone-like singularities in the…

Unlike $\mathbb{R}^{3}$, the homogeneous spaces $\mathbb{E}(-1,\tau)$ have a great variety of entire vertical minimal graphs. In this paper we explore conditions which guarantees that a minimal surface in $\mathbb{E}(-1,\tau)$ is such a…

We investigate the possibility of embedding minimal abelian surfaces in smooth toric 4-folds with Picard number 2. The existence of such an embedding imposes conditions on the 4-fold, which we partly describe. On the other hand, we exhibit…

A non-square-tiled Veech surface has finitely many periodic points, i.e., points with finite orbit under the affine automorphism group. We present an algorithm that inputs a non-square-tiled Veech surface and outputs its set of periodic…

The emergence of trapped surfaces in solutions to the Einstein field equations is intimately tied to the well-posedness properties of the corresponding Cauchy problem in the low regularity regime. In this paper, we study the question of…

For a given spatial graph $\mathcal{G} \subset \mathbb{R}^3$, we would like to find a closed orientable surface $\mathcal{S}$ embedded in $\mathbb{R}^3$ in which $\mathcal{G}$ is cellular embedded. However, for general $\mathcal{G}$ this is…

Inspired by an argument of Ros [15] -- we use the L\'{o}pez-Ros deformation to give another proof of the fact -- due to Meeks and Wolf [13] -- that the only smooth, connected, singly-periodic minimal surfaces in $\Real^3$ with the area…