Smooth Compact Manifold Research Articles

On the isometric version of Whitney's strong embedding theorem

We prove a version of Whitney's strong embedding theorem for isometric embeddings within the general setting of the Nash-Kuiper h-principle. More precisely, we show that any n-dimensional smooth compact manifold admits infinitely many global isometric embeddings into 2n-dimensional Euclidean space, of Hölder class C1,θ with θ<1/3 for n=2 and θ<(n+2)−1 for n≥3. The proof is performed by Nash-Kuiper's convex integration construction and applying the gluing technique of the authors on short embeddings with small amplitude.

Approximated harmonic maps with tension fields in Zygmund class

Suppose that u is a map from D8 to a compact smooth Riemannian manifold N with bounded energy. We show that there exists a constant λ>0 which depends only on N and E(u,D8) such that if the tension field τ belongs to Zygmund class Llnλ⁡L(D8), then the Hopf differential of u belongs to the Zygmund class Lln3⁡L(D1) and the norm ‖h‖Lln3⁡L(D1) depends only on N,E(u,D8) and ‖τ‖Llnλ⁡L(D8). As a direct corollary, we obtain the energy identity and necklessness of a blow-up sequence un with bounded energy E(un) and bounded τ(un) in Llnλ⁡L(D8).

The 𝐿𝑝 restriction bounds for Neumann data on surface

Abstract Let { u λ } \{u_{\lambda}\} be a sequence of L 2 L^{2} -normalized Laplacian eigenfunctions on a compact two-dimensional smooth Riemanniann manifold ( M , g ) (M,g) . We seek to get an L p L^{p} restriction bound of the Neumann data λ − 1 ⁢ ∂ ν u λ | γ \lambda^{-1}\partial_{\nu}u_{\lambda}|_{\gamma} along a unit geodesic 𝛾. Using the 𝑇- T ∗ T^{*} argument, one can transfer the problem to an estimate of the norm of a Fourier integral operator and show that such bound is O ⁢ ( λ − 1 p + 3 2 ) O(\lambda^{-\frac{1}{p}+\frac{3}{2}}) . The stationary phase theorem plays the crucial role in our proof. Moreover, this upper bound is shown to be optimal.

Central limit theorem for intrinsic Fréchet means in smooth compact Riemannian manifolds

We prove a central limit theorem (CLT) for the Fréchet mean of independent and identically distributed observations in a compact Riemannian manifold assuming that the population Fréchet mean is unique. Previous general CLT results in this setting have assumed that the cut locus of the Fréchet mean lies outside the support of the population distribution. In this paper we present a CLT under some mild technical conditions on the manifold plus the following assumption on the population distribution: in a neighbourhood of the cut locus of the population Fréchet mean, the population distribution is absolutely continuous with respect to the volume measure on the manifold and in this neighhbourhood the Radon–Nikodym derivative has a version that is continuous. So far as we are aware, the CLT given here is the first which allows the cut locus to have co-dimension one or two when it is included in the support of the distribution. A key part of the proof is establishing an asymptotic approximation for the parallel transport of a certain vector field. Whether or not a non-standard term arises in the CLT depends on whether the co-dimension of the cut locus is one or greater than one: in the former case a non-standard term appears but not in the latter case. This is the first paper to give a general and explicit expression for the non-standard term which arises when the co-dimension of the cut locus is one.

Existence of singular isoperimetric regions in 8-dimensional manifolds

It is well known that isoperimetric regions in a smooth compact (n+1)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(n+1)$$\\end{document}-manifold are themselves smooth, up to a closed set of codimension at most 8. In this note, we construct an 8-dimensional compact smooth manifold whose unique isoperimetric region with half volume that of the manifold exhibits two isolated singularities. This stands in contrast with the situation in which a manifold is a space form, where isoperimetric regions are smooth in every dimension.

Spectral asymptotics for linear elasticity: the case of mixed boundary conditions

We establish two-term spectral asymptotics for the operator of linear elasticity with mixed boundary conditions on a smooth compact Riemannian manifold of arbitrary dimension. We illustrate our results by explicit examples in dimension two and three, thus verifying our general formulae both analytically and numerically.

