On maximal surfaces in the space of oriented geodesics of hyperbolic 3-space

We study area-stationary, or maximal, surfaces in the space ${\mathbb L}({\mathbb H}^3)$ of oriented geodesics of hyperbolic 3-space, endowed with the canonical neutral K\"ahler structure. We prove that every holomorphic curve in ${\mathbb L}({\mathbb H}^3)$ is a maximal surface. We then classify Lagrangian maximal surfaces $\Sigma$ in ${\mathbb L}({\mathbb H}^3)$ and prove that the family of parallel surfaces in ${\mathbb H}^3$ orthogonal to the geodesics $\gamma\in\Sigma$ form a family of equidistant tubes around a geodesic.Comment: 16 pages, AMS-Late

A new geometric structure on tangent bundles

For a Riemannian manifold $(N,g)$, we construct a scalar flat metric $G$ in the tangent bundle $TN$. It is locally conformally flat if and only if either, $N$ is a 2-dimensional manifold or, $(N,g)$ is a real space form. It is also shown that $G$ is locally symmetric if and only if $g$ is locally symmetric. We then study submanifolds in $TN$ and, in particular, find the conditions for a curve to be geodesic. The conditions for a Lagrangian graph to be minimal or Hamiltonian minimal in the tangent bundle $T{\mathbb R}^n$ of the Euclidean real space ${\mathbb R}^n$ are studied. Finally, using the cross product in ${\mathbb R}^3$ we show that the space of oriented lines in ${\mathbb R}^3$ can be minimally isometrically embedded in $T{\mathbb R}^3$.Comment: 23 pages, AMS-Te

Overcoming Probabilistic Faults in Disoriented Linear Search

We consider search by mobile agents for a hidden, idle target, placed on the infinite line. Feasible solutions are agent trajectories in which all agents reach the target sooner or later. A special feature of our problem is that the agents are $p$-faulty, meaning that every attempt to change direction is an independent Bernoulli trial with known probability $p$, where $p$ is the probability that a turn fails. We are looking for agent trajectories that minimize the worst-case expected termination time, relative to competitive analysis. First, we study linear search with one deterministic $p$-faulty agent, i.e., with no access to random oracles, $p\in (0,1/2)$. For this problem, we provide trajectories that leverage the probabilistic faults into an algorithmic advantage. Our strongest result pertains to a search algorithm (deterministic, aside from the adversarial probabilistic faults) which, as $p\to 0$, has optimal performance $4.59112+\epsilon$, up to the additive term $\epsilon$ that can be arbitrarily small. Additionally, it has performance less than $9$ for $p\leq 0.390388$. When $p\to 1/2$, our algorithm has performance $\Theta(1/(1-2p))$, which we also show is optimal up to a constant factor. Second, we consider linear search with two $p$-faulty agents, $p\in (0,1/2)$, for which we provide three algorithms of different advantages, all with a bounded competitive ratio even as $p\rightarrow 1/2$. Indeed, for this problem, we show how the agents can simulate the trajectory of any $0$-faulty agent (deterministic or randomized), independently of the underlying communication model. As a result, searching with two agents allows for a solution with a competitive ratio of $9+\epsilon$, or a competitive ratio of $4.59112+\epsilon$. Our final contribution is a novel algorithm for searching with two $p$-faulty agents that achieves a competitive ratio $3+4\sqrt{p(1-p)}$