Smooth Compact Riemannian Manifold Research Articles

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.

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 .

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

For 2 ≤ p > 4 2\leq p>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).

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.

Weyl remainders: an application of geodesic beams

We obtain new quantitative estimates on Weyl Law remainders under dynamical assumptions on the geodesic flow. On a smooth compact Riemannian manifold (M, g) of dimension n, let Pi _lambda denote the kernel of the spectral projector for the Laplacian, mathbb {1}_{[0,lambda ^2]}(-Delta _g). Assuming only that the set of near periodic geodesics over {W}subset M has small measure, we prove that as lambda rightarrow infty ∫WΠλ(x,x)dx=(2π)-nvolRn(B)volg(W)λn+O(λn-1logλ),\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} \\int _{{W}} \\Pi _\\lambda (x,x)dx=(2\\pi )^{-n}{{\\,\ extrm{vol}\\,}}_{_{{\\mathbb {R}}^n}}\\!(B){{\\,\ extrm{vol}\\,}}_g({W})\\,\\lambda ^n+O\\Big (\\frac{\\lambda ^{n-1}}{\\log \\lambda }\\Big ), \\end{aligned}$$\\end{document}where B is the unit ball. One consequence of this result is that the improved remainder holds on all product manifolds, in particular giving improved estimates for the eigenvalue counting function in the product setup. Our results also include logarithmic gains on asymptotics for the off-diagonal spectral projector Pi _lambda (x,y) under the assumption that the set of geodesics that pass near both x and y has small measure, and quantitative improvements for Kuznecov sums under non-looping type assumptions. The key technique used in our study of the spectral projector is that of geodesic beams.

A twist in sharp Sobolev inequalities with lower order remainder terms

Abstract Let ( M , g ) {(M,g)} be a smooth compact Riemannian manifold of dimension n ≥ 3 {n\geq 3} . Let also A be a smooth symmetrical positive ( 0 , 2 ) {(0,2)} -tensor field in M. By the Sobolev embedding theorem, we can write that there exist K , B > 0 {K,B>0} such that for any u ∈ H 1 ⁢ ( M ) {u\in H^{1}(M)} , (0.1) ∥ u ∥ L 2 ⋆ 2 ≤ K ⁢ ∥ ∇ A ⁡ u ∥ L 2 2 + B ⁢ ∥ u ∥ L 1 2 \|u\|_{L^{2^{\star}}}^{2}\leq K\|\nabla_{A}u\|_{L^{2}}^{2}+B\|u\|_{L^{1}}^{2} where H 1 ⁢ ( M ) {H^{1}(M)} is the standard Sobolev space of functions in L 2 {L^{2}} with one derivative in L 2 {L^{2}} , | ∇ A ⁡ u | 2 = A ⁢ ( ∇ ⁡ u , ∇ ⁡ u ) {|\nabla_{A}u|^{2}=A(\nabla u,\nabla u)} and 2 ⋆ {2^{\star}} is the critical Sobolev exponent for H 1 {H^{1}} . We compute in this paper the value of the best possible K in (0.1) and investigate the validity of the corresponding sharp inequality.

An upper bound on the revised first Betti number and a torus stability result for RCD spaces

We prove an upper bound on the rank of the abelianised revised fundamental group (called "revised first Betti number") of a compact $RCD^{*}(K,N)$ space, in the same spirit of the celebrated Gromov-Gallot upper bound on the first Betti number for a smooth compact Riemannian manifold with Ricci curvature bounded below. When the synthetic lower Ricci bound is close enough to (negative) zero and the aforementioned upper bound on the revised first Betti number is saturated (i.e. equal to the integer part of $N$, denoted by $\lfloor N \rfloor$), then we establish a torus stability result stating that the space is $\lfloor N \rfloor$-rectifiable as a metric measure space, and a finite cover must be mGH-close to an $\lfloor N \rfloor$-dimensional flat torus; moreover, in case $N$ is an integer, we prove that the space itself is bi-H\"older homeomorphic to a flat torus. This second result extends to the class of non-smooth $RCD^{*}(-\delta, N)$ spaces a celebrated torus stability theorem by Colding (later refined by Cheeger-Colding).

Aubin Type Almost sharp Moser–Trudinger Inequality Revisited

We give a new proof of the almost sharp Moser–Trudinger inequality on smooth compact Riemannian manifolds based on the sharp Moser inequality on Euclidean spaces and generalize it to Riemannian manifolds with continuous metrics and to higher order Sobolev spaces on manifolds with boundary under several boundary conditions. These generalizations can be applied to fourth order Q curvature equations in dimension 4.

СОБСТВЕННЫЕ ЗНАЧЕНИЯ И СОБСТВЕННЫЕ ФУНКЦИИ ОПЕРАТОРА ЛАПЛАСА В КВАДРАТЕ И В КРУГЕ С КРАЕВЫМ УСЛОВИЕМ ВЕНТЦЕЛЯ

Abstract. Recently, in the mathematical literature, the Wentzel boundary condition has been considered from two points of view. In the first case, let us call it a classical case, this condition is an equation containing a linear combination of the values of the function and its derivatives at the boundary of the domain. Meanwhile, the function itself also satisfies an equation with an elliptic operator given in the domain. In the second, neoclassical case, the Wentzel condition is an equation with the Laplace–Beltrami operator defined on the boundary of the domain, understood as a smooth compact Riemannian manifold without an edge; and the external effect is represented by the normal derivative of the function specified in the domain. The paper considers the properties of the Laplace operator with the Wentzel boundary condition in the neoclassical sense. In particular, eigenvalues and eigenfunctions of the Laplace operator are constructed for a system of Wentzel equations in a circle and in a square.

