Tubular Neighborhood Research Articles

On the regularity of the distance near the boundary of an obstacle

We study the regularity of the Euclidean distance function from a given point-wise target of a n-dimensional vector space in the presence of a compact obstacle bounded by a smooth hypersurface. It is known that such a function is semiconcave with fractional modulus one half. We provide a geometrical explanation of the exponent one half. Furthermore, under a natural (weak) assumption on the position of the point-wise target relatively to the obstacle, we show that there exists a point in the boundary of the obstacle so that no better regularity result holds near such a point. As a consequence of this result, we show that the Euclidean metric cannot be extended to a tubular neighborhood of the obstacle, as a Riemannian metric, keeping the property that the associated distances coincide outside the obstacle.

On Wilson’s theorem about domains of attraction and tubular neighborhoods

In this paper, we show that the domain of attraction of a compact asymptotically stable submanifold in a finite-dimensional smooth manifold of an autonomous system is homeomorphic to the submanifold’s tubular neighborhood. The compactness of the submanifold is crucial, without which this result is false; two counterexamples are provided to demonstrate this. • We show a fundamental result about the domain of attraction of a compact asymptotically stable submanifold. • We provide counterexamples to emphasize the key role of the compactness of the submanifold. • The aforementioned result refines the existing one and facilitates the topological analysis of autonomous systems.

A gluing construction of projective K3 surfaces

We construct a non-Kummer projective K3 surface $X$ which admits compact Levi-flats by holomorphically patching two open complex surfaces obtained as the complements of tubular neighborhoods of elliptic curves embedded in blow-ups of the projective plane at nine general points.

A countable partition for singular flows, and its application on the entropy theory

In this paper, we construct a countable partition $${\cal A}$$ for flows with hyperbolic singularities by introducing a new cross section at each singularity. Such partition forms a Kakutani tower in a neighborhood of the singularity, and turns out to have finite metric entropy for every invariant probability measure. Moreover, each element of $${{\cal A}^\infty}$$ will stay in a scaled tubular neighborhood for arbitrarily long time. This new construction enables us to study entropy theory for singular flows away from homoclinic tangencies, and show that the entropy function is upper semi-continuous with respect to both invariant measures and the flows, provided that the limiting measure satisfies μ(Sing(X)) = 0.

Spectral Properties of Relativistic Quantum Waveguides

We make a spectral analysis of the massive Dirac operator in a tubular neighborhood of an unbounded planar curve,subject to infinite mass boundary conditions. Under general assumptions on the curvature, we locate the essential spectrum and derive an effective Hamiltonian on the base curve which approximates the original operator in the thin-strip limit. We also investigate the existence of bound states in the non-relativistic limit and give a geometric quantitative condition for the bound states to exist.

Measurement of irreversibility and entropy production via the tubular ensemble

The appealing theoretical measure of irreversibility in a stochastic process, as the ratio of the probabilities of a trajectory and its time reversal, cannot be accessed directly in experiments since the probability of a single trajectory is zero. We regularize this definition by considering, instead, the limiting ratio of probabilities for trajectories to remain in the tubular neighborhood of a smooth path and its time reversal. The resulting pathwise medium entropy production agrees with the formal expression from stochastic thermodynamics and can be obtained from measurable tube probabilities. Estimating the latter from numerically sampled trajectories for Langevin dynamics yields excellent agreement with theory. By combining our measurement of pathwise entropy production with a Markov chain Monte Carlo algorithm, we infer the entropy-production distribution for a transition path ensemble directly from short recorded trajectories. Our work enables the measurement of irreversibility along individual paths and path ensembles in a model-free manner.

Corrected trapezoidal rules for singular implicit boundary integrals

We present new higher-order quadratures for a family of boundary integral operators re-derived using the approach introduced in Kublik et al. (2013) [7]. In this formulation, a boundary integral over a smooth, closed hypersurface is transformed into an equivalent volume integral defined in a sufficiently thin tubular neighborhood of the surface. The volumetric formulation makes it possible to use the simple trapezoidal rule on uniform Cartesian grids and relieves the need to use parameterization for developing quadrature. Consequently, typical point singularities in a layer potential extend along the surface's normal lines. We propose new higher-order corrections to the trapezoidal rule on the grid nodes around the singularities. This correction is based on local decompositions of the singularity and is dependent on the angle of approach to the singularity relative to the surface's principal curvature directions. The proposed decomposition, combined with the volumetric formulation, leads to a special quadrature error cancellation.

