Smooth Submanifold Research Articles

Generalized Hamiltonian systems on subvarieties: constant rank case

For the constraint variety in symplectic manifold, the solvable Hamiltonian vector fields on the constraint are investigated. According to P.A.M. Dirac [3], the space of solvable Hamiltonian systems is determined by the geometric restriction of the symplectic form to the constraint. Solvability condition of the generalized Hamiltonian systems is extended to singular varieties and applied under some assumption on singularities. The constraint being a smooth submanifold in a symplectic space was considered in [6]. In this paper, we investigate the solvability of generalized Hamiltonian systems and the constraint invariants on singular constraints in the constant rank case.

A note on O6 intersections in AdS flux vacua

The DGKT-CFI construction of AdS flux vacua in type IIA string theory has interesting features such as classical moduli stabilization and a parametric scale separation between the Hubble scale and the Kaluza-Klein scale. A possible worry regarding the consistency of these vacua is that pathologies could arise due to intersections of the O6-planes, which are not well understood in the 10D solution. In this note, we show in explicit examples that such intersections are absent if one compactifies on smooth Calabi-Yau manifolds instead of toroidal orbifolds. In particular, we show that the blow-up of the T6/ℤ3 orbifold yields a single O6-plane which wraps a smooth submanifold without any (self-)intersections. On the other hand, blowing up the T6/ℤ32\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ {\\mathbb{Z}}_3^2 $$\\end{document} orbifold seems to yield an O6-plane which self-intersects. However, we show that this is due to an inconsistent orientifold involution in the original DGKT model. Imposing a consistent involution again yields a single smooth O6-plane without self-intersections on the blown-up manifold.

Disk-band graphs in the theory of framed tangles

A disk-band graph is a smooth compact two-dimensional manifold with boundary, partitioned into handles; the partition contains only index zero and index one handles and imitates the structure of the graph. (Index zero handles are analogues of the vertices of the graph, index one handles are analogues of the graph edges.) A disk-band graph is called spatial if it is a smooth submanifold of three-dimensional Euclidean space. A tangle is usually understood as a smooth compact one-dimensional submanifold of the standard three-dimensional ball that intersects the boundary of the ball orthogonally, only along its boundary, the intersection is contained in the equator. We call a tangle framed if it is equipped with a smooth field of normal straight lines. It is well known that there is a reduction of the problem of isotopic classification of spatial disk-band graphs to the problem of isotopic classification of framed tangles. This work focuses on the application of (abstract) disk-band graphs to study the set of isotopic classes of framed tangles.

Седловые связки

We prove that all vector fields that close to a field with the same saddle connections form a smooth Banach submanifold. Provide a sufficient condition for appearing of saddle connections in a generic family. Also we prove that after a generic perturbation of a monodromial hyperbolic polycycle formed by n saddle connections at least n limit cycles appear.

On the Quantum Deformations of Associative Sato Grassmannian Algebras and the Related Matrix Problems

We analyze the Lie algebraic structures related to the quantum deformation of the Sato Grassmannian, reducing the problem to studying co-adjoint orbits of the affine Lie subalgebra of the specially constructed loop diffeomorphism group of tori. The constructed countable hierarchy of linear matrix problems made it possible, in part, to describe some kinds of Frobenius manifolds within the Dubrovin-type reformulation of the well-known WDVV associativity equations, previously derived in topological field theory. In particular, we state that these equations are equivalent to some bi-Hamiltonian flows on a smooth functional submanifold with respect to two compatible Poisson structures, generating a countable hierarchy of commuting to each other’s hydrodynamic flows. We also studied the inverse problem aspects of the quantum Grassmannian deformation Lie algebraic structures, related with the well-known countable hierarchy of the higher nonlinear Schrödinger-type completely integrable evolution flows.

Nonlocal critical exponent singular problems under mixed Dirichlet-Neumann boundary conditions

In this paper, we study the following singular problem, under mixed Dirichlet-Neumann boundary conditions, and involving the fractional Laplacian(Pλ){(−Δ)su=λu−q+u2s⁎−1,u>0in Ω,A(u)=0on∂Ω=∑D∪∑N, where Ω⊂RN is a bounded domain with smooth boundary ∂Ω, 1/2<s<1, λ>0 is a real parameter, 0<q<1, N>2s, 2s⁎=2N/(N−2s) andA(u)=uX∑D+∂νuX∑N,∂ν=∂∂ν. Here ∑D, ∑N are smooth (N−1) dimensional submanifolds of ∂Ω such that ∑D∪∑N=∂Ω, ∑D∩∑N=∅ and ∑D∩∑N‾=τ′ is a smooth (N−2) dimensional submanifold of ∂Ω. Within a suitable range of λ, we establish existence of at least two opposite energy solutions for (Pλ) using the standard Nehari manifold technique.

