Smooth Riemannian Manifold Research Articles

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).

Higher Cheeger ratios of features in Laplace-Beltrami eigenfunctions

This paper investigates links between the eigenvalues and eigenfunctions of the Laplace-Beltrami operator, and the higher Cheeger constants of smooth Riemannian manifolds, possibly weighted and/or with boundary. The higher Cheeger constants give a loose description of the major geometric features of a manifold. We give a constructive upper bound on the higher Cheeger constants, in terms of the eigenvalue of any eigenfunction with the corresponding number of nodal domains. Specifically, we show that for each such eigenfunction, a positive-measure collection of its superlevel sets have their Cheeger ratios bounded above in terms of the corresponding eigenvalue.Some manifolds have their major features entwined across several eigenfunctions, and no single eigenfunction contains all the major features. In this case, there may exist carefully chosen linear combinations of the eigenfunctions, each with large values on a single feature, and small values elsewhere. We can then apply a soft-thresholding operator to these linear combinations to obtain new functions, each supported on a single feature. We show that the Cheeger ratios of the level sets of these functions also give an upper bound on the Laplace-Beltrami eigenvalues. We extend these level set results to nonautonomous dynamical systems, and show that the dynamic Laplacian eigenfunctions reveal sets with small dynamic Cheeger ratios.

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.

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.

Small spheres with prescribed nonconstant mean curvature in Riemannian manifolds

Given a function f on a smooth Riemannian manifold without boundary, we prove that if p∈M is a non-degenerate critical point of f, then a neighborhood of p contains a foliation by spheres with mean curvature proportional to f. This foliation is essentially unique. The nondegeneracy assumption can be substantially relaxed, at the expense of losing the property that the family of spheres with prescribed mean curvature defines a foliation.

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.

Convex functions defined on metric spaces are pulled back to subharmonic ones by harmonic maps

If u:Ω⊂Rd→X\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$u: \\Omega \\subset \\mathbb {R}^d \\rightarrow \ extrm{X}$$\\end{document} is a harmonic map valued in a metric space X\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extrm{X}$$\\end{document} and E:X→R\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extsf{E}: \ extrm{X}\\rightarrow \\mathbb {R}$$\\end{document} is a convex function, in the sense that it generates an EVI0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extrm{EVI}_0$$\\end{document}-gradient flow, we prove that the pullback E∘u:Ω→R\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extsf{E}\\circ u: \\Omega \\rightarrow \\mathbb {R}$$\\end{document} is subharmonic. This property was known in the smooth Riemannian manifold setting or with curvature restrictions on X\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extrm{X}$$\\end{document}, while we prove it here in full generality. In addition, we establish generalized maximum principles, in the sense that the Lq\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^q$$\\end{document} norm of E∘u\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extsf{E}\\circ u$$\\end{document} on ∂Ω\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\partial \\Omega $$\\end{document} controls the Lp\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^p$$\\end{document} norm of E∘u\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extsf{E}\\circ u$$\\end{document} in Ω\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Omega $$\\end{document} for some well-chosen exponents p≥q\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$p \\ge q$$\\end{document}, including the case p=q=+∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$p=q=+\\infty $$\\end{document}. In particular, our results apply when E\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extsf{E}$$\\end{document} is a geodesically convex entropy over the Wasserstein space, and thus settle some conjectures of Brenier (Optimal transportation and applications (Martina Franca, 2001), volume 1813 of lecture notes in mathematics, Springer, Berlin, pp 91–121, 2003).

Optimization Method of the Clamping Force for Large Cabin Parts

In order to realize the stable clamping of large cabin parts, this paper studied the clamping force optimization of the clamping mechanism for large-mass and large-size cabin parts. Firstly, three kinds of contact models are introduced. Then, a clamping matrix is constructed for a particular clamping configuration. The nonlinear friction cone constraint at the contact point is transformed into a linear affine constraint in a smooth Riemannian manifold using the special structure of the positive definite symmetric matrix. Finally, a large cabin part used in the aerospace field is used as an example for calculating and simulating the clamping force optimization. Different optimization algorithms are used to calculate the initial value which is put into the gradient flow optimization method of the clamping force for optimization calculation. Meanwhile, the normal clamping force value of a 2 t object is measured using relevant experimental equipment. The simulation results and experimental results show that the gradient flow optimization method of the clamping force can quickly complete the clamping force optimization of large-mass and large-size cabin parts. The actual measured value of the normal clamping force is close to the simulated convergence value. The distribution of the normal force of the clamping mechanisms and the convergence value of the clamping force for each clamping mechanism can provide some references for determining the output clamping force of the clamping mechanism and confirming a reasonable distribution of the clamping mechanism. It also confirms the feasibility and effectiveness of this method applied to the clamping force optimization for a large axial part.

On the Boundary Behavior of Mass-Minimizing Integral Currents

Let Σ \Sigma be a smooth Riemannian manifold, Γ ⊂ Σ \Gamma \subset \Sigma a smooth closed oriented submanifold of codimension higher than 2 2 and T T an integral area-minimizing current in Σ \Sigma which bounds Γ \Gamma . We prove that the set of regular points of T T at the boundary is dense in Γ \Gamma . Prior to our theorem the existence of any regular point was not known, except for some special choice of Σ \Sigma and Γ \Gamma . As a corollary of our theorem we answer to a question in Almgren’s Almgren’s big regularity paper from 2000 showing that, if Γ \Gamma is connected, then T T has at least one point p p of multiplicity 1 2 \frac {1}{2} , namely there is a neighborhood of the point p p where T T is a classical submanifold with boundary Γ \Gamma ; we generalize Almgren’s connectivity theorem showing that the support of T T is always connected if Γ \Gamma is connected; we conclude a structural result on T T when Γ \Gamma consists of more than one connected component, generalizing a previous theorem proved by Hardt and Simon in 1979 when Σ = R m + 1 \Sigma = \mathbb R^{m+1} and T T is m m -dimensional.

