Related papers: Adding one handle to half-plane layers

We construct harmonic diffeomorphisms from the complex plane $C$ onto any Hadamard surface $M$ whose curvature is bounded above by a negative constant. For that, we prove a Jenkins-Serrin type theorem for minimal graphs in $M\times R$ over…

We describe the family of minimal graphs on strips with boundary values $\pm\infty$ disposed alternately on edges of length one, and whose conjugate graphs are contained in horizontal slabs of width one in $\mathbb{R}^3$. We can obtain as…

Laurent Hauswirth and Harold Rosenberg developed the theory of minimal surfaces with finite total curvature in $\H^2\times\R$. They showed that the total curvature of one such a surface must be a non-negative integer multiple of $-2\pi$.…

We study the Dirichlet problem for minimal surface systems in arbitrary dimension and codimension via mean curvature flow, and obtain the existence of minimal graphs over arbitrary mean convex bounded $C^2$ domains for a large class of…

Given a tiling $\mathcal{T}$ of the plane by straight edge polygons, which is invariant by two independent translations, we construct a family of embedded triply periodic minimal surfaces which desingularizes $\mathcal{T}\times\mathbb{R}$.…

We show the existence of various families of properly embedded singly periodic minimal surfaces in R^3 with finite arbitrary genus and Scherk type ends in the quotient. The proof of our results is based on the gluing of small perturbations…

We prove the existence of a family of embedded doubly periodic minimal surfaces of (quotient) genus $g$ with orthogonal ends that generalizes the classical doubly periodic surface of Scherk and the genus-one Scherk surface of Karcher. The…

There exist two new embedded minimal surfaces, asymptotic to the helicoid. One is periodic, with quotient (by orientation-preserving translations) of genus one. The other is nonperiodic of genus one.

We develop Teichmuller theoretical methods to construct new minimal surfaces in $\BE^3$ by adding handles and planar ends to existing minimal surfaces in $\BE^3$. We exhibit this method on an interesting class of minimal surfaces which are…

We give a positive answer to M. Traizet's open question about the existence of complete embedded minimal surfaces with Scherk-ends without planar geodesics. In the singly periodic case, these examples get close to an extension of Traizet's…

We construct new examples of immersed minimal surfaces with catenoid ends and finite total curvature, of both genus zero and higher genus. In the genus zero case, we classify all such surfaces with at most $2n+1$ ends, and with symmetry…

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…

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…

In this paper, we study existence and uniqueness of solutions to Jenkins-Serrin type problems on domains in a Riemannian surface. In the case of unbounded domains, the study is focused on the hyperbolic plane.

We consider a Riemannian submersion from a 3-manifold $\mathbb{E}$ to a surface $M$, both connected and orientable, whose fibers are the integral curves of a Killing vector field without zeros, not necessarily unitary. We solve the…

We propose an alternative condition for the solvability of the Dirichlet problem for the minimal surface equation that applies to non-mean convex domains. We introduce a structural condition, obtained from a second-order ordinary…

We present a general construction of embedded minimal and constant mean curvature surfaces in $\mathbb{S}^n$ and one-phase free boundaries joined by a smooth interpolation by capillary hypersurfaces. This framework recovers all known…

We construct embedded minimal surfaces which are $n$-periodic in $\mathbb{R}^n$. They are new for codimension $n-2\ge 2$. We start with a Jordan curve of edges of the $n$-dimensional cube. It bounds a Plateau minimal disk which Schwarz…

We consider the question of existence of embedded doubly periodic minimal surfaces in Euclidean 3-space with Scherk-type ends, surfaces that topologically are Scherk's doubly periodic surface with handles added in various ways. We extend…

We study minimal graphs in the homogeneous Riemannian 3-manifold $\widetilde{PSL_2(\mathbb{R})}$ and we give examples of invariant surfaces. We derive a gradient estimate for solutions of the minimal surface equation in this space and…