Tubular Neighborhood Research Articles

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.

Poisson structures of near-symplectic manifolds and their cohomology

We connect Poisson and near-symplectic geometry by showing that there is a singular Poisson structure on a near-symplectic 4-manifold. The Poisson structure π is defined on the tubular neighborhood of the singular locus Zω of the 2-form ω; it is of maximal rank 4, and it vanishes on a degeneracy set containing Zω. We compute its smooth Poisson cohomology, which depends on the modular vector field, and it is finite dimensional. We conclude with a discussion on the relation between the Poisson structure π and the overtwisted contact structure associated with a near-symplectic 4-manifold.

Twisted Alexander modules of hyperplane arrangement complements

We study torsion properties of the twisted Alexander modules of the affine complement M of a complex essential hyperplane arrangement, as well as those of punctured stratified tubular neighborhoods of complex essential hyperplane arrangements. We investigate divisibility properties between the twisted Alexander polynomials of the two spaces, compute the (first) twisted Alexander polynomial of a punctured stratified tubular neighborhood of an essential line arrangement, and study the possible roots of the twisted Alexander polynomials of both the complement and the punctured stratified tubular neighborhood of an essential hyperplane arrangement in higher dimensions. We apply our results to distinguish non-homeomorphic homotopy equivalent arrangement complements. We also relate the twisted Alexander polynomials of M with the corresponding twisted homology jump loci.

On sharp fronts and almost-sharp fronts for singular SQG

In this paper we consider a family of active scalars with a velocity field given by u=Λ−1+α∇⊥θ, for α∈(0,1). This family of equations is a more singular version of the two-dimensional Surface Quasi-Geostrophic (SQG) equation, which would correspond to α=0.We consider the evolution of sharp fronts by studying families of almost-sharp fronts. These are smooth solutions with simple geometry in which a sharp transition in the solution occurs in a tubular neighbourhood (of size δ). We study their evolution and that of compatible curves, and introduce the notion of a spine for which we obtain improved evolution results, gaining a full power (of δ) compared to other compatible curves.

Shrinking targets for the geodesic flow on geometrically finite hyperbolic manifolds

&lt;p style='text-indent:20px;'&gt;Let &lt;inline-formula&gt;&lt;tex-math id="M1"&gt;\begin{document}$ \mathscr{M} $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt; be a geometrically finite hyperbolic manifold. We present a very general theorem on the shrinking target problem for the geodesic flow, using its exponential mixing. This includes a strengthening of Sullivan's logarithm law for the excursion rate of the geodesic flow. More generally, we prove logarithm laws for the first hitting time for shrinking cusp neighborhoods, shrinking tubular neighborhoods of a closed geodesic, and shrinking metric balls, as well as give quantitative estimates for the time a generic geodesic spends in such shrinking targets.&lt;/p&gt;

External Bifurcations of Double Heterodimensional Cycles with One Orbit Flip

In this paper, external bifurcations of heterodimensional cycles connecting three saddle points with one orbit flip, in the shape of “∞”, are studied in three-dimensional vector field. We construct a poincaré return map between returning points in a transverse section by establishing a locally active coordinate system in the tubular neighborhood of unperturbed double heterodimensional cycles, through which the bifurcation equations are obtained under different conditions. Near the double heterodimensional cycles, the authors prove the preservation of “∞”-shape double heterodimensional cycles and the existence of the second and third shape heterodimensional cycle and a large 1-heteroclinic cycle connecting with P1 and P3. 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.

Taming the pseudoholomorphic beasts in ℝ × (S1× S2)

For a closed oriented smooth 4-manifold X with $b^2_+(X)>0$, the Seiberg-Witten invariants are well-defined. Taubes' "SW=Gr" theorem asserts that if X carries a symplectic form then these invariants are equal to well-defined counts of pseudoholomorphic curves, Taubes' Gromov invariants. In the absence of a symplectic form there are still nontrivial closed self-dual 2-forms which vanish along a disjoint union of circles and are symplectic elsewhere. This paper and its sequel describes well-defined counts of pseudoholomorphic curves in the complement of the zero-set of such "near-symplectic" forms, and it is shown that they recover the Seiberg-Witten invariants (mod 2). This is an extension of Taubes' "SW=Gr" theorem to non-symplectic 4-manifolds. The main result of this paper asserts the following. Given a suitable near-symplectic form w, a tubular neighborhood N of its zero-set, and a generic w-compatible almost complex structure J on X-N, there are well-defined counts of J-holomorphic curves in a completion of the symplectic cobordism (X-N, w) which are asymptotic to Reeb orbits on the ends. They can be packaged together to form "near-symplectic" Gromov invariants as a function of spin-c structures on X.

