arXiv subjects
Publications and source records attributed to Francisco Martín.
We give examples of proper minimal immersions in Euclidean space with very rapid area growth. The first is a proper embedding into $\bf{R}^4$ that yields a stable minimal surface, while the second is a proper immersion into $\bf{R}^3$. These results are motivated by [CM1] that shows that proper minimal submanifolds confined in space satisfy strong structural constraints.
We show that mean curvature flow translators may exhibit non-removable singularities at infinity, due to jump discontinuities in their asymptotic profiles, and that oscillation can persist so as to yield a continuum of subsequential limit tangent planes. Nonetheless, we prove that as time $t\to\pm \infty$, any finite entropy, finite genus, embedded, collapsed translating soliton in $\mathbb{R}^3$ converges to a uniquely determined collection of planes. This requires global analysis of quasilinear soliton equations with non-perturbative drifts, which we analyze via sharp non-standard elliptic decay estimates for the drift Laplacian, implying improvements on the Evans-Spruck and Ecker-Huisken estimates in the soliton setting, and exploiting a link from potential theory of the Yukawa equation to heat flows with $L^\infty$-data on non-compact slice curves of these solitons. The structure theorem follows: such solitons decompose at infinity into standard regions asymptotic to planes or grim reaper cylinders. As one application, we classify collapsed translators of entropy two with empty limits as $t\to +\infty$.
We complete the classification of semigraphical translators for mean curvature flow in $\mathbb{R}^3$ that was initiated by Hoffman-Martín-White. Specifically, we show that there is no solution to the translator equation on the upper half-plane with alternating positive and negative infinite boundary values, and we prove the uniqueness of pitchfork and helicoid translators. The proofs use Morse-Radó theory for translators and an angular maximum principle.
Accurate traversability estimation is essential for safe and effective navigation of outdoor robots operating in complex environments. This paper introduces a novel experience-based method that allows robots to autonomously learn which terrains are traversable based on prior navigation experience, without relying on extensive pre-labeled datasets. The approach integrates elevation and texture data into multi-layered grid maps, which are processed using a variational autoencoder (VAE) trained on a generic texture dataset. During an initial teleoperated phase, the robot collects sensory data while moving around the environment. These experiences are encoded into compact feature vectors and clustered using the BIRCH algorithm to represent traversable terrain areas efficiently. In deployment, the robot compares new terrain patches to its learned feature clusters to assess traversability in real time. The proposed method does not require training with data from the targeted scenarios, generalizes across diverse surfaces and platforms, and dynamically adapts as new terrains are encountered. Extensive evaluations on both synthetic benchmarks and real-world scenarios with wheeled and legged robots demonstrate its effectiveness, robustness, and superior adaptability compared to state-of-the-art approaches.
We study translating solitons for the mean curvature flow, $Σ^2\subseteq\mathbb{R}^3$ which are contained in slabs, and are of finite genus and finite entropy. As a first consequence of our results, we can enumerate connected components of slices to define asymptotic invariants $ω^\pm(Σ)\in\mathbb{N}$, which count the numbers of "wings''. Analyzing these, we give a method for computing the entropies $λ(Σ)$ via a simple formula involving the wing numbers, which in particular shows that for this class of solitons the entropy is quantized into integer steps. Finally, combining the concept of wing numbers with Morse theory for minimal surfaces, we prove the uniqueness theorem that if $Σ$ is a complete embedded simply connected translating soliton contained in a slab with entropy $λ(Σ)=3$ and containing a vertical line, then $Σ$ is one of the translating pitchforks of Hoffman-Martín-White
Obstructions to the existence of spacelike solitons depending on the growth of the mean curvature $H$ are proved for Lorentzian products $(M\times \mathbb{R}, \bar g=g_M-dt^2)$ with lowerly bounded curvature. The role of these bounds for both the completeness of the soliton $Σ$ and the applicability of the Omori-Yau principle for the drift Laplacian, is underlined. The differences between bounds in terms of the intrinsic distance $g$ of the soliton and the distance $g_M$ in the ambiance are analyzed, and lead to a revision of classic results on completeness for spacelike submanifolds. In particular, primary bounds, including affine $g$-bounds and logarithmic $g_M$-bounds, become enough to ensure both completeness and Omori-Yau's. Therefore, they become an obstruction to the existence of solitons when the ambiance Ricci is non-negative. These results, illustrated with a detailed example, deepen the uniqueness of solutions of elliptic equations which are not uniformly elliptic, providing insights into the interplay between mean curvature growth and the global properties of this geometric flow.
We prove a conjecture by Hoffman, White, and the first author regarding the uniqueness of pitchfork and helicoid translators of the mean curvature flow in $\mathbb{R}^3$. We employ an arc-counting argument motivated by Morse-Radó theory for translators and a rotational maximum principle. Applications to the classification of semigraphical translators in $\mathbb{R}^3$ and their limits are discussed, strengthening compactness results of the first author with Hoffman-White and with Gama-Moller.
We construct a family of complete, properly embedded, annular translators $M$ such that $M$ lies in a slab and is invariant under reflections in the vertical coordinate planes. Each translator in the family is asymptotic as $z\to -\infty$ to four vertical planes $\{y= \pm b\}$ and $\{y= \pm B\}$, where $0 0$, there is a translator in the family with inner width $b$ and with necksize $s$. (We also show that there are no translators with inner width $<π/2$ having the properties of the examples we construct.)