Uniqueness for stochastic scalar conservation laws on Riemannian manifolds revisited

We revise a uniqueness question for the scalar conservation law with stochastic forcing du + divgf(x,u)dt = ?(x,u)dWt, x ? M, t ? 0 on a smooth compact Riemannian manifold (M,g) whereWt is the Wiener process and x ? f(x,?) is a vector field on M for each ? ? R. We introduce admissibility conditions, derive the kinetic formulation and use it to prove uniqueness in a more straight-forward way than in the existing literature.

Linear isoperimetric inequality for normal and integral currents in compact subanalytic sets

The isoperimetric inequality for a smooth compact Riemannian manifold $A$ provides a positive ${\bf c}(A)$, so that for any $k+1$ dimensional integral current $S_0$ in $A$ there exists an integral current $ S$ in $A$ with $\partial S=\partial S_0$ and ${\bf M}(S)\leq {\bf c}(A){\bf M}(\partial S)^{(k+1)/k}$. Although such an inequality still holds for any compact Lipschitz neighborhood retract $A$, it may fail in case $A$ contains a single polynomial singularity. Here, replacing $(k+1)/k$ by $1$, we find that a linear inequality ${\bf M}(S)\leq {\bf c}(A){\bf M}(\partial S)$ is valid for any compact algebraic, semi-algebraic, or even subanalytic set $A$. In such a set, this linear inequality holds not only for integral currents, which have $\boldsymbol{Z}$ coefficients, but also for normal currents having $\boldsymbol{R}$ coefficients and generally for normal flat chains with coefficients in any complete normed abelian group. A relative version for a subanalytic pair $B\subset A$ is also true, and there are applications to variational and metric properties of subanalytic sets.

Local stable and unstable sets for positive entropy C1 dynamical systems

For any C1 diffeomorphism on a smooth compact Riemannian manifold that admits an ergodic measure with positive entropy, a lower bound of the Hausdorff dimension for the local stable and unstable sets is given in terms of the measure-theoretic entropy and the maximal Lyapunov exponent. The main line of our approach to this result is under the setting of topological dynamical systems, which is also applicable to infinite-dimensional C1 dynamical systems.

Partial Regularity of Harmonic Maps From Alexandrov Spaces

Abstract In this paper, we prove the Lipschitz regularity of continuous harmonic maps from a finite-dimensional Alexandrov space to a compact smooth Riemannian manifold. This solves a conjecture of F. H. Lin in [38]. The proof extends the argument of Huang-Wang [28].

Casimir pistons with generalized boundary conditions: a step forward

In this work we study the Casimir effect for massless scalar fields propagating in a piston geometry of the type $I\times N$ where $I$ is an interval of the real line and $N$ is a smooth compact Riemannian manifold. Our analysis represents a generalization of previous results obtained for pistons configurations as we consider all possible boundary conditions that are allowed to be imposed on the scalar fields. We employ the spectral zeta function formalism in the framework of scattering theory in order to obtain an expression for the Casimir energy and the corresponding Casimir force on the piston. We provide explicit results for the Casimir force when the manifold $N$ is a $d$-dimensional sphere and a disk.

A Remark on the Concentration Compactness Principle in Critical Dimension

AbstractWe prove some refinements of the concentration compactness principle for Sobolev space W1, n on a smooth compact Riemannian manifold of dimension n. As an application, we extend Aubin's theorem for functions on with zero first‐order moments of the area element to the higher‐order moments case. Our arguments are very flexible and can be easily modified for functions satisfying various boundary conditions or belonging to higher‐order Sobolev spaces. © 2020 Wiley Periodicals LLC.

Compactness of scalar-flat conformal metrics on low-dimensional manifolds with constant mean curvature on boundary

We concern C2-compactness of the solution set of the boundary Yamabe problem on smooth compact Riemannian manifolds with boundary provided that their dimensions are 4, 5 or 6. By conducting a quantitative analysis of a linear equation associated with the problem, we prove that the trace-free second fundamental form must vanish at possible blow-up points of a sequence of blowing-up solutions. Applying this result and the positive mass theorem, we deduce the C2-compactness for all 4-manifolds (which may be non-umbilic). For the 5-dimensional case, we also establish that a sum of the second-order derivatives of the trace-free second fundamental form is non-negative at possible blow-up points. We essentially use this fact to obtain the C2-compactness for all 5-manifolds. Finally, we show that the C2-compactness on 6-manifolds is true if the trace-free second fundamental form on the boundary never vanishes.

Pollicott–Ruelle spectrum and Witten Laplacians

We study the asymptotic behavior of eigenvalues and eigenmodes of the Witten Laplacian on a smooth compact Riemannian manifold without boundary. We show that they converge to the Pollicott–Ruelle spectral data of the corresponding gradient flow acting on appropriate anisotropic Sobolev spaces. As an application of our methods, we also construct a natural family of quasimodes satisfying the Witten–Helffer–Sjöstrand tunneling formulas and the Fukaya conjecture on Witten deformation of the wedge product.

Liouville-type theorems on manifolds with nonnegative curvature and strictly convex boundary

We prove some Liouville type theorems on smooth compact Riemannian manifolds with nonnegative sectional curvature and strictly convex boundary. This gives a nonlinear generalization in low dimension of the recent sharp lower bound of the first Steklov eigenvalue by Xia-Xiong and verifies partially a conjecture by the third author. As a consequence, we derive several sharp Sobolev trace inequalities on these manifolds.