We obtain 2 and 3 parameter families of new minimal surfaces in the hyperbolic space $\mathbb{H}^3$, by applying Darboux transformations i.e., conformal Ribaucour transformations, to Mori’s spherical catenoids in $\mathbb{H}^3$. Depending on the values of the parameters, the minimal surfaces can have any finite number of closed curves in the boundary at infinity of $\mathbb{H}^3$ or an infinite number of such curves. In particular, we obtain minimal surfaces periodic in one variable, with certain symmetries, whose parametrization is defined in $\mathbb{R}^2 \setminus \mathcal{C}$ , where $\mathcal{C}$ is either a disjoint union of Jordan curves or a non closed regular curve. A connected region bounded by such a Jordan curve, generates a complete minimal surface, whose boundary at infinity of $\mathbb{H}^3$ is a closed curve. A connected unbounded domain of $\mathbb{R}^2 \setminus \mathcal{C}$ generates a non complete immersed minimal surface whose boundary at infinity consists of a finite number of closed curves. This is a joint work with Ningwei Cui.

We consider the partial differential equations whose solutions characterize the linear Weingarten surfaces in space forms. This is a class of seven equations that includes the elliptic sinh-Gordon, the elliptic cosh-Gordon, the sine-Gordon, the Liouville and the Laplace equations. Given a solution of such a differential equation, we will show that the geometric Ribaucour transformation for linear Weingarten surfaces generates a Bäcklund type transformation, which is an integrable system that provides families of new solutions of the same differential equation. The permutability property provides superposition formulae which give new solutions algebraically. Explicit examples will be included.
Conferencia dentro del programa de doctorado Matemáticas

We consider curves in $\\Rset^{2n}$ endowed with the standard symplectic structure. We introduce the concept of symplectic arc length for curves. We construct an adapted symplectic Frenet frame and we define $2n-1$ local differential invariants that we call symplectic curvatures of the curve. > We prove that, up to a rigid symplectic motion of $\\Rset^{2n}$, there exists a unique curve with prescribed symplectic curvatures. We characterize the curves in $\\Rset^{4}$ with constant symplectic curvatures.