Geometric Notions Research Articles

Partial differential equations and variational methods for geometric processing of images

This paper arose from a minisymposium held in 2018 at the 9th International Conference on Curves and Surface in Arcachon, France, and organized by Simon Masnou and Carola-Bibiane Schonlieb. This minisymposium featured a variety of recent developments of geometric partial differential equations and variational models which are directly or indirectly related to several problems in image and data processing. The current paper gathers three contributions which are in connection with the talks of three minisymposium speakers: Blanche Buet, Jean-Marie Mirebeau, and Yves van Gennip. The first contribution (Section 1) by Yves van Gennip provides a short overview of recent activity in the field of PDEs on graphs, without aiming to be exhaustive. The main focus is on techniques related to the graph Ginzburg–Landau variational model, but some other research in the field is also mentioned at the end of the section. The second contribution (Section 2), written by Jean-Marie Mirebeau, Francois Desquilbet, Johann Dreo, and Frederic Barbaresco presents a recent numerical method devoted to computing curves that globally minimize an energy featuring both a data driven term, and a second order curvature penalizing term. Applications to image segmentation are discussed, and recent progress on radar network configuration, in which the optimal curves represent an opponent’s trajectories, is described in detail. Lastly, Section 3 is devoted to a work by Blanche Buet, Gian Paolo Leonardi, and Simon Masnou on the definition and the approximation of weak curvatures for a large class of generalized surfaces, and in particular for point clouds, based on the geometric measure theoretic notion of varifolds.

Long hitting time for translation flows and L-shaped billiards

We consider the flow in direction $ \theta $ on a translation surface and we study the asymptotic behavior for $ r\to 0 $ of the time needed by orbits to hit the $ r $-neighborhood of a prescribed point, or more precisely the exponent of the corresponding power law, which is known as hitting time. For flat tori the limsup of hitting time is equal to the Diophantine type of the direction $ \theta $. In higher genus, we consider a generalized geometric notion of Diophantine type of a direction $ \theta $ and we seek for relations with hitting time. For genus two surfaces with just one conical singularity we prove that the limsup of hitting time is always less or equal to the square of the Diophantine type. For any square-tiled surface with the same topology the Diophantine type itself is a lower bound, and any value between the two bounds can be realized, moreover this holds also for a larger class of origamis satisfying a specific topological assumption. Finally, for the so-called Eierlegende Wollmilchsau origami, the equality between limsup of hitting time and Diophantine type subsists. Our results apply to L-shaped billiards.

Open Problems Column

This issue's Open Problem Column is by Stephen Fenner, Fred Green, and Steven Homer. It is titled: Fixed-Parameter Extrapolation and Aperiodic Order: Open Problems. It is not on xedparameter tractability. It is on a generalization of the geometric notion of convexity.

Metric and Geometric Relaxations of Self-Contracted Curves

The metric notion of a self-contracted curve (respectively, self-expanded curve, if we reverse the orientation) is hereby extended in a natural way. Two new classes of curves arise from this extension, both depending on a parameter, a specific value of which corresponds to the class of self-expanded curves. The first class is obtained via a straightforward metric generalization of the metric inequality that defines self-expandedness, while the second one is based on the (weaker) geometric notion of the so-called cone property (eel-curve). In this work, we show that these two classes are different; in particular, curves from these two classes may have different asymptotic behavior. We also study rectifiability of these curves in the Euclidean space, with emphasis in the planar case.

Normal crossings singularities for symplectic topology

We introduce topological notions of normal crossings symplectic divisor and variety and establish that they are equivalent, in a suitable sense, to the desired geometric notions. Our proposed concept of equivalence of associated topological and geometric notions fits ideally with important constructions in symplectic topology. This partially answers Gromov's question on the feasibility of defining singular symplectic (sub)varieties and lays foundation for rich developments in the future. In subsequent papers, we establish a smoothability criterion for symplectic normal crossings varieties, in the process providing the multifold symplectic sum envisioned by Gromov, and introduce symplectic analogues of logarithmic structures in the context of normal crossings symplectic divisors.

Min-max property in metric spaces with convex structure

In the setting of convex metric spaces, we introduce the two geometric notions of uniform convexity in every direction as well as sequential convexity. They are used to study a concept of proximal normal structure. We also consider the class of noncyclic relatively nonexpansive mappings and analyze the min-max property for such mappings. As an application of our main results we conclude with some best proximity pair theorems for noncyclic mappings.

