Non-empty Boundary Research Articles

Boundary topology and rigidity results for generalized (λ, n + m)-Einstein manifolds

In this note, we deal with compact generalized (λ, n + m)-Einstein manifolds with nonempty boundary. More precisely, we extend some known results about compact static manifolds for (λ, n + m)-Einstein manifolds, giving topological classifications for its boundary. As a consequence, we obtain some characterizations of the round hemispheres.

Estimates of the Laplacian Spectrum and Bounds of Topological Invariants for Riemannian Manifolds with Boundary II

Abstract We present some estimate of the Laplacian Spectrum and of Topological Invariants for Riemannian manifold with pinched sectional curvature and with non-empty and non-convex boundary with finite injectivity radius. These estimates do not depend directly on the the lower bound of the boundary injectivity radius but on the bounds of the curvatures of the manifold and its boundary.

Strolling along gravitational vacua

We consider General Relativity (GR) on a space-time whose spatial slices are compact manifolds M with non-empty boundary ∂M. We argue that this theory has a non-trivial space of ‘vacua’, consisting of spatial metrics obtained by an action on a reference flat metric by diffeomorpisms that are non-trivial at the boundary. In an adiabatic limit the Einstein equations reduce to geodesic motion on this space of vacua with respect to a particular pseudo-Riemannian metric that we identify. We show how the momentum constraint implies that this metric is fully determined by data on the boundary ∂M only, while the Hamiltonian constraint forces the geodesics to be null. We comment on how the conserved momenta of the geodesic motion correspond to an infinite set of conserved boundary charges of GR in this setup.

Self-Adjoint Local Boundary Problems on Compact Surfaces. I. Spectral Flow

The paper deals with first order self-adjoint elliptic differential operators on a smooth compact oriented surface with non-empty boundary. We consider such operators with self-adjoint local boundary conditions. The paper is focused on paths in the space of such operators connecting two operators conjugated by a unitary automorphism. The first result is the computation of the spectral flow for such paths in terms of the topological data over the boundary. The second result is the universality of the spectral flow: we show that the spectral flow is a universal additive invariant for such paths, if the vanishing on paths of invertible operators is required. In the next paper of the series we generalize these results to families of such operators parametrized by points of an arbitrary compact space instead of an interval. The integer-valued spectral flow is replaced then by the family index taking values in the $K^1$-group of the base space.

$L^p$ Hardy inequality on $C^{1,\gamma}$ domains

We consider the $L^p$ Hardy inequality involving the distance to the boundary of a domain in the $n$-dimensional Euclidean space with nonempty compact boundary. We extend the validity of known existence and non-existence results, as well as the appropriate tight decay estimates for the corresponding minimizers, from the case of domains of class $C^2$ to the case of domains of class $C^{1,\gamma}$ with $\gamma \in (0,1]$. We consider both bounded and exterior domains. The upper and lower estimates for the minimizers in the case of exterior domains and the corresponding related non-existence result seem to be new even for $C^2$-domains.

Maximizing Steklov eigenvalues on surfaces

We study the Steklov eigenvalue functionals $\sigma_k (\Sigma, g) L_g (\partial \Sigma)$ on smooth surfaces with non-empty boundary. We prove that, under some natural gap assumptions, these functionals do admit maximal metrics which come with an associated minimal surface with free boundary from $\Sigma$ into some Euclidean ball, generalizing previous results by Fraser and Schoen in [“Sharp eigenvalue bounds and minimal surfaces in the ball,” Invent. Math., 203(3):823–890, 2016].

Derived equivalences of gentle algebras via Fukaya categories

Following the approach of Haiden–Katzarkov–Kontsevich (Publ Math Inst Hautes Études Sci 126:247–318, 2017), to any homologically smooth mathbb {Z}-graded gentle algebra A we associate a triple (Sigma _A, Lambda _A; eta _A), where Sigma _A is an oriented smooth surface with non-empty boundary, Lambda _A is a set of stops on partial Sigma _A and eta _A is a line field on Sigma _A, such that the derived category of perfect dg-modules of A is equivalent to the partially wrapped Fukaya category of (Sigma _A, Lambda _A ;eta _A). Modifying arguments of Johnson and Kawazumi, we classify the orbit decomposition of the action of the (symplectic) mapping class group of Sigma _A on the homotopy classes of line fields. As a result we obtain a sufficient criterion for homologically smooth graded gentle algebras to be derived equivalent. Our criterion uses numerical invariants generalizing those given by Avella–Alaminos–Geiss in Avella et al. (J Pure Appl Algebra 212(1):228–243, 2008), as well as some other numerical invariants. As an application, we find many new cases when the AAG-invariants determine the derived Morita class. As another application, we establish some derived equivalences between the stacky nodal curves considered in Lekili and Polishchuk (J Topology 11:615–444, 2018)

The smooth Riemannian extension problem

Given a metrically complete Riemannian manifold $(M,g)$ with smooth nonempty boundary and assuming that one of its curvatures is subject to a certain bound, we address the problem of whether it is possibile to realize $(M,g)$ as a domain inside a geodesically complete Riemannian manifold $(M',g')$ without boundary, by preserving the same curvature bounds. In this direction we provide three kind of results: (1) a general existence theorem showing that it is always possible to obtain a geodesically complete Riemannian extension without curvature constraints; (2) various topological obstructions to the existence of a complete Riemannian extension with prescribed sectional and Ricci curvature bounds; (3) some existence results of complete Riemannian extensions with sectional and Ricci curvature bounds, mostly in the presence of a convexity condition on the boundary.