Nonsmooth manifold decompositions

We study the structure induced on a smooth manifold by a continuous selection of smooth functions. In case such selection is suitably generic, it provides a stratification of the manifold, whose strata are algebraically defined smooth submanifolds. When the continuous selection has nondegenerate critical points, stratification descends to the local topological structure. We analyze this structure for the maximum of three smooth functions on a 4-manifold, which provides a new perspective on the theory of trisections.

Reconstruction of random geometric graphs: Breaking the [formula omitted] distortion barrier

Embedding graphs in a geographical or latent space, i.e. inferring locations for vertices in Euclidean space or on a smooth manifold or submanifold, is a common task in network analysis, statistical inference, and graph visualization. We consider the classic model of random geometric graphs where n points are scattered uniformly in a square of area n, and two points have an edge between them if and only if their Euclidean distance is less than r. The reconstruction problem then consists of inferring the vertex positions, up to the symmetries of the square, given only the adjacency matrix of the resulting graph. We give an algorithm that, if r=nα for any 0<α<1/2, with high probability reconstructs the vertex positions with a maximum error of O(nβ) where β=1/2−(4/3)α, until α≥3/8 where β=0 and the error becomes O(logn). This improves over earlier results, which were unable to reconstruct with error less than r. Our method estimates Euclidean distances using a hybrid of graph distances and short-range estimates based on the number of common neighbors. We extend our results to the surface of the sphere in R3 and to hypercubes in any constant fixed dimension.

Minimax estimation of distances on a surface and minimax manifold learning in the isometric-to-convex setting

Abstract We start by considering the problem of estimating intrinsic distances on a smooth submanifold. We show that minimax optimality can be obtained via a reconstruction of the surface, and discuss the use of a particular mesh construction—the tangential Delaunay complex—for that purpose. We then turn to manifold learning and argue that a variant of Isomap where the distances are instead computed on a reconstructed surface is minimax optimal for the isometric variant of the problem.

Fitting a manifold of large reach to noisy data

Let [Formula: see text] be a [Formula: see text]-smooth compact submanifold of dimension [Formula: see text]. Assume that the volume of [Formula: see text] is at most [Formula: see text] and the reach (i.e. the normal injectivity radius) of [Formula: see text] is greater than [Formula: see text]. Moreover, let [Formula: see text] be a probability measure on [Formula: see text] whose density on [Formula: see text] is a strictly positive Lipschitz-smooth function. Let [Formula: see text], [Formula: see text] be [Formula: see text] independent random samples from distribution [Formula: see text]. Also, let [Formula: see text], [Formula: see text] be independent random samples from a Gaussian random variable in [Formula: see text] having covariance [Formula: see text], where [Formula: see text] is less than a certain specified function of [Formula: see text] and [Formula: see text]. We assume that we are given the data points [Formula: see text] [Formula: see text], modeling random points of [Formula: see text] with measurement noise. We develop an algorithm which produces from these data, with high probability, a [Formula: see text] dimensional submanifold [Formula: see text] whose Hausdorff distance to [Formula: see text] is less than [Formula: see text] for [Formula: see text] and whose reach is greater than [Formula: see text] with universal constants [Formula: see text]. The number [Formula: see text] of random samples required depends almost linearly on [Formula: see text], polynomially on [Formula: see text] and exponentially on [Formula: see text].

Lower bounds on the low-distortion embedding dimension of submanifolds of Rn

Let M be a smooth submanifold of Rn equipped with the Euclidean (chordal) metric. This note considers the smallest dimension m for which there exists a bi-Lipschitz function f:M↦Rm with bi-Lipschitz constants close to one. The main result bounds the best achievable embedding dimension m below in terms of the Lipschitz constants of f as well as the reach, volume, diameter, and dimension of M. This new lower bound is then applied to show that prior upper bounds for m via random matrices by Eftekhari and Wakin [6] are near-optimal in a wide range of settings. This supports random linear maps as being nearly as efficient as nonlinear maps at reducing the ambient dimension for manifold data in many situations. Along the way, we also prove results concerning the impossibility of achieving better nonlinear measurement maps with the Restricted Isometry Property (RIP) in compressive sensing applications.

Transnormal Functions and Focal Varieties on Finsler Manifolds

In this paper, we study transnormal functions and their level sets and focal varieties on complete Finsler manifolds. We prove that the focal varieties of a $$C^{2}$$ transnormal function are smooth submanifolds and each regular level set is a tube over either of the focal varieties.

On the growth of generalized Fourier coefficients of restricted eigenfunctions

Let (M, g) be a smooth, compact, Riemannian manifold and a sequence of L 2-normalized Laplace eigenfunctions on M. For a smooth submanifold we consider the growth of the restricted eigenfunctions by testing them against a sequence of functions on H whose wavefront set avoids That is, we study what we call the generalized Fourier coefficients: We give an explicit bound on these coefficients depending on how the defect measures for the two collections of functions and ψh relate. This allows us to get a little– o improvement whenever the collection of recurrent directions over the wavefront set of ψh is small. To obtain our estimates, we utilize geodesic beam techniques.

