# Investigación / Research (Actualizado en mayo de 2013 / Updated at May 2013)

## Artículos / Papers

• Complete bounded embedded complex curves in $$\mathbb{C}^2$$ (with Francisco J. López). Preprint, May 2013.
• We prove that any convex domain of $$\mathbb{C}^2$$ carries properly embedded complete complex curves. In particular, there exist complete bounded embedded complex curves in $$\mathbb{C}^2$$.

• Null curves and directed immersions of open Riemann surfaces (with Franc Forstneric). Preprint, October 2012.
• We study holomorphic immersions of open Riemann surfaces into $$\mathbb{C}^n$$ whose derivative lies in a conical algebraic subvariety $$A$$ of $$\mathbb{C}^n$$ that is smooth away from the origin. We establish a basic structure theorem for the set of all $$A$$-immersions of a bordered Riemann surface, and prove several approximation and desingularization theorems. As application we show that every open Riemann surface is biholomorphic to a properly embedded null curve in $$\mathbb{C}^3$$.

• Every bordered Riemann surface is a complete proper curve in a ball (with Franc Forstneric). Math. Ann., in press.
• We prove that every bordered Riemann surface admits a complete proper holomorphic immersion into a ball of $$\mathbb{C}^2$$, and a complete proper holomorphic embedding into a ball of $$\mathbb{C}^3$$. We point out how the technique developed in this paper can also be generalized in order to construct complete proper holomorphic immersions of any bordered Riemann surface into any Stein manifold of dimension at least $$2$$. ArXiv.

• The Minkowski problem, new constant curvature surfaces in $$\mathbb{R}^3$$, and some applications (with Rabah Souam). Preprint, April 2012.
• We classify the family of surfaces in $$\mathbb{R}^3$$ with constant Gauss curvature $$K=1$$, extrinsic conformal structure a circular domain, and Gauss map a diffeomorphism onto a finitely punctured sphere. We derive applications to the generalized Minkowski problem, existence of harmonic diffeomorphisms between domains of $$\mathbb{S}^2$$, existence of capillary surfaces in $$\mathbb{R}^3$$, and a Hessian equation of Monge-Ampère type.

• Properness of associated minimal surfaces (with Francisco J. López). Trans. Amer. Math. Soc., in press.
• We show that any open Riemann surface admits a conformal minimal immersion into $$\mathbb{R}^3$$ with vanishing flux, such that the immersion itself and a given finite family of its associated immersions properly project into $$\mathbb{R}^2$$.

• Harmonic diffeomorphisms between domains in the Euclidean 2-sphere (with Rabah Souam). Comment. Math. Helv., in press.
• We prove existence of circular domains $$U$$ and harmonic diffeomorphisms $$U\to\mathbb{S}^2-\{p_1,\ldots,p_n\}$$, for any given subset $$\{p_1,\ldots,p_n\}\subset\mathbb{S}^2$$ and natural number $$n\geq 2$$. They are obtained as vertical projection of maximal graphs over $$\mathbb{S}^2-\{p_1,\ldots,p_n\}$$ in the Lorentzian product $$\mathbb{S}^2\times\mathbb{R}_1$$.

• Compact complete null curves in Complex 3-space (with Francisco J. López). Israel J. Math., in press.
• We prove that for any open orientable surface $$S$$ of finite topology, there exist a Riemann surface $$\mathcal{M}$$, a relatively compact domain $$M\subset\mathcal{M}$$ homeomorphic to $$S$$, and a continuous map $$X : \overline{M}\to\mathbb{C}^3$$ whose restriction to $$M$$ is a complete null holomorphic immersion. ArXiv.

• Proper holomorphic embeddings of Riemann surfaces with arbitrary topology into $$\mathbb{C}^2$$ (with Francisco J. López). J. Geom. Anal., in press.
• We prove existence of properly embedded complex curves in $$\mathbb{C}^2$$ with arbitrary topological type. ArXiv.

• On harmonic quasiconformal immersions of surfaces in $$\mathbb{R}^3$$ (with Francisco J. López). Trans. Amer. Math. Soc. 365 (2013), no. 4, 1711-1742.
• We study the global properties of harmonically immersed Riemann surfaces in $$\mathbb{R}^3$$. We focus on the geometry of complete harmonic immersions with quasiconformal Gauss map, and in particular, of those with finite total curvature. ArXiv.

• Inmersiones armónicas de superficies de Riemann en $$\mathbb{R}^3$$ (Spanish) (with Francisco J. López). Florentino García Santos: In memoriam. Universidad de Granada, 2011, p. 1-8.
• We announce here the results in the paper just above.

• Harmonic mappings and conformal minimal immersions of Riemann surfaces into $$\mathbb{R}^N$$ (with Isabel Fernández and Francisco J. López). Calc. Var. Partial Differential Equations 47 (2013), no. 1-2, 227-242.
• We show complete minimal surfaces in $$\mathbb{R}^N$$, $$N>3$$, with arbitrary conformal structure and $$N-2$$ prescribed coordinate functions. We derive applications regarding the Gauss map of minimal surfaces in $$\mathbb{R}^N$$ and the existence of complete embedded non-properly immersed minimal surfaces in $$\mathbb{R}^4$$. ArXiv.