Nonnegatively curved quotient spaces with boundary

Let M be a compact nonnegatively curved Riemannian manifold admitting an isometric action by a compact Lie group $$\mathsf {G}$$ in a way that the quotient space $$M/\mathsf {G}$$ has nonempty boundary. Let $$\pi : M \rightarrow M/\mathsf {G}$$ denote the quotient map and B be any boundary stratum of $$M/\mathsf {G}$$. Via a specific soul construction for $$M/\mathsf {G}$$, we construct a smooth closed submanifold N of M such that $$M {\setminus } \pi ^{-1}(B)$$ is diffeomorphic to the normal bundle of N. As an application we show that a simply connected torus manifold admitting an invariant metric of nonnegative curvature is rationally elliptic.

Estimates of the Laplacian Spectrum and Bounds of Topological Invariants for Riemannian Manifolds with Boundary

Abstract We set out to obtain estimates of the Laplacian Spectrum of Riemannian manifolds with non-empty boundary. This was achieved using standard doubled manifold techniques. In simple terms, we pasted two copies of the same manifold along their common boundary thereby obtaining a Riemannian manifold with empty boundary and with a C 0 −metric. This made it possible to adapt some estimates of the spectrum dependent on the volume or genus of the manifold as calculated in recent years by several authors. In order to extend further estimates that depend on the curvature, it is necessary to regularize the metric of the doubled manifold so that the new metric is isometric to that of each copy and such that the curvature has a finite lower bound. Controlling the curvature in this way also makes estimates of topological invariants available.

Virtual 3-Manifolds of Complexity 1 and 2

Matveev in 2009 introduced the notion of virtual 3-manifold, which generalizes the classical notion of 3-manifold. A virtual 3-manifold is an equivalence class of so-called special polyhedra. Each virtual 3-manifold determines a 3-manifold with nonempty boundary and ℝP2-singularities. Many invariants of manifolds, such as Turaev–Viro invariants, can be extended to virtual 3-manifolds. The complexity of a virtual 3-manifold is k if its equivalence class contains a special polyhedron with k true vertices and contains no special polyhedra with fewer true vertices. In this paper, we give a complete list of virtual 3-manifolds of complexity 1 and present two-sided bounds for the number of virtual 3-manifolds of complexity 2. The question of the complete classification for virtual 3-manifolds of complexity 2 remains open.

Hyperbolic groups with boundary an n-dimensional Sierpinski space

For [Formula: see text], we show that if [Formula: see text] is a torsion-free hyperbolic group whose visual boundary [Formula: see text] is an [Formula: see text]-dimensional Sierpinski space, then [Formula: see text] for some aspherical [Formula: see text]-manifold [Formula: see text] with non-empty boundary. Concerning the converse, we construct, for each [Formula: see text], examples of aspherical manifolds with boundary, whose fundamental group [Formula: see text] is hyperbolic, but with visual boundary [Formula: see text] not homeomorphic to [Formula: see text]. Our examples even support (metric) negative curvature, and have totally geodesic boundary.

The volume and Chern–Simons invariant of a Dehn-filled manifold

For a compact 3-manifold N with non-empty boundary, Zickert [12] gave a combinatorial formula for computing the volume and Chern–Simons invariant of a boundary parabolic representation π1(N)→PSL(2,C). In this paper, we introduce a notion of deformed Ptolemy varieties and extend the formula of Zickert to a representation that is not necessarily boundary parabolic. This allows us to compute the volume and Chern–Simons invariant of a PSL(2,C)-representation of a closed 3-manifold.

Improved parametrix in the glancing region for the interior Dirichlet-to-Neumann map

We study the semiclassical microlocal structure of the Dirichlet-to-Neumann map for an arbitrary compact Riemannian manifold with a nonempty smooth boundary. We build a new, improved parametrix in the glancing region compaired with that one built in [Vodev, Commun. Math. Phys. 2015;336(3):1141–1166; Vodev, Asymptotic Anal. 2018;106:147–168]. We also study the way in which the parametrix depends on the refraction index. As a consequence, we improve the transmission eigenvalue-free regions obtained in (Vodev, Asymptotic Anal. 2018;106:147–168) in the isotropic case when the restrictions of the refraction indices on the boundary coincide.