Discrete projective minimal surfaces

We propose a natural discretisation scheme for classical projective minimal surfaces. We follow the classical geometric characterisation and classification of projective minimal surfaces and introduce at each step canonical discrete models of the associated geometric notions and objects. Thus, we introduce discrete analogues of classical Lie quadrics and their envelopes and classify discrete projective minimal surfaces according to the cardinality of the class of envelopes. This leads to discrete versions of Godeaux-Rozet, Demoulin and Tzitzéica surfaces. The latter class of surfaces requires the introduction of certain discrete line congruences which may also be employed in the classification of discrete projective minimal surfaces. The classification scheme is based on the notion of discrete surfaces which are in asymptotic correspondence. In this context, we set down a discrete analogue of a classical theorem which states that an envelope (of the Lie quadrics) of a surface is in asymptotic correspondence with the surface if and only if the surface is either projective minimal or a Q surface. Accordingly, we present a geometric definition of discrete Q surfaces and their relatives, namely discrete counterparts of classical semi-Q, complex, doubly Q and doubly complex surfaces.

On New Structures in the Theory of Fully Nonlinear Equations

We describe the current state of the theory of equations with m-Hessian stationary and evolution operators. It is quite important that new algebraic and geometric notions appear in this theory. In the present work, a list of those notions is provided. Among them, the notion of m-positivity of matrices is quite important; we provide a proof of an analog of Sylvester’s criterion for such matrices. From this criterion, we easily obtain necessary and sufficient conditions for existence of classical solutions of the first initial boundary-value problem for m-Hessian evolution equations. The asymptotic behavior of m-Hessian evolutions in a semibounded cylinder is considered as well.

Extracting Geometry from Quantum Spacetime

Any acceptable quantum gravity theory must allow us to recover the classical spacetime in the appropriate limit. Moreover, the spacetime geometrical notions should be intrinsically tied to the behavior of the matter that probes them. We consider some difficulties that would be confronted in attempting such an enterprise. The problems we uncover seem to go beyond the technical level to the point of questioning the overall feasibility of the project. The main issue is related to the fact that, in the quantum theory, it is impossible to assign a trajectory to a physical object, and, on the other hand, according to the basic tenets of the geometrization of gravity, it is precisely the trajectories of free localized objects that define the spacetime geometry. The insights gained in this analysis should be relevant to those interested in the quest for a quantum theory of gravity and might help refocus some of its goals.

Branches of the black hole wave function need not contain firewalls

We discuss the branching structure of the quantum-gravitational wave function that describes the evaporation of a black hole. A global wave function which initially describes a classical Schwarzschild geometry is continually decohered into distinct semiclassical branches by the emission of Hawking radiation. The laws of quantum mechanics dictate that the wave function evolves unitarily, but this unitary evolution is only manifest when considering the global description of the wave function; it is not implemented by time evolution on a single semiclassical branch. Conversely, geometric notions like the position or smoothness of a horizon only make sense on the level of individual branches. We consider the implications of this picture for probes of black holes by classical observers in definite geometries, like those involved in the AMPS construction. We argue that individual branches can describe semiclassical geometries free of firewalls, even as the global wave function evolves unitarily. We show that the pointer states of infalling detectors that are robust under Hamiltonian evolution are distinct from, and incompatible with, those of exterior detectors stationary with respect to the black hole horizon, in the sense that the pointer bases are related to each other via nontrivial transformations that mix the system, apparatus, and environment. This result describes a Hilbert-space version of black hole complementarity.

Factorization homology I: Higher categories

We construct a pairing, which we call factorization homology, between framed manifolds and higher categories. The essential geometric notion is that of a vari-framing of a stratified manifold, which is a framing on each stratum together with a coherent system of compatibilities of framings along links between strata. Our main result constructs labeling systems on disk-stratified vari-framed n-manifolds from (∞,n)-categories. These (∞,n)-categories, in contrast with the literature to date, are not required to have adjoints. This allows the following conceptual definition: the factorization homology∫MC of a framed n-manifold M with coefficients in an (∞,n)-category C is the classifying space of C-labeled disk-stratifications over M. The core calculation underlying our main result is the following: for any disk-stratified manifold, the space of conically smooth diffeomorphisms which preserve a vari-framing is discrete.

Kähler manifolds of quasi-constant holomorphic sectional curvature and generalized Sasakian space forms

In the present paper, two geometric notions, namely Kahler manifolds of quasi-constant holomorphic sectional curvature and generalized Sasakian space forms, are related to each other, for the first time. Some conditions under which each of these structures induces the other one, are provided here. Several results are obtained on direct products (which are special cases of Naveira’s classification), warped products or hypersurfaces of manifolds and relevant examples are included. A result of Niebergall and Ryan is generalized here. Some necessary and sufficient conditions for Einsteinian hypersurfaces are given at the end.

Pythagorean Property and Asymptotic Relatively Nonexpansive Mappings

The aim of this article is to study the existence of best proximity pairs for asymptotic relatively nonexpansive mappings using a geometric notion of the Pythagorean property. In this way, we establish an existence theorem under some different conditions with respect to the same result of Rajesh and Veeramani (Numer Funct Anal Optim 37:80–91, 2016). We also prove a best proximity point theorem for cyclic relatively nonexpansive mappings which are compact in the setting of reflexive Busemann convex spaces. An illustrative example will be presented to show the usability of our conclusions.