Strange attractors in a dynamical system inspired by a seasonally forced SIR model

We analyze a multiparameter periodically-forced dynamical system inspired in the SIR endemic model. We show that the condition on the basic reproduction numberR0<1 is not sufficient to guarantee the elimination of Infectious individuals due to a backward bifurcation. Using the theory of rank-one attractors, for an open subset in the space of parameters where R0<1, the flow exhibits persistent strange attractors. These sets are not confined to a tubular neighborhood in the phase space, shadow the ghost of a two-dimensional invariant torus and are numerically observable. Although numerical experiments have already suggested that periodically-forced biological models may exhibit observable chaos, a rigorous proof was not given before. Our results agree well with the empirical belief that intense seasonality induces chaos.This work provides a preliminary investigation of the interplay between seasonality, deterministic dynamics and the prevalence of strange attractors in a nonlinear forced system inspired by biology.

Tubular Excision and Steklov Eigenvalues

Given a closed manifold M and a closed connected submanifold \(N\subset M\) of positive codimension, we study the Steklov spectrum of the domain \(\varOmega _\varepsilon \subset M\) obtained by removing the tubular neighborhood of size \(\varepsilon \) around N. All nonzero eigenvalues in the mid-frequency range tend to infinity at a rate which depends only on the codimension of N in M. Eigenvalues above the mid-frequency range are also described: they tend to infinity following an unbounded sequence of clusters. This construction is then applied to obtain manifolds with unbounded perimeter-normalized spectral gap and to show the necessity of using the injectivity radius in some known isoperimetric-type upper bounds.

Construction of maximal functions associated with skewed cylinders generated by incompressible flows and applications

We construct a maximal function associated with a family of skewed cylinders. These cylinders, which are defined as tubular neighborhoods of trajectories of a mollified flow, appear in the study of fluid equations such as the Navier–Stokes equations and the Euler equations. We define a maximal function subordinate to these cylinders and show it is of weak type (1, 1) and strong type (p, p) by a covering lemma. As an application, we give an alternative proof for the higher-derivatives estimate of smooth solutions to the three-dimensional Navier–Stokes equations.

Boundary regularity and stability for spaces with Ricci bounded below

This paper studies the structure and stability of boundaries in noncollapsed {{,mathrm{RCD},}}(K,N) spaces, that is, metric-measure spaces (X,{mathsf {d}},{mathscr {H}}^N) with Ricci curvature bounded below. Our main structural result is that the boundary partial X is homeomorphic to a manifold away from a set of codimension 2, and is N-1 rectifiable. Along the way, we show effective measure bounds on the boundary and its tubular neighborhoods. These results are new even for Gromov–Hausdorff limits (M_i^N,{mathsf {d}}_{g_i},p_i) rightarrow (X,{mathsf {d}},p) of smooth manifolds with boundary, and require new techniques beyond those needed to prove the analogous statements for the regular set, in particular when it comes to the manifold structure of the boundary partial X. The key local result is an varepsilon -regularity theorem, which tells us that if a ball B_{2}(p)subset X is sufficiently close to a half space B_{2}(0)subset {mathbb {R}}^N_+ in the Gromov–Hausdorff sense, then B_1(p) is biHölder to an open set of {mathbb {R}}^N_+. In particular, partial X is itself homeomorphic to B_1(0^{N-1}) near B_1(p). Further, the boundary partial X is N-1 rectifiable and the boundary measure is Ahlfors regular on B_1(p) with volume close to the Euclidean volume. Our second collection of results involve the stability of the boundary with respect to noncollapsed mGH convergence X_irightarrow X. Specifically, we show a boundary volume convergence which tells us that the N-1 Hausdorff measures on the boundaries converge to the limit Hausdorff measure on partial X. We will see that a consequence of this is that if the X_i are boundary free then so is X.

