Related papers: Genus-One Helicoids from a Variational Point of Vi…
The singly periodic genus-one helicoid was in the origin of the discovery of the first example of a complete minimal surface with finite topology but infinite total curvature, the celebrated Hoffman-Karcher-Wei's genus one helicoid. The…
We partially resolve a conjecture of Meeks on the asymptotic behavior of minimal surfaces in $\mathbb{R}^3$ with quadratic area growth.
We present a new construction of embedded minimal surfaces in hyperbolic space with $3$ asymptotically totally geodesic ends and arbitrary finite genus.
A very interesting problem in the classical theory of minimal surfaces consists of the classification of such surfaces under some geometrical and topological constraints. In this short paper, we give a brief summary of the known…
In this paper, we use the conjugate surface construction to prove the existence of certain non-periodic symmetric immersed minimal surfaces. These surfaces have finite total curvature and embedded catenoid ends, and they have positive genus…
For all $m \in \mathbb N - \{0\}$, we prove the existence of a one dimensional family of genus $m$, constant mean curvature (equal to 1) surfaces which are complete, immersed in $\mathbb R^3$ and have two Delaunay ends asymptotic to…
We construct the first examples of complete, properly embedded minimal surfaces in $\mathbb{H}^2 \times \mathbb{R}$ with finite total curvature and positive genus. These are constructed by gluing copies of horizontal catenoids or other…
We prove the existence of pure braids with arbitrarily many strands which are small, i.e. they contain no closed incompressible surface in the complement which is not boundary parallel. This implies the existence of irreducible non-Haken…
We show that a 3-manifold containing an incompressible surface has topologically minimal surfaces of arbitrary high genus.
We prove that orientable index one minimal surfaces in spherical space forms with large fundamental group have genus at most two. This confirms a conjecture of R. Schoen for an infinite class of 3-manifolds.
In this paper, we show the existence of a timelike minimal surface with an arbitrary number of weak complete ends. Then, we discuss the asymptotic behaviour of the simple ends and the topology of the singularity set of the constructed…
In this paper we describe a new deformation that connects minimal disks with planar ends with minimal disks with helicoidal ends. In this way, we are able to construct a family of complete minimal surfaces with helicoidal ends that contains…
We consider families of embedded, screw motion invariant minimal surfaces in $\R^3$ which limit to parking garage structures. We derive balance equations for the nodal limit and regenerate to obtain surfaces corresponding to solutions. We…
In this paper, we study Alexandrov-embedded r-noids with genus 1 and horizontal ends. Such minimal surfaces are of two types and we build several examples of the first one. We prove that if a polygon bounds an immersed polygonal disk, it is…
A triangulated piecewise-linear minimal surface in Euclidean 3-space defined using a variational characterization is critical for area amongst all continuous piecewise-linear variations with compact support that preserve the simplicial…
We consider closed acylindrical surfaces in 3-manifolds and in knot and link complements, and show that the genus of these surfaces is bounded linearly by the number of tetrahedra in the triangulation of the manifold and by the number of…
We describe a method to show that certain elliptic surfaces do not admit purely inseparable multisections (equivalently, that genus one curves over function fields admit no points over the perfect closure of the base field) and use it to…
We discover a family of closed, embedded minimal surfaces in the three-dimensional round sphere which includes new examples with low genus. The existence proof relies on an equivariant min-max procedure applied to a novel sweepout which is…
A classical result by Marston Morse asserts that on some ellipsoids of ${\mathbb R}^3$ there exists exactly 3 closed and simple geodesics. The goal of this presentation is to prove that this rigidity result does not extend to higher…
We show that an embedded minimal disk in R^3 with large curvature is bilipschitz with a piece of a helicoid. Additionally, a simplified proof of the uniqueness of the helicoid is provided.