Geometric Classification of Graph C*-algebras over Finite Graphs

AbstractWe address the classification problem for graph C*-algebras of finite graphs (finitely many edges and vertices), containing the class of Cuntz-Krieger algebras as a prominent special case. Contrasting earlier work, we do not assume that the graphs satisfy the standard condition (K), so that the graph C*-algebras may come with uncountably many ideals.We find that in this generality, stable isomorphism of graph C*-algebras does not coincide with the geometric notion of Cuntz move equivalence. However, adding a modest condition on the graphs, the two notions are proved to be mutually equivalent and equivalent to the C*-algebras having isomorphicK-theories. This proves in turn that under this condition, the graph C*-algebras are in fact classifiable byK-theory, providing, in particular, complete classification when the C* - algebras in question are either of real rank zero or type I/postliminal. The key ingredient in obtaining these results is a characterization of Cuntz move equivalence using the adjacency matrices of the graphs.Our results are applied to discuss the classification problem for the quantumlens spaces defined by Hong and Szymański, and to complete the classification of graph C*-algebras associated with all simple graphs with four vertices or less.

Clearance-driven motion planning for mobile robots with differential constraints

SUMMARYThis paper presents an approach that integrates the geometric notion of clearance (distance to the closest obstacle) into sampling-based motion planning to enable a robot to safely navigate in challenging environments. To reach the goal destination, the robot must obey geometric and differential constraints that arise from the underlying motion dynamics and the characteristics of the environment. To produce safe paths, the proposed approach expands a motion tree of collision-free and dynamically feasible motions while maintaining locally maximal clearance. In distinction from related work, rather than explicitly constructing the medial axis, the proposed approach imposes a grid or a triangular tessellation over the free space and uses the clearance information to construct a weighted graph where edges that connect regions with low clearance have high cost. Minimum-cost paths over this graph produce high-clearance routes that tend to follow the medial axis without requiring its explicit construction. A key aspect of the proposed approach is a route-following component which efficiently expands the motion tree to closely follow such high-clearance routes. When expansion along the current route becomes difficult, edges in the tessellation are penalized in order to promote motion-tree expansions along alternative high-clearance routes to the goal. Experiments using vehicle models with second-order dynamics demonstrate that the robot is able to successfully navigate in complex environments. Comparisons to the state-of-the-art show computational speedups of one or more orders of magnitude.

Kähler-Einstein metrics: Old and New

Abstract We present classical and recent results on Kähler-Einstein metrics on compact complex manifolds, focusing on existence, obstructions and relations to algebraic geometric notions of stability (K-stability). These are the notes for the SMI course "Kähler-Einstein metrics" given by C.S. in Cortona (Italy), May 2017. The material is not intended to be original.

Issues Concerning Isometric Deformations of Planar Regions to Curved Surfaces

In this work, we make a distinction between the differential geometric notion of an isometry relationship among two dimensional surfaces embedded in three-dimensional point space and the continuum mechanical notion of an isometric deformation of a two dimensional material surface. We illustrate the importance of separating the abstract theory of surfaces in differential geometry and their related differential geometric features from the physical notion of a material surface which is subject to a deformation from a given reference configuration. In differential geometry, while two surfaces may be isometric, the mapping between them that characterizes the isometry is simply a mapping between the points of the surfaces and not necessarily between corresponding material particles of a single deformed material surface.We review two equivalent characterizations of a smooth isometric deformation of a flat material surface into a curved surface, and emphasize the requirement that the referential directrix and rulings, and their deformed counterparts, must provide a basis for establishing a complete curvilinear coordinate covering of the material surface in both the reference and deformed states. Because this covering requirement has been overlooked in recent publications concerning the isometric bending of ribbons, we illustrate its importance in properly defining the deformation of a ribbon in the two examples of a flat rectangular material strip that is isometrically deformed into either (i) a portion of a circular cylindrical surface, or (ii) a portion of a circular conical surface. We then show how the accurate calculation of the bending energy in these two examples is influenced by this oversight. In example (i), the curvature along the generators of the deformed surface, generally helical in form, is constant. In this special circumstance overlooking the covering requirement, as has been done in the literature by integrating the specific bending energy, dependent only on the curvature, over a domain on the supporting circular cylindrical surface equal in area, though not equal in geometric form, to that of the deformed ribbon, gives the correct bending energy result. In example (ii), the curvature along a generator of the cone is not constant and the calculation of the bending energy is, indeed, compromised by this oversight.The historically important dimensional reductions that Sadowsky and Wunderlich introduced to study the bending energy and the equilibrium configurations of isometrically deformed rectangular strips have gained classical notoriety within the subject of elastic ribbons and Möbius bands. We show that the Sadowsky and Wunderlich functionals also overlook the covering requirement and that the exact bending energy is underestimated by these functionals, the Sadowsky functional being the lowest. We then show that the error in using these functionals can be great for a rectangular strip of given length ell and width w, depending on the form of the isometric deformation and the size of the half-length-to-width ratio w/2ell. The Sadowsky functional is meant to apply to strips for which w/2ell is sufficiently small, in which case the covering requirement is of little consequence, and for such strips it yields an acceptable approximation of the actual bending energy. In such cases the Wunderlich functional shows only an incremental improvement over the Sadowsky calculation. While the Wunderlich functional is meant to apply accurately for all strips, without restricting the size of w/2ell, we show that in overlooking the covering requirement it greatly underestimates the actual bending energy for many isometrically deformed ribbons. In particular, we show relative errors between the exact, the Wunderlich, and the Sadowsky calculations of the bending energy as a function of w/2ell for the case of a rectangular strip which is isometrically deformed into a portion of a right circular conical surface, and we observe that the error in approximating the exact bending energy by the Wunderlich functional for reasonable ratios w/2ell is large and unacceptable. We then give an example of the isometric deformation of a rectangular strip whose Wunderlich functional predicts zero bending energy but for which the exact bending energy can be as large as one pleases.Finally, contrary to suggestions in the literature, we argue that Kirchhoff rod theory does not generally apply to the study of the isometric deformation of a thin rectangular strip because for this class of problems the through thickness dimension of the strip is assumed to be infinitesimal as compared to its width w. For Kirchhoff rod theory to apply, these dimensions must be comparable.