Euler-like vector fields, normal forms, and isotropic embeddings

As shown by Bursztyn et al. (2019), germs of tubular neighborhood embeddings for submanifolds N⊆M are in one–one correspondence with germs of Euler-like vector fields near N. In many contexts, this reduces the proof of ‘normal forms results’ for geometric structures to the construction of an Euler-like vector field compatible with the given structure. We illustrate this principle in a variety of examples, including the Morse–Bott lemma, Weinstein’s Lagrangian embedding theorem, and Zung’s linearization theorem for proper Lie groupoids. In the second part of this article, we extend the theory to a weighted context, with an application to isotropic embeddings.

Ritz method for transition paths and quasipotentials of rare diffusive events

The probability of trajectories of weakly diffusive processes to remain in the tubular neighbourhood of a smooth path is given by the Freidlin-Wentzell-Graham theory of large deviations. The most probable path between two states (the instanton) and the leading term in the logarithm of the process transition density (the quasipotential) are obtained from the minimum of the Freidlin-Wentzell action functional. Here we present a Ritz method that searches for the minimum in a space of paths constructed from a global basis of Chebyshev polynomials. The action is reduced, thereby, to a multivariate function of the basis coefficients, whose minimum can be found by nonlinear optimization. For minimisation regardless of path duration, this procedure is most effective when applied to a reparametrisation-invariant "on-shell" action, which is obtained by exploiting a Noether symmetry and is a generalisation of the scalar work [Olender and Elber, 1997] for gradient dynamics and the geometric action [Heyman and Vanden-Eijnden, 2008] for non-gradient dynamics. Our approach provides an alternative to chain-of-states methods for minimum energy paths and saddlepoints of complex energy landscapes and to Hamilton-Jacobi methods for the stationary quasipotential of circulatory fields. We demonstrate spectral convergence for three benchmark problems involving the Muller-Brown potential, the Maier-Stein force field and the Egger weather model.

On metrics with minimal singularities of line bundles whose stable base loci admit holomorphic tubular neighborhoods

We investigate the minimal singularities of metrics on a big line bundle $L$ over a projective manifold when the stable base locus $Y$ of $L$ is a submanifold of codimension $r\geq 1$. Under some assumptions on the normal bundle and a neighborhood of $Y$, we give a explicit description of the minimal singularity of metrics on $L$. We apply this result to study a higher (co-)dimensional analogue of Zariski's example, in which the line bundle $L$ is not semi-ample, however it is nef and big.

On Bennequin-type inequalities for links in tight contact 3-manifolds

We prove that a version of the Thurston–Bennequin inequality holds for Legendrian and transverse links in a rational homology contact 3-sphere [Formula: see text], whenever [Formula: see text] is tight. More specifically, we show that the self-linking number of a transverse link [Formula: see text] in [Formula: see text], such that the boundary of its tubular neighborhood consists of incompressible tori, is bounded by the Thurston norm [Formula: see text] of [Formula: see text]. A similar inequality is given for Legendrian links by using the notions of positive and negative transverse push-off. We apply this bound to compute the tau-invariant for every strongly quasi-positive link in [Formula: see text]. This is done by proving that our inequality is sharp for this family of smooth links. Moreover, we use a stronger Bennequin inequality, for links in the tight 3-sphere, to generalize this result to quasi-positive links and determine their maximal self-linking number.

Stability, analyticity, and maximal regularity for parabolic finite element problems on smooth domains

In this paper, we consider the finite element semidiscretization for a parabolic problem on a smooth domain $\Omega \subset \mathbb {R}^N$ with the Neumann boundary condition. We emphasize that the domain can be nonconvex in general. We discretize the this problem by the finite element method by constructing a family of polygonal or polyhedral domains $\{ \Omega _h \}_h$ that approximate the original domain $\Omega$. The aim of this study is to derive the smoothing property for the discrete parabolic semigroup and the maximal regularity for the discrete elliptic operator. The main difficulty is the effect of the boundary-skin (symmetric difference) $\Omega \bigtriangleup \Omega _h$. In order to address the effect of the boundary-skin, we introduce the tubular neighborhood of the original boundary $\partial \Omega$.