BIFURCATIONS OF DOUBLE HETERODIMENSIONAL CYCLES WITH THREE SADDLE POINTS

In this paper, bifurcations of double heterodimensional cycles of an "∞" shape consisting of two saddles of (1,2) type and one saddle of (2,1) type are studied in three dimensional vector field. We discuss the gaps between returning points in transverse sections by establishing a local active coordinate system in the tubular neighborhood of unperturbed double heterodimensional cycles, through which the preservation of "∞"-shape double heterodimensional cycles is proved. We then get the existence of a new heteroclinic cycle consisting of two saddles of (1,2) type and one saddle of (2,1) type, which is composed of one big orbit linking <i>p</i>₁, <i>p</i>₃ and two orbits linking <i>p</i>₃, <i>p</i>₂ and <i>p</i>₂, <i>p</i>₁ respectively, and another heterodimensional cycle consisting of one saddle <i>p</i>₁ of (2,1) type and one saddle <i>p</i>₂ of (1,2) type, which is composed of one orbit starting from <i>p</i>₁ to <i>p</i>₂ and another orbit starting from <i>p</i>₂ to <i>p</i>₁. Moreover, the 1-fold and 2-fold large 1-heteroclinic cycle consisting of two saddles <i>p</i>₁ and <i>p</i>₃ of (1,2) type is also presented. As well as the coexistence of a 1-fold large 1-heteroclinic cycle and the "∞"-shape double heterodimensional cycles and the coexistence conditions are also given in the parameter space.

Topology of ambient manifolds of non-singular Morse – Smale flows with three periodic orbits

The purpose of this study is to establish the topological properties of three-dimensional manifolds which admit Morse – Smale flows without fixed points (non-singular or NMS-flows) and give examples of such manifolds that are not lens spaces. Despite the fact that it is known that any such manifold is a union of circular handles, their topology can be investigated additionally and refined in the case of a small number of orbits. For example, in the case of a flow with two non-twisted (having a tubular neighborhood homeomorphic to a solid torus) orbits, the topology of such manifolds is established exactly: any ambient manifold of an NMS-flow with two orbits is a lens space. Previously, it was believed that all prime manifolds admitting NMS-flows with at most three non-twisted orbits have the same topology. Methods. In this paper, we consider suspensions over Morse – Smale diffeomorphisms with three periodic orbits. These suspensions, in turn, are NMS-flows with three periodic trajectories. Universal coverings of the ambient manifolds of these flows and lens spaces are considered. Results. In this paper, we present a countable set of pairwise distinct simple 3-manifolds admitting NMS-flows with exactly three non-twisted orbits. Conclusion. From the results of this paper it follows that there is a countable set of pairwise distinct three-dimensional manifolds other than lens spaces, which refutes the previously published result that any simple orientable manifold admitting an NMS-flow with at most three orbits is lens space.

A spinning construction for virtual 1-knots and 2-knots, and the fiberwise and welded equivalence of virtual 1-knots