Dimension estimates and approximation in non-uniformly hyperbolic systems

Abstract Let $f: M\rightarrow M$ be a $C^{1+\alpha }$ diffeomorphism on an $m_0$ -dimensional compact smooth Riemannian manifold M and $\mu $ a hyperbolic ergodic f-invariant probability measure. This paper obtains an upper bound for the stable (unstable) pointwise dimension of $\mu $ , which is given by the unique solution of an equation involving the sub-additive measure-theoretic pressure. If $\mu $ is a Sinai–Ruelle–Bowen (SRB) measure, then the Kaplan–Yorke conjecture is true under some additional conditions and the Lyapunov dimension of $\mu $ can be approximated gradually by the Hausdorff dimension of a sequence of hyperbolic sets $\{\Lambda _n\}_{n\geq 1}$ . The limit behaviour of the Carathéodory singular dimension of $\Lambda _n$ on the unstable manifold with respect to the super-additive singular valued potential is also studied.

$C^{\ast}$-algebraic approach to the principal symbol. III

We treat the notion of principal symbol mapping on a compact smooth manifold as a \ast -homomorphism of C^{\ast} -algebras. Principal symbol mapping is built from the ground, without referring to the pseudodifferential calculus on the manifold. Our concrete approach allows us to extend Connes trace theorem for compact Riemannian manifolds.

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.

Determining Lamé coefficients by the elastic Dirichlet-to-Neumann map on a Riemannian manifold

For the Lamé operator with variable coefficients λ and µ on a smooth compact Riemannian manifold (M, g) with smooth boundary , we give an explicit expression for the full symbol of the elastic Dirichlet-to-Neumann map . We show that uniquely determines the partial derivatives of all orders of the Lamé coefficients λ and µ on . Moreover, for a nonempty smooth open subset , suppose that the manifold and the Lamé coefficients are real analytic up to Γ, we prove that uniquely determines the Lamé coefficients on the whole manifold .

Error bounds for a least squares meshless finite difference method on closed manifolds

We present an error bound for a least squares version of the kernel based meshless finite difference method for elliptic differential equations on smooth compact manifolds of arbitrary dimension without boundary. In particular, we obtain sufficient conditions for the convergence of this method. Numerical examples are provided for the equation -Delta _{mathcal {M}} u + u = f on the 2- and 3-spheres, where Delta _{mathcal {M}} is the Laplace-Beltrami operator.

Continuity of families of Calderón projections

We consider a continuous family of linear elliptic differential operators of arbitrary order over a smooth compact manifold with boundary. Assuming constant dimension of the spaces of inner solutions, we prove that the orthogonalized Calderón projections of the underlying family of elliptic operators form a continuous family of projections. Hence, its images (the Cauchy data spaces) form a continuous family of closed subspaces in the relevant Sobolev spaces. We use only elementary tools and classical results: basic manipulations of operator graphs and other closed subspaces in Banach spaces, elliptic regularity, Green's formula and trace theorems for Sobolev spaces, well-posed boundary conditions, duality of spaces and operators in Hilbert space, and the interpolation theorem for operators in Sobolev spaces.

Miscellaneous on commutants mod normed ideals and quasicentral modulus $I$

We define commutants mod normed ideals associated with compact smooth manifolds with boundary. The results about the K -theory of these operator algebras include an exact sequence for the connected sum of manifolds, derived from the Mayer–Vietoris sequence. We also make a few remarks about bicommutants mod normed ideals and about the quasicentral modulus for the quasinormed p -Schatten–von Neumann classes 0 &lt; p &lt; 1.

Eigenfunction restriction estimates for curves with nonvanishing geodesic curvatures in compact Riemannian surfaces with nonpositive curvature

For 2 ≤ p &gt; 4 2\leq p&gt;4 , we study the L p L^p norms of restrictions of eigenfunctions of the Laplace-Beltrami operator on smooth compact 2 2 -dimensional Riemannian manifolds. Burq, Gérard, and Tzvetkov [Duke Math. J. 138 (2007), pp. 445–486], and Hu [Forum Math. 21 (2009), pp. 1021–1052] found the eigenfunction estimates restricted to a curve with nonvanishing geodesic curvatures. We will explain how the proof of the known estimates helps us to consider the case where the given smooth compact Riemannian manifold has nonpositive sectional curvatures. For p = 4 p=4 , we will also obtain a logarithmic analogous estimate, by using arguments in Xi and Zhang [Comm. Math. Phys. 350 (2017), pp. 1299–1325], Sogge [Math. Res. Lett. 24 (2017), pp. 549–570], and Bourgain [Geom. Funct. Anal. 1 (1991), pp. 147–187].

