arXiv subjects

Magdalena Rodriguez

Publications and source records attributed to Magdalena Rodriguez.

5 recordsLinked to original sources

arXiv2021

Slab Theorem and Halfspace Theorem for constant mean curvature surfaces in $\mathbb H^2\times\mathbb R$

We prove that a properly embedded annular end of a surface in $\mathbb H^2\times\mathbb R$ with constant mean curvature $0<H\leq \frac{1}{2}$ can not be contained in any horizontal slab. Moreover, we show that a properly embedded surface with constant mean curvature $0<H\leq \frac{1}{2}$ contained in $\mathbb H^2\times[0,+\infty)$ and with finite topology is necessarily a graph over a simply connected domain of $\mathbb H^2$. For the case $H=\frac{1}{2}$, the graph is entire.

math.DG↗

arXiv2016

Minimal graphs over Riemannian surfaces and harmonic diffeomorphisms

We construct a parabolic entire minimal graph $S$ over a finite topology complete Riemannian surface $Σ$ of curvature $-1$ and infinite area (thus of non-parabolic conformal type). The vertical projection of this graph yields a harmonic diffeomorphism from $S$ onto $Σ$. The proof uses the theory of divergence lines to construct minimal graphs. We also generalize a theorem of R. Schoen. Let $g_1$ and $g_2$ be two complete metrics on a orientable surface $S$ with compact boundary and suppose $$\int_{S_r^2}K_{g_2}^-dσ_{g_2}\le C\ln(2+r)$$ for some $C>0$ and all $r>0$. If there is a harmonic diffeomorphism from $(S,g_1)$ to $(S,g_2)$, then $(S,g_1)$ is parabolic.

math.DG↗

arXiv2012

Non-proper complete minimal surfaces embedded in H^2 x R

Examples of complete minimal surfaces properly embedded in H^2 x R have been extensively studied and the literature contains a plethora of nontrivial ones. In this paper we construct a large class of examples of complete minimal surfaces embedded in H^2 x R, not necessarily proper, which are invariant by a vertical translation or by a hyperbolic or parabolic screw motion. In particular, we construct a large family of non-proper complete minimal disks embedded in H^2 x R invariant by a vertical translation and a hyperbolic screw motion and whose importance is twofold. They have finite total curvature in the quotient of H^2 x R by the isometry, thus highlighting a different behaviour from minimal surfaces embedded in R^3 satisfying the same properties. And they show that the Calabi-Yau conjectures do not hold for embedded minimal surfaces in H^2 x R.

math.DG↗

arXiv2012

Simply-connected minimal surfaces with finite total curvature in $\H^2\times\R$

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 $-2π$. The first examples appearing in this context are vertical geodesic planes and Scherk minimal graphs over ideal polygonal domains. Other non simply-connected examples have been constructed recently. In the present paper, we show that the only complete minimal surfaces in $\H^2\times\R$ of total curvature $-2π$ are Scherk minimal graphs over ideal quadrilaterals. We also construct properly embedded simply-connected minimal surfaces with total curvature $-4kπ$, for any integer $k\geq 1$, which are not Scherk minimal graphs over ideal polygonal domains.

math.DG↗

arXiv2008

An end-to-end-construction for singly periodic minimal surfaces

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 of pieces of already known minimal surfaces.

math.DG↗