Secondary Polytope and Secondary Power Diagram

An ingenious construction of Gel’fand, Kapranov, and Zelevinsky [5] geometrizes the triangulations of a point configuration, such that all coherent triangulations form a convex polytope, the so-called secondary polytope. The secondary polytope can be treated as a weighted Delaunay triangulation in the space of all possible coherent triangulations. Naturally, it should have a dual diagram. In this work, we explicitly construct the secondary power diagram, which is the power diagram of the space of all possible power diagrams with nonempty boundary cells. Secondary power diagram gives an alternative proof for the classical secondary polytope theorem based on Alexandrov theorem. Furthermore, secondary power diagram theory shows one can transform a nondegenerated coherent triangulation to another nondegenerated coherent triangulation by a sequence of bistellar modifications, such that all the intermediate triangulations are nondegenerated and coherent. As an application of this theory, we propose an algorithm to triangulate a special class of 3d nonconvex polyhedra without using additional vertices. We prove that this algorithm terminates in $$O({{n}^{3}})$$ time.

Remarks on homotopy equivalence of configuration spaces of a polyhedron

We show that the configuration space F_n(M) of n particles in a compact connected PL manifold M with nonempty boundary partial M is homotopy equivalent to the configuration space F_n({mathrm{Int}}, M) where . Actually we prove some generalization of this result for polyhedra. Similar results recently have been obtained independently for topological manifolds by Zapata (Collision-free motion planning on manifolds with boundary, 2017. arXiv:1710.00293), using different techniques. We also address the question of whether a compact PL manifold M can be approximated up to homotopy type by discrete configuration spaces defined combinatorially via a simplicial subdivision of M.

Some isolation results for f-harmonic maps on weighted Riemannian manifolds with boundary

In this paper, we study f -harmonic (respectively, ( p , f ) -harmonic) maps on a compact weighted Riemannian manifold of nonempty boundary and of positive Bakry–Émery Ricci curvature. We first establish a Bochner–Reilly formula for such maps and deduce therefrom some immediate isolation results. In addition, using a weighted Sobolev inequality obtained in [12] , we prove that, under some energy level depending on the Bakry–Émery curvature of the initial manifold and on the sectional curvature of the final one, the only f -harmonic (respectively, ( p , f ) -harmonic) maps are the constant ones. A gap property of the energy density of f -harmonic maps from a weighted Riemannian manifold to a unit sphere is also obtained.

Singular solutions of elliptic equations on a perturbed cone

In this work we obtain positive singular solutions of{−Δu(y)=u(y)p in y∈Ωt,u=0 on y∈∂Ωt, where Ωt is a sufficiently small C2,α perturbation of the cone Ω:={x∈RN:x=rθ,r>0,θ∈S} where S⊂SN−1 has a smooth nonempty boundary and where p>1 satisfies suitable conditions. By singular solution we mean the solution is singular at the ‘vertex of the perturbed cone’. We also consider some other perturbations of the equation on the unperturbed cone Ω and here we use a different class of function spaces.

The mapping class group orbits in the framings of compact surfaces

AbstractWe compute the mapping class group orbits in the homotopy set of framings of a compact connected oriented surface with non-empty boundary. In the case g≥2, the computation is some modification of Johnson’s results (D. Johnson, Spin structures and quadratic forms on surfaces, J. London Math. Soc. (2)22 (1980), 365–373; D. Johnson, An abelian quotient of the mapping class group ℐg, Math. Ann.249 (1980), 225–242) and certain arguments on the Arf invariant, while we need an extra invariant for the genus 1 case. In addition, we discuss how this invariant behaves in the relative case, which Randal-Williams (O. Randal-Williams, Homology of the moduli spaces and mapping class groups of framed, r-Spin and Pin surfaces, J. Topology7 (2014), 155–186) studied for g≥2.

Two low-order nonconforming finite element methods for the Stokes flow in three dimensions

Abstract In this paper, we propose two low-order nonconforming finite element methods (FEMs) for the three-dimensional Stokes flow that generalize the nonconforming FEM of Kouhia & Stenberg (1995, A linear nonconforming finite element method for nearly incompressible elasticity and Stokes flow. Comput. Methods Appl. Mech. Eng, 124, 195–212). The finite element spaces proposed in this paper consist of two globally continuous components (one piecewise affine and one enriched component) and one component that is continuous at the midpoints of interior faces. We prove that the discrete Korn inequality and a discrete inf–sup condition hold uniformly in the mesh size and also for a nonempty Neumann boundary. Based on these two results, we show the well-posedness of the discrete problem. Two counterexamples prove that there is no direct generalization of the Kouhia–Stenberg FEM to three space dimensions: the finite element space with one nonconforming and two conforming piecewise affine components does not satisfy a discrete inf–sup condition with piecewise constant pressure approximations, while finite element functions with two nonconforming and one conforming component do not satisfy a discrete Korn inequality.