Two-Term Spectral Asymptotics in Linear Elasticity

We establish the two-term spectral asymptotics for boundary value problems of linear elasticity on a smooth compact Riemannian manifold of arbitrary dimension. We also present some illustrative examples and give a historical overview of the subject. In particular, we correct erroneous results published by Liu (J Geom Anal 31:10164–10193, 2021).

Yamabe Invariants, Homogeneous Spaces, and Rational Complex Surfaces

The Yamabe invariant is a diffeomorphism invariant of smooth compact manifolds that arises from the normalized Einstein-Hilbert functional. This article highlights the manner in which one compelling open problem regarding the Yamabe invariant appears to be closely tied to static potentials and the first eigenvalue of the Laplacian.

Algebras of Smooth Functions and Holography of Traversing Flows

Let X be a smooth compact manifold and v a vector field on X which admits a smooth function f : X ! R such that df(v) &gt; 0. Let @X be the boundary of X. We denote by C1(X) the algebra of smooth functions on X and by C1(@X) the algebra of smooth functions on @X. With the help of (v; f), we introduce two subalgebras A(v) and B(f) of C1(@X) and prove (under mild hypotheses) that C1(X) _ A(v) ^B(f), the topological tensor product. Thus the topological algebras A(v) and B(f), viewed as boundary data, allow for a reconstruction of C1(X). As a result, A(v) and B(f) allow for the recovery of the smooth topological type of the bulk X.

Spectral invariants of the magnetic Dirichlet-to-Neumann map on Riemannian manifolds

This paper is devoted to investigating the heat trace asymptotic expansion associated with the magnetic Steklov problem on a smooth compact Riemannian manifold (Ω, g) with smooth boundary ∂Ω. By computing the full symbol of the magnetic Dirichlet-to-Neumann map M, we establish an effective procedure, by which we can calculate all the coefficients a0, a1, …, an−1 of the asymptotic expansion. In particular, we explicitly give the first four coefficients a0, a1, a2, and a3. They are spectral invariants, which provide precise information concerning the volume and curvatures of the boundary ∂Ω and some physical quantities.

Deep learning delay coordinate dynamics for chaotic attractors from partial observable data.

A common problem in time-series analysis is to predict dynamics with only scalar or partial observations of the underlying dynamical system. For data on a smooth compact manifold, Takens' theorem proves a time-delayed embedding of the partial state is diffeomorphic to the attractor, although for chaotic and highly nonlinear systems, learning these delay coordinate mappings is challenging. We utilize deep artificial neural networks (ANNs) to learn discrete time maps and continuous time flows of the partial state. Given training data for the full state, we also learn a reconstruction map. Thus, predictions of a time series can be made from the current state and several previous observations with embedding parameters determined from time-series analysis. The state space for time evolution is of comparable dimension to reduced order manifold models. These are advantages over recurrent neural network models, which require a high-dimensional internal state or additional memory terms and hyperparameters. We demonstrate the capacity of deep ANNs to predict chaotic behavior from a scalar observation on a manifold of dimension three via the Lorenz system. We also consider multivariate observations on the Kuramoto-Sivashinsky equation, where the observation dimension required for accurately reproducing dynamics increases with the manifold dimension via the spatial extent of the system.

Geometric mean of probability measures and geodesics of Fisher information metric

AbstractThe space of all probability measures having positive density function on a connected compact smooth manifold M, denoted by , carries the Fisher information metric G. We define the geometric mean of probability measures by the aid of which we investigate information geometry of , equipped with G. We show that a geodesic segment joining arbitrary probability measures μ1 and μ2 is expressed by using the normalized geometric mean of its endpoints. As an application, we show that any two points of can be joined by a unique geodesic. Moreover, we prove that the function ℓ defined by , , , gives the Riemannian distance function on . It is shown that geodesics are all minimal.