Reduced optimal models via cross Gramian for continuous linear time‐invariant systems

In this study, the authors focus on the problem of model order reduction (MOR) suitable for continuous linear time-invariant (LTI) systems. Specifically, for single-input single-output (SISO) LTI systems, a couple of MOR algorithms via the cross Gramian are presented. The authors first give a cross Gramian-based optimal MOR iterative algorithm to reduce the SISO system. Further, the authors explore the optimisation problem on the Stiefel manifold. Based on the geometric notions on this manifold, a conjugate gradient iterative algorithm (CGIAHM) is proposed. The conjugate gradient direction used to search for the minimiser is derived by applying the notion of vector transport. It is worth mentioning that the conjugate gradient used in the CGIAHM algorithm is a decent direction for the cost function due to the ingenious construction of this algorithm. In addition, the authors’ algorithms are extended to solve the optimal MOR problem for general LTI systems. Finally, two numerical examples demonstrate the effectiveness of their algorithms.

Boundary-bulk relation in topological orders

In this paper, we study the relation between an anomaly-free n+1D topological order, which are often called n+1D topological order in physics literature, and its nD gapped boundary phases. We argue that the n+1D bulk anomaly-free topological order for a given nD gapped boundary phase is unique. This uniqueness defines the notion of the “bulk” for a given gapped boundary phase. In this paper, we show that the n+1D “bulk” phase is given by the “center” of the nD boundary phase. In other words, the geometric notion of the “bulk” corresponds precisely to the algebraic notion of the “center”. We achieve this by first introducing the notion of a morphism between two (potentially anomalous) topological orders of the same dimension, then proving that the notion of the “bulk” satisfies the same universal property as that of the “center” of an algebra in mathematics, i.e. “bulk = center”. The entire argument does not require us to know the precise mathematical description of a (potentially anomalous) topological order. This result leads to concrete physical predictions.

Output Regulation by Error Dynamic Feedback in Hybrid Linear Systems with State Jumps

This work deals with a regulation problem for hybrid linear systems which exhibit a continuous-time state motion, ruled by the so-called flow dynamics, except at isolated points of the time axis, where the state has discontinuities governed by a jump behavior. Jump time instants are not a-priori known and may be unequally spaced, the only admissibility constraint being that the set of time intervals between consecutive jumps has a fixed positive lower bound. The considered problem consists in finding a hybrid error feedback compensator that forces the output of a given hybrid plant to asymptotically follow a reference trajectory generated by a hybrid exogenous system, while achieving global asymptotic stability of the closed-loop dynamics, for all the admissible sequences of jump times. The problem is investigated from a structural point of view, using geometric notions and properties. A sufficient condition for the existence of solutions is first stated in an implicit form, by considering the overall compensated system. This result is instrumental in giving a sufficient constructive condition which refers to the output-difference connection of the plant and the exogenous generator. The conditions are expressed in terms of hybrid invariant and controlled invariant subspaces of the systems at issue as well as of their stability and stabilizability properties, respectively. This approach provides a viable algorithmic procedure for the synthesis of solutions, if any exist.