Social Robots need to be safe and reliable to share their space with humans. This paper reports on the first results of a research project that aims to create more safe and reliable, intelligent autonomous robots by investigating the implications and interactions between cybersecurity and safety. We propose creating a robotic intrusion prevention system (RIPS) that follows a novel approach to detect and mitigate intrusions in cognitive social robot systems and other cyber-physical systems. The RIPS detects threats at the robotic communication level and enables mitigation of the cyber-physical threats by using System Modes to define what part of the robotic system reduces or limits its functionality while the system is compromised. We demonstrate the validity of our approach by applying it to a cognitive architecture running in a real social robot that preserves the privacy and safety of humans while facing several cyber attack situations.
For a class of functions (called minimal Radó functions) that arise naturally in minimal surface theory, we bound the number of interior critical points (counting multiplicity) in terms of the boundary data and the Euler characteristic of the domain of the function.
The intelligent robotics community usually organizes knowledge into symbolic and sub-symbolic levels. These two levels establish the set of symbols and rules for manipulating knowledge based on their (symbol system - dictionary). Thus, the correspondences -- Grounding or knowledge representation -- require specific software techniques for anchoring continuous and discrete state variables between these two levels. This paper presents the design and evaluation of an Open Source tool called KANT(Knowledge mAnagemeNT) to let different components of the system architecture controlling the robot query, save, edit, and delete the data from the Knowledge Base without having to worry about the type and the implementation of the source data. Using KANT, components managing subsymbolic information can smoothly interact with symbolic components. Besides, implementation mechanisms used in KANT, such as the use of in-memory and non-SQL databases, improve the performance of the knowledge management systems in ROS middleware, as shown by the evaluations presented in this work.
We construct a one-parameter family of singly periodic translating solutions to mean curvature flow that converge as the period tends to $0$ to the union of a grim reaper surface and a plane that bisects it lengthwise. The surfaces are semigraphical: they are properly embedded, and, after removing a discrete collection of vertical lines, they are graphs. We also provide a nearly complete classification of semigraphical translators.
The purpose of these notes is to provide an introduction to those who want to learn more about translating solitons for the mean curvature flow in $\mathbb{R}^3$, particularly those which are complete graphs over domains in $\mathbb{R}^2$. In this paper we describe a full classification of complete translating graphs in $\mathbb{R}^3$.
Autonomous robots need to plan the tasks they carry out to fulfill their missions. The missions' increasing complexity does not let human designers anticipate all the possible situations, so traditional control systems based on state machines are not enough. This paper contains a description of the ROS2 Planning System (PlanSys2 in short), a framework for symbolic planning that incorporates novel approaches for execution on robots working in demanding environments. PlanSys2 aims to be the reference task planning framework in ROS2, the latest version of the {\em de facto} standard in robotics software development. Among its main features, it can be highlighted the optimized execution, based on Behavior Trees, of plans through a new actions auction protocol and its multi-robot planning capabilities. It already has a small but growing community of users and developers, and this document is a summary of the design and capabilities of this project.
Robots need task planning to sequence and execute actions toward achieving their goals. On the other hand, Behavior Trees provide a mathematical model for specifying plan execution in an intrinsically composable, reactive, and robust way. PDDL (Planning Domain Definition Language) has become the standard description language for most planners. In this paper, we present a novel algorithm to systematically create behavior trees from PDDL plans to execute them. This approach uses the execution graph of the plan to generate a behavior tree. The most remarkable contribution of this approach is the algorithm to build a Behavior Tree that optimizes its execution by paralyzing actions, applicable to any plan, taking into account the actions' causal relationships. We demonstrate the improvement in the execution of plans in mobile robots using the ROS2 Planning System framework.
Developments in mobile robot navigation have enabled robots to operate in warehouses, retail stores, and on sidewalks around pedestrians. Various navigation solutions have been proposed, though few as widely adopted as ROS Navigation. 10 years on, it is still one of the most popular navigation solutions. Yet, ROS Navigation has failed to keep up with modern trends. We propose the new navigation solution, Navigation2, which builds on the successful legacy of ROS Navigation. Navigation2 uses a behavior tree for navigator task orchestration and employs new methods designed for dynamic environments applicable to a wider variety of modern sensors. It is built on top of ROS2, a secure message passing framework suitable for safety critical applications and program lifecycle management. We present experiments in a campus setting utilizing Navigation2 to operate safely alongside students over a marathon as an extension of the experiment proposed in Eppstein et al. The Navigation2 system is freely available at https://github.com/ros-planning/navigation2 with a rich community and instructions.
We prove existence and uniqueness for a two-parameter family of translators for mean curvature flow. We get additional examples by taking limits at the boundary of the parameter space. Some of the translators resemble well-known minimal surfaces (Scherk's doubly periodic minimal surfaces, helicoids), but others have no minimal surface analogs. A one-parameter subfamily of the examples (the pitchforks) have finite topology and quadratic area growth, and thus might arise as blowups at singularities of initially smooth, closed surfaces flowing by mean curvature flow.
This note provides some new perspectives and calculations regarding an interesting known family of minimal surfaces in $\mathbb{H}^2 \times \mathbb{R}$. The surfaces in this family are the catenoids, parabolic catenoids and tall rectangles. Each is foliated by either circles, horocycles or circular arcs in horizontal copies of $\mathbb{H}^2$. All of these surfaces are well-known, but the emphasis here is on their unifying features and the fact that they lie in a single continuous family. We also initiate a study of the Jacobi operator on the parabolic catenoid, and compute the Jacobi fields associated to deformations to either of the two other types of surfaces in this family.