Schwarz Solvers and Preconditioners for the Closest Point Method

The discretization of surface intrinsic elliptic partial differential equations (PDEs) poses interesting challenges not seen in flat spaces. The discretization of these PDEs typically proceeds by either parametrizing the surface, triangulating the surface, or embedding the surface in a higher dimensional flat space. The closest point method (CPM) is an embedding method that represents surfaces using a function that maps points in the embedding space to their closest points on the surface. In the CPM, this mapping also serves as an extension operator that brings surface intrinsic data onto the embedding space, allowing PDEs to be numerically approximated by standard methods in a narrow tubular neighborhood of the surface. We focus on numerically approximating the positive Helmholtz equation, $\left(c-\Delta_{\mathcal{S}}\right)u=f,~c\in\mathbb{R}^+$, by the CPM paired with finite differences. This yields a large, sparse, and nonsymmetric system to be solved. Herein, we develop restricted additive Schwarz (RAS) and optimized restricted additive Schwarz (ORAS) solvers and preconditioners for this discrete system. In particular, we develop a general strategy for computing overlapping partitions of the computational domain and defining the corresponding Dirichlet and Robin transmission conditions. We demonstrate that the convergence of the ORAS solvers and preconditioners can be improved by using a modified transmission condition where more than two overlapping subdomains meet. Numerical experiments are provided for a variety of analytical and triangulated surfaces. We find that ORAS solvers and preconditioners outperform their RAS counterparts and that, as expected using domain decomposition (DD) as a preconditioner rather than as a solver gives faster convergence. The methods exhibit good parallel scalability over the range of process counts tested.

The surface diffusion and the Willmore flow for uniformly regular hypersurfaces

We consider the surface diffusion and Willmore flows acting on a general class of (possibly non–compact) hypersurfaces parameterized over a uniformly regular reference manifold possessing a tubular neighborhood with uniform radius. The surface diffusion and Willmore flows each give rise to a fourth–order quasilinear parabolic equation with nonlinear terms satisfying a specific singular structure. We establish well–posedness of both flows for initial surfaces that are $ C^{1+\alpha} $–regular and parameterized over a uniformly regular hypersurface. For the Willmore flow, we also show long–term existence for initial surfaces which are $ C^{1+\alpha} $–close to a sphere, and we prove that these solutions become spherical as time goes to infinity.

A Second order minimality condition for a free-boundary problem

The goal of this paper is to derive in the two-dimensional case necessary and sufficient minimality conditions in terms of the second variation for the functional \[ v\mapsto\int_{\Omega}\big(|\nabla v|^{2}+\chi_{\{v>0\}}Q^{2} \big)\,dx, \] introduced in a classical paper of Alt and Caffarelli. For a special choice of $Q$ this includes water waves. The second variation is obtained by computing the second derivative of the functional along suitable variations of the free boundary. It is proved that the strict positivity of the second variation gives a sufficient condition for local minimality. Also, it is shown that smooth critical points are local minimizers in a small tubular neighborhood of the free-boundary.

Tubular neighborhoods of orbits of power-logarithmic germs

We consider a class of power-logarithmic germs. We call them parabolic Dulac germs, as they appear as Dulac germs (first-return germs) of hyperbolic polycycles. In view of formal or analytic characterization of such a germ f by fractal properties of several of its orbits, we study the tubular $$\varepsilon $$ -neighborhoods of orbits of f with initial points $$x_0$$ . We denote by $$A_f(x_0,\varepsilon )$$ the length of such a tubular $$\varepsilon $$ -neighborhood. We show that, even if f is an analytic germ, the function $$\varepsilon \mapsto A_f(x_0,\varepsilon )$$ does not have a full asymptotic expansion in $$\varepsilon $$ in the scale of powers and (iterated) logarithms. Hence, this partial asymptotic expansion cannot contain necessary information for analytic classification. In order to overcome this problem, we introduce a new notion: the continuous time length of the $$\varepsilon $$ -neighborhood $$A^c_f(x_0,\varepsilon )$$ . We show that this function has a full transasymptotic expansion in $$\varepsilon $$ in the power, iterated logarithm scale. Moreover, its asymptotic expansion extends the initial, existing part of the asymptotic expansion of the classical length $$\varepsilon \mapsto A_f(x_0,\varepsilon )$$ . Finally, we prove that this initial part of the asymptotic expansion determines the class of formal conjugacy of the Dulac germ f.