Convexity estimates for high codimension mean curvature flow

We consider the flow by mean curvature of smooth n-dimensional submanifolds of Rn+k\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathbb {R}}^{n+k}$$\\end{document}, k≥2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$k \\ge 2$$\\end{document}, which are compact and quadratically pinched. We establish that such flows are asymptotically convex, that is, the first eigenvalue of the second fundamental form in the principal mean curvature direction blows up at a strictly slower rate than the mean curvature vector. This generalises the convexity estimate of Huisken–Sinestrari to higher codimensions. By combining our estimate with work of Naff, we conclude that singularity models for the flow are convex ancient solutions of codimension one.

Multiple solutions for some strongly degenerate second order elliptic equations

We consider a boundary value problem in a bounded domain involving a degenerate operator of the form L ( u ) = − div ( a ( x ) ∇ u ) and a suitable nonlinearity f. The function a vanishes on smooth 1-codimensional submanifolds of Ω where it is not allowed to be C 2 . By using weighted Sobolev spaces we are still able to find existence of solutions which vanish, in the trace sense, on the set where a vanishes.

On the Algebra of Operators Corresponding to the Union of Smooth Submanifolds

For a pair of smooth transversally intersecting submanifolds in some enveloping smooth manifold, we study the algebra generated by pseudodifferential operators and (co)boundary operators corresponding to submanifolds. We establish that such an algebra has 18 types of generating elements. For operators from this algebra, we define the concept of symbol and obtain the composition formula.

Multi-bump solutions for nonlinear elliptic equations involving critical Sobolev exponents

We study the existence of multi-bump solutions of −Δu−λu=up,u>0inΩt,u=0on∂Ωt,where λ is a real constant, N≥3, p=N+2N−2 , and Ωt is a domain that expands as t→∞. It is known that the topology of the domain Ωt affects the existence and non-existence of solutions. In this article, we prove the existence of multi-bump solutions for more general domains; for example, Ωtball≡{tq+v|q∈M,v∈(TqM)⊥,|v|<1} or Ωtann={tq+v|q∈M,v∈(TqM)⊥,a<|v|<a+1}, where a>0 and M is a k-dimensional compact smooth submanifold of RN without boundary (k=1,⋯,N−1).

Representation formulas and pointwise properties for Barron functions

We study the natural function space for infinitely wide two-layer neural networks with ReLU activation (Barron space) and establish different representation formulae. In two cases, we describe the space explicitly up to isomorphism. Using a convenient representation, we study the pointwise properties of two-layer networks and show that functions whose singular set is fractal or curved (for example distance functions from smooth submanifolds) cannot be represented by infinitely wide two-layer networks with finite path-norm. We use this structure theorem to show that the only $$C^1$$ -diffeomorphisms which preserve Barron space are affine. Furthermore, we show that every Barron function can be decomposed as the sum of a bounded and a positively one-homogeneous function and that there exist Barron functions which decay rapidly at infinity and are globally Lebesgue-integrable. This result suggests that two-layer neural networks may be able to approximate a greater variety of functions than commonly believed.

Local rigidity, symplectic homeomorphisms, and coisotropic submanifolds

We introduce the notion of a point on a locally closed subset of a symplectic manifold being ‘locally rigid’ with respect to that subset, prove that this notion is invariant under symplectic homeomorphisms, and show that coisotropic submanifolds are distinguished among all smooth submanifolds by the property that all of their points are locally rigid. This yields a simplified proof of the Humilière–Leclercq–Seyfaddini theorem on the C 0 $C^0$ -rigidity of coisotropic submanifolds. Connections are also made to the ‘rigid locus’ that has previously been used in the study of Chekanov–Hofer pseudo-metrics on orbits of closed subsets under the Hamiltonian diffeomorphism group.

Multi-Bump Standing Waves for Nonlinear Schrödinger Equations with a General Nonlinearity: The Topological Effect of Potential Wells

Abstract In this article, we are interested in multi-bump solutions of the singularly perturbed problem - ε 2 ⁢ Δ ⁢ v + V ⁢ ( x ) ⁢ v = f ⁢ ( v ) in ⁢ ℝ N . -\varepsilon^{2}\Delta v+V(x)v=f(v)\quad\text{in }\mathbb{R}^{N}. Extending previous results, we prove the existence of multi-bump solutions for an optimal class of nonlinearities f satisfying the Berestycki–Lions conditions and, notably, also for more general classes of potential wells than those previously studied. We devise two novel topological arguments to deal with general classes of potential wells. Our results prove the existence of multi-bump solutions in which the centers of bumps converge toward potential wells as ε → 0 {\varepsilon\rightarrow 0} . Examples of potential wells include the following: the union of two compact smooth submanifolds of ℝ N {\mathbb{R}^{N}} where these two submanifolds meet at the origin and an embedded topological submanifold of ℝ N {\mathbb{R}^{N}} .