Related papers: Holomorphic Cubic Differentials and Minimal Lagran…

In this paper, we give a lower bound for the spectrum of the Laplacian on minimal hypersurfaces immersed into $H^m \times R$. As an application, in dimension 2, we prove that a complete minimal surface with finite total extrinsic curvature…

In this paper, we study the elliptic Weingarten surfaces of minimal type immersed in the warped product space $\mathbb{R} \times_{h} \mathbb{R}$, when $h$ is a $C^{1}$-function in $\mathbb{R}^{2}$ with radial symmetry. That is, surfaces…

We show that the Clifford torus and the totally geodesic real projective plane RP^2 in the complex projective plane CP^2 are the unique Hamiltonian stable minimal Lagrangian compact surfaces of CP^2 with genus less than or equal to 4, when…

We continue our investigations into Toda's algorithm [14,3]; a Weierstrass-type representation of Gauss curvature $K=-1$ surfaces in $\mathbb{R}^3$. We show that $C^0$ input potentials correspond in an appealing way to a special new class…

Given an oriented Riemannian surface $(\Sigma, g)$, its tangent bundle $T\Sigma$ enjoys a natural pseudo-K\"{a}hler structure, that is the combination of a complex structure $\J$, a pseudo-metric $\G$ with neutral signature and a symplectic…

Let $f\in W^{3,1}_{\mathrm{loc}}(\Omega)$ be a function defined on a connected open subset $\Omega\subseteq\mathbb R^2$. We will show that its graph is contained in a quadratic surface if and only if $f$ is a weak solution to a certain…

The complex projective space $\mathbb C P^2$ of complex dimension $2$ has a Spin$^c$ structure carrying K\"ahlerian Killing spinors. The restriction of one of these K\"ahlerian Killing spinors to a surface $M^2$ characterizes the isometric…

We study Lagrangian immersions in the nearly K\"ahler $\mathbb{S}^6$ which are warped product manifolds of a $1$-dimensional base and a surface. Apart from the totally geodesic ones, they are either of constant sectional curvature…

Given a finite unbranched covering of a nonsingular projective scheme we analyse the morphism between moduli spaces of sheaves induced by pullback. We have a closer look at cyclic coverings and, in particular, at canonical coverings of…

We propose a new method for the construction of Hamiltonian-minimal and minimal Lagrangian immersions of some manifolds in $C^n$ and in $CP^n$. By this method one can construct, in particular, immersions of such manifolds as the generalized…

Let $M^n$ be either a simply connected space form or a rank-one symmetric space of noncompact type. We consider Weingarten hypersurfaces of $M\times\mathbb R$, which are those whose principal curvatures $k_1,\dots ,k_n$ and angle function…

We prove that an infinite Riemann surface $X$ is parabolic ($X\in O_G$) if and only if the union of the horizontal trajectories of any integrable holomorphic quadratic differential that are cross-cuts is of zero measure. Then we establish…

Let $M$ be an open Riemann surface and let $\Lambda\subset M$ be a closed discrete subset. In this paper, we prove the existence of complete conformal minimal immersions $M\to\mathbb{R}^n$, $n\ge 3$, with prescribed values on $\Lambda$ and…

We give a definition of isoclinic parametric surfaces in $\mathbb{R}^4_2$ and prove that such an isoclinic conformal immersion comes from two holomorphic functions. A Cauchy problem was proposed and solved, namely: construct an isoclinic…

We construct geometrically a homeomorphism between the moduli space of polynomial quadratic differentials on the complex plane and light-like polygons in the 2-dimensional Einstein Universe. As an application, we find a class of minimal…

In this paper we establish new Calabi-Bernstein results for maximal surfaces immersed into a Lorentzian product space of the form $M^2\times\mathbb{R}_1$, where $M^2$ is a connected Riemannian surface and $M^2\times\mathbb{R}_1$ is endowed…

We define a new theory of discrete Riemann surfaces and present its basic results. The key idea is to consider not only a cellular decomposition of a surface, but the union with its dual. Discrete holomorphy is defined by a straightforward…

We develop a loop group (DPW-type) representation for minimal Lagrangian surfaces in the complex quadric $Q_{2}\cong \mathbb S^{2}\times \mathbb S^{2}$, formulated via a flat family of connections $\{\nabla^\lambda\}_{\lambda\in \mathbb…

We introduce a new method to construct a large family of Lagrangian surfaces in complex Euclidean plane by means of two planar curves making use of their usual product as complex functions and integrating the Hermitian product of their…

We construct a moduli space for Riemann surfaces that is universal in the sense that it represents compact Riemann surfaces of any finite genus. This moduli space is stratifed according to genus, and it carries a metric and a measure that…