Spun-knots (respectively, spinning tori) in [Formula: see text] made from classical 1-knots compose an important class of 2-knots (respectively, embedded tori) contained in [Formula: see text]. Virtual 1-knots are generalizations of classical 1-knots. We generalize these constructions to the virtual 1-knot case by using what we call, in this paper, the spinning construction of submanifolds. The construction proceeds as follows: For a virtual 1-knot [Formula: see text], take an embedded circle [Formula: see text] contained in (a closed oriented surface [Formula: see text])×(a closed interval [Formula: see text]), where [Formula: see text] is called a representing surface in virtual 1-knot theory. Embed [Formula: see text] in [Formula: see text] by an embedding map [Formula: see text], and let [Formula: see text] stand for [Formula: see text] Regard the tubular neighborhood of [Formula: see text] in [Formula: see text] as the result of rotating [Formula: see text] around [Formula: see text]. Rotate [Formula: see text] together then with [Formula: see text]. When [Formula: see text], we obtain an embedded torus [Formula: see text]. We prove the following: The embedding type [Formula: see text] in [Formula: see text] depends only on [Formula: see text], and does not depend on [Formula: see text]. Furthermore, the submanifolds, [Formula: see text] and “the embedded torus made from [Formula: see text] by using Satoh’s method”, of [Formula: see text] are isotopic. Fiberwise equivalence of diagrams refers to fiberwise equivalence of tori in 4-space that lie over the diagrams. We prove that two virtual 1-knot diagrams [Formula: see text] and [Formula: see text] are fiberwise equivalent if and only if [Formula: see text] and [Formula: see text] are rotational welded equivalent (see the body of the paper for this definition). We generalize the construction in the virtual 1-knot case written in the first paragraph, and we also succeed to make a consistent construction of one-dimensional-higher submanifolds from any virtual two-dimensional knot. Note that Satoh’s method says nothing about the virtual 2-knot case. Rourke’s interpretation of Satoh’s method is that one puts “fiber-circles” on each point of each virtual 1-knot diagram. If there is no virtual branch point in a virtual 2-knot diagram, our way gives such fiber-circles to each point of the virtual 2-knot diagram. Furthermore we prove the following: If a virtual 2-knot diagram [Formula: see text] has a virtual branch point, [Formula: see text] cannot be covered by such fiber-circles. Hence Rourke’s method cannot be generalized to the virtual 2-knot case. Only the spinning construction introduced in this paper works for now.

Covariant series in the normal neighborhood of a submanifold

We consider the covariant series in a some normal neighborhood of a submanifold. Such a neighborhood is a generalization of the normal neighborhood of a point. We discuss how the coefficients of the covariant Taylor series of an arbitrary tensor field can be expressed in terms covariant derivatives of the torsion, Riemann curvature and the field under consideration. We also discuss the algorithm of calculating coefficients of a pseudo-Riemannian metric with respect to the corresponding metric connection without torsion. As an example, we calculate the covariant expansion of the Schwarzschild metric in the normal tubular neighborhood of a circular orbit up to fifth order using the Fermi coordinate system.

Existence, uniqueness and regularity of the projection onto differentiable manifolds

We investigate the maximal open domain {mathscr {E}}(M) on which the orthogonal projection map p onto a subset Msubseteq {{mathbb {R}}}^d can be defined and study essential properties of p. We prove that if M is a C^1 submanifold of {{mathbb {R}}}^d satisfying a Lipschitz condition on the tangent spaces, then {mathscr {E}}(M) can be described by a lower semi-continuous function, named frontier function. We show that this frontier function is continuous if M is C^2 or if the topological skeleton of M^c is closed and we provide an example showing that the frontier function need not be continuous in general. We demonstrate that, for a C^k-submanifold M with kge 2, the projection map is C^{k-1} on {mathscr {E}}(M), and we obtain a differentiation formula for the projection map which is used to discuss boundedness of its higher order differentials on tubular neighborhoods. A sufficient condition for the inclusion Msubseteq {mathscr {E}}(M) is that M is a C^1 submanifold whose tangent spaces satisfy a local Lipschitz condition. We prove in a new way that this condition is also necessary. More precisely, if M is a topological submanifold with Msubseteq {mathscr {E}}(M), then M must be C^1 and its tangent spaces satisfy the same local Lipschitz condition. A final section is devoted to highlighting some relations between {mathscr {E}}(M) and the topological skeleton of M^c.