• Null curves in $$\mathbb{C}^3$$ and Calabi-Yau conjectures (with Francisco J. López). Math. Ann. 355 (2013), no. 2, 429-455.
• We show complete bounded null holomorphic curves in $$\mathbb{C}^3$$ with arbitrary topological type. We derive similar existence results for null holomorphic curves in $${\rm SL}(2,\mathbb{C})$$, Bryant surfaces in $$\mathbb{H}^3$$, complex curves in $$\mathbb{C}^2$$, and minimal surfaces in $$\mathbb{R}^3$$. ArXiv.

• Complete minimal surfaces and harmonic functions (with Isabel Fernández and Francisco J. López). Comment. Math. Helv. 87 (2012), no. 4, 891-904.
• We construct complete minimal surfaces in $$\mathbb{R}^3$$ with any given conformal structure and a prescribed harmonic coordinate function. As a consequence, complete minimal surfaces with arbitrary conformal structure and whose Gauss map misses two points are shown. ArXiv.

• Minimal surfaces in $$\mathbb{R}^3$$ properly projecting into $$\mathbb{R}^2$$ (with Francisco J. López). J. Differential Geom. 90 (2012), no. 3, 351-382.
• We prove a Runge-Mergelyan type theorem for minimal surfaces in $$\mathbb{R}^3$$. This provides a flexible method for constructing examples. As application we show minimal surfaces in $$\mathbb{R}^3$$, with arbitrary conformal structure, properly projecting into a plane. We also show proper hyperbolic minimal surfaces with non-empty boundary, lying over a negative sublinear graph in $$\mathbb{R}^3$$. ArXiv.

• Proper harmonic maps from hyperbolic Riemann surfaces into the Euclidean plane (with José A. Gálvez). Results Math. 60 (2011), no. 1-4, 487-505.
• We show existence of such Riemann surfaces and harmonic maps. ArXiv.

• Complete minimal surfaces in $$\mathbb{R}^3$$ with a prescribed coordinate function (with Isabel Fernández). Differ. Geom. Appl. 29 (2011), s. 1, S9-S15.
• Any non-constant harmonic function on either the unit complex disc or the complex plane is a coordinate function of a complete conformal minimal immersion into $$\mathbb{R}^3$$. ArXiv.

• Compact complete proper minimal immersions in strictly convex bounded regular domains of $$\mathbb{R}^3$$. AIP Conf. Proc. 1260 (2010), 105-111.
• Surfaces as those in the paper just below can be constructed to be proper in any given strictly convex bounded regular domain of $$\mathbb{R}^3$$. pdf.

• Compact complete minimal immersions in $$\mathbb{R}^3$$. Trans. Amer. Math. Soc. 362 (2010), no. 8, 4063-4076.
• We find, for any arbitrary finite topological type, a compact Riemann surface $$\mathcal{M}$$, a relatively compact domain $$M\subset\mathcal{M}$$ with the fixed topological type, and a conformal complete minimal immersion $$X : M\to \mathbb{R}^3$$ extending continuously to $$\overline{M}$$. ArXiv.

• On the Calabi-Yau problem for maximal surfaces in $$\mathbb{L}^3$$. Differ. Geom. Appl. 26 (2008), no. 6, 625-634.
• We show weakly complete maximal surfaces in Minkowski space which are bounded by a hyperboloid. ArXiv.

• Limit sets for complete minimal immersions (with Nikolai Nadirashvili). Math. Z. 258 (2008), no. 1, 107-113.
• We study the behaviour of the limit set of complete proper compact minimal immersions in a domain $$G\subset \mathbb{R}^3$$ with $$\mathcal{C}^2$$ boundary. ArXiv.

• On the existence of a proper conformal maximal disk in $$\mathbb{L}^3$$. Differ. Geom. Appl. 26 (2008), no. 2, 151-168.
• Such a surface exists. ArXiv.

• Density theorems for complete minimal surfaces in $$\mathbb{R}^3$$ (with Leonor Ferrer and Francisco Martín). Geom. Funct. Anal. 18 (2008), no. 1, 1-49.
• We prove that complete surfaces are dense in the space of all minimal surfaces in $$\mathbb{R}^3$$ endowed with the topology of $$\mathcal{C}^k$$ convergence on compact sets, for any $$k\in\mathbb{N}$$. As application, we show a properly immersed minimal surface in $$\mathbb{R}^3$$ with uncountably many ends. ArXiv.

• A uniqueness theorem for the singly periodic genus-one helicoid (with Leonor Ferrer and Francisco Martín). Trans. Amer. Math. Soc. 359 (2007), no. 6, 2819-2829.
• We give such a theorem provided the existence of one symmetry. ArXiv.

• Recent progresses in the Calabi-Yau problem for minimal surfaces. Mat. Contemp. 30 (2006), 29-40.
• This is a short survey on the construction of complete bounded minimal surfaces in $$\mathbb{R}^3$$. pdf.

• On the singly periodic genus one helicoid (with Leonor Ferrer and Francisco Martín). Differ. Geom. Dyn. Syst. 8 (2006), 1-7.
• We write down here some Weierstrass representations and a uniqueness theorem for the singly periodic genus one helicoid. pdf.