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This article is motivated by a problem posed by David A. Singer in 1999 and by the classical Euler elastic curves. We study spacelike and timelike curves in the Lorentz-Minkowski plane 𝕃2 whose curvature is expressed in terms of the Lorentzian pseudodistance to fixed geodesics. In this way, we get a complete description of all the elastic curves in 𝕃2 and provide the Lorentzian versions of catenaries and grim-reaper curves. We show several uniqueness results for them in terms of their geometric linear momentum. In addition, we are able to get arc-length parametrizations of all the aforementioned curves and they are depicted graphically.

We show the existence of a $2$-parameter family of properly Alexandrov-embedded surfaces with constant mean curvature $0\leq H\leq \frac {1}{2}$ in ${\mathbb {H}^2\times \mathbb {R}}$. They are symmetric with respect to a horizontal slice and $k$ vertical planes disposed symmetrically and extend the so-called minimal saddle towers and $k$-noids. We show that the orientation plays a fundamental role when $H>0$ by analyzing their conjugate minimal surfaces in $\widetilde {\textrm {SL}}_2(\mathbb {R})$ or $\textrm {Nil}_3$. We also discover new complete examples that we call $(H,k)$-nodoids, whose $k$ ends are asymptotic to vertical cylinders over curves of geodesic curvature $2H$ from the convex side, often giving rise to non-embedded examples if $H>0$. In the discussion of embeddedness of the constructed examples, we prove that the Krust property does not hold for any $H>0$, that is, there are minimal graphs over convex domains in $\widetilde {\textrm {SL}}_2(\mathbb {R})$, $\textrm {Nil}_3$ or the Berger spheres, whose conjugate surfaces with constant mean curvature $H$ in $\mathbb {H}^2\times \mathbb {R}$ are not graphs.

For each k ≥ 3, we construct a 1-parameter family of complete Alexandrov-embedded minimal surfaces in the Riemannian product space H 2 × R with genus 1 and k embedded ends asymptotic to vertical planes. We also obtain complete minimal surfaces with genus 1 and 2k ends in the quotient of H 2 × R by an arbitrary vertical translation. They all have dihedral symmetry with respect to k vertical planes, as well as finite total curvature −4kπ. Finally, we provide examples of complete properly embedded minimal surfaces with infinitely many ends, each of them asymptotic to a vertical plane and with finite total curvature.

Motivated by the classical Euler elastic curves, David A. Singer posed in 1999 the problem of determining a plane curve whose curvature is given in terms of its position. We propound the same question in the Lorentz-Minkowski plane, focusing on spacelike and timelike curves. In this article, we study those curves in {{\mathbb{L}}}^{2} whose curvature depends on the Lorentzian pseudodistance from the origin, and those ones whose curvature depends on the Lorentzian pseudodistance through the horizontal or vertical geodesic to a fixed lightlike geodesic. Making use of the notions of geometric angular momentum (with respect to the origin) and geometric linear momentum (with respect to the fixed lightlike geodesic), respectively, we get two abstract integrability results to determine such curves through quadratures. In this way, we find out several new families of Lorentzian spiral, special elastic and grim-reaper curves whose intrinsic equations are expressed in terms of elementary functions. In addition, we provide uniqueness results for the generatrix curve of the Enneper surface of second kind and for Lorentzian versions of some well-known curves in the Euclidean setting, like the Bernoulli lemniscate, the cardioid, the sinusoidal spirals and some non-degenerate conics. We are able to get arc-length parametrizations of them and they are depicted graphically.