Manifolds of amphiphilic bilayers: Stability up to the boundary

We consider the mass preserving L 2 -gradient flow of the strong scaling of the functionalized Cahn Hilliard gradient flow and establish the nonlinear stability of a manifold comprised of quasi-equilibrium bilayer distributions up to the manifold's boundary. In the limit of thin but non-zero interfacial width, ε ≪ 1 , the bilayer manifold is parameterized by meandering modes that describe the interfacial evolution. The normal coercivity of the manifold is limited by “pearling” modes that control the structure of the profile near the interface, these are weakly damped and can lead to the dynamic rupture of the interface. Amphiphilic interfaces may decrease energy by either lengthening or shortening, depending upon the amphiphilic mass distributed in the bulk. We introduce an implicitly defined parameterization of the interfacial shape that uncouples the length change from the parameters describing the shape and introduce a nonlinear projection onto the manifold from a surrounding neighborhood. The bilayer manifold has asymptotically large but finite dimension tuned to maximize normal coercivity while preserving the wave-number gap between the meandering and the pearling modes. Modulo a pearling stability assumption, we show that the manifold attracts nearby orbits into a tubular neighborhood about itself so long as the interfacial shape remains sufficiently smooth and far from self-intersection. In a companion paper, [8] , we identify open sets of initial data whose orbits converge to circular equilibrium after a significant transient, and derive a singularly perturbed interfacial evolution comprised of motion against curvature regularized by an asymptotically weak Willmore term.

Cohomology with local coefficients and knotted manifolds

We show how the classical notions of cohomology with local coefficients, CW-complex, covering space, homeomorphism equivalence, simple homotopy equivalence, tubular neighbourhood, and spinning can be encoded on a computer and used to calculate ambient isotopy invariants of continuous embeddings N↪M of one topological manifold into another. More specifically, we describe an algorithm for computing the homology Hn(X,A) and cohomology Hn(X,A) of a finite connected CW-complex X with local coefficients in a Zπ1X-module A when A is finitely generated over Z. It can be used, in particular, to compute the integral cohomology Hn(X˜H,Z) and induced homomorphism Hn(X,Z)→Hn(X˜H,Z) for the covering map p:X˜H→X associated to a finite index subgroup H<π1X, as well as the corresponding homology homomorphism. We illustrate an open-source implementation of the algorithm by using it to show that: (i) the degree 2 homology group H2(X˜H,Z) distinguishes between the homotopy types of the complements X⊂R4 of the spun Hopf link and Satoh's tube map of the welded Hopf link (these two complements having isomorphic fundamental groups and integral homology); (ii) the degree 1 homology homomorphism H1(p−1(B),Z)→H1(X˜H,Z) distinguishes between the homeomorphism types of the complements X⊂R3 of the granny knot and the reef knot, where B⊂X is the knot boundary (these two complements again having isomorphic fundamental groups and integral homology). Our open source implementation allows the user to experiment with further examples of knots, knotted surfaces, and other embeddings of spaces. We conclude the paper with an explanation of how the cohomology algorithm also provides an approach to computing the set [W,X]ϕ of based homotopy classes of maps f:W→X of finite CW-complexes over a fixed group homomorphism π1f=ϕ in the case where dim⁡W=n, π1X is finite and πiX=0 for 2≤i≤n−1.

Bordifications of hyperplane arrangements and their curve complexes

The complement of an arrangement of hyperplanes in $\mathbb C^n$ has a natural bordification to a manifold with corners formed by removing (or "blowing up") tubular neighborhoods of the hyperplanes and certain of their intersections. When the arrangement is the complexification of a real simplicial arrangement, the bordification closely resembles Harvey's bordification of moduli space. We prove that the faces of the universal cover of the bordification are parameterized by the simplices of a simplicial complex $\mathcal{C}$, the vertices of which are the irreducible "parabolic subgroups" of the fundamental group of the arrangement complement. So, the complex $\mathcal{C}$ plays a similar role for an arrangement complement as the curve complex does for moduli space. Also, in analogy with curve complexes and with spherical buildings, we prove that $\mathcal{C}$ has the homotopy type of a wedge of spheres.

Anisotropic tubular neighborhoods of sets

Let E subset {{mathbb {R}}}^N be a compact set and Csubset {{mathbb {R}}}^N be a convex body with 0in mathrm{int},C. We prove that the topological boundary of the anisotropic enlargement E+rC is contained in a finite union of Lipschitz surfaces. We also investigate the regularity of the volume function V_E(r):=|E+rC| proving a formula for the right and the left derivatives at any r>0 which implies that V_E is of class C^1 up to a countable set completely characterized. Moreover, some properties on the second derivative of V_E are proved.