Ordered exponential and its features in Yang–Mills effective action

In this paper, we discuss some non-trivial relations for ordered exponentials on smooth Riemannian manifolds. As an example of application, we study the dependence of the four-dimensional quantum Yang–Mills effective action on the special gauge transformation with respect to the background field. Also, we formulate some open questions about a structure of divergences for a special type of regularization in the presence of the background field formalism.

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 .

Geometry of sample spaces

In statistics, independent, identically distributed random samples do not carry a natural ordering, and their statistics are typically invariant with respect to permutations of their order. Thus, an n-sample in a space M can be considered as an element of the quotient space of Mn modulo the permutation group. The present paper takes this definition of sample space and the related concept of orbit types as a starting point for developing a geometric perspective on statistics. We aim at deriving a general mathematical setting for studying the behavior of empirical and population means in spaces ranging from smooth Riemannian manifolds to general stratified spaces.We fully describe the orbifold and path-metric structure of the sample space when M is a manifold or path-metric space, respectively. These results are non-trivial even when M is Euclidean. We show that the infinite sample space exists in a Gromov–Hausdorff type sense and coincides with the Wasserstein space of probability distributions on M. We exhibit Fréchet means and k-means as metric projections onto 1-skeleta or k-skeleta in Wasserstein space, and we define a new and more general notion of polymeans. This geometric characterization via metric projections applies equally to sample and population means, and we use it to establish asymptotic properties of polymeans such as consistency and asymptotic normality.

Low-Resolution MMSE Equalization for Massive MU-MIMO Systems

Finite-alphabet equalization that represents a spatial equalization matrix with low-resolution coefficients is a promising technique to reduce the power consumption, processing delay, and circuit area of baseband processing in the context of all-digital massive multiuser multiple-input multiple-output uplink systems. However, to minimize the performance loss caused by a coarse-resolution spatial equalization matrix, its coefficients must be carefully designed in the minimum mean-square error sense to achieve the desired bit error rate (BER) performance, which unfortunately constitutes an NP-hard optimization problem. To tackle this problem, we first reformulate the finite-alphabet equalization design problem as an unconstrained optimization on a smooth Riemannian manifold. Then we propose an algorithm based on Riemannian manifold optimization (RMO) to solve the reformulated problem. Based on simulation results, the proposed 2-bit RMO-assisted equalizer outperforms its state-of-the-art counterparts while maintaining the same asymptotic complexity. In addition, the proposed 2-bit RMOassisted equalizer exhibits a loss of only 1.47 dB at a BER of 10 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">-4</sup> compared to an unquantized linear minimum mean squared error equalizer.

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).

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.

Area-minimizing hypersurfaces in manifolds of Ricci curvature bounded below

Abstract In this paper, we study area-minimizing hypersurfaces in manifolds of Ricci curvature bounded below with Cheeger–Colding theory. Let N i {N_{i}} be a sequence of smooth manifolds with Ricci curvature ≥ - n ⁢ κ 2 {\geq-n\kappa^{2}} on B 1 + κ ′ ⁢ ( p i ) {B_{1+\kappa^{\prime}}(p_{i})} for constants κ ≥ 0 {\kappa\geq 0} , κ ′ &gt; 0 {\kappa^{\prime}&gt;0} , and volume of B 1 ⁢ ( p i ) {B_{1}(p_{i})} has a positive uniformly lower bound. Assume B 1 ⁢ ( p i ) {B_{1}(p_{i})} converges to a metric ball B 1 ⁢ ( p ∞ ) {B_{1}(p_{\infty})} in the Gromov–Hausdorff sense. For a sequence of area-minimizing hypersurfaces M i {M_{i}} in B 1 ⁢ ( p i ) {B_{1}(p_{i})} with ∂ ⁡ M i ⊂ ∂ ⁡ B 1 ⁢ ( p i ) {\partial M_{i}\subset\partial B_{1}(p_{i})} , we prove the continuity for the volume function of area-minimizing hypersurfaces equipped with the induced Hausdorff topology. In particular, each limit M ∞ {M_{\infty}} of M i {M_{i}} is area-minimizing in B 1 ⁢ ( p ∞ ) {B_{1}(p_{\infty})} provided B 1 ⁢ ( p ∞ ) {B_{1}(p_{\infty})} is a smooth Riemannian manifold. By blowing up argument, we get sharp dimensional estimates for the singular set of M ∞ {M_{\infty}} in ℛ {\mathcal{R}} , and 𝒮 ∩ M ∞ {\mathcal{S}\cap M_{\infty}} . Here, ℛ {\mathcal{R}} and 𝒮 {\mathcal{S}} are the regular and singular parts of B 1 ⁢ ( p ∞ ) {B_{1}(p_{\infty})} , respectively.

Study on the Space Geodesy

Geodesic is a kind of shortest path among all curves in a metric space, which originally appeared in the Gaussian period. Since then, it was extensively applied in various branches of mathematics and physics, such as Riemannian geometry, digital geometry, Einstein’s relativity, etc. This thesis mainly discusses geodesic definitions in Euclidean space and those in smooth manifolds after introducing the basic theory of smooth manifolds. At first, the thesis applies three ways to define geodesics on a surface, including the geodesic curvature method, the shortest distance method, and the relation of the principal normal of a curve and the normal vector of a surface. Then, the paper introduces some basic concepts in Riemannian manifolds. Finally, it gives a general definition of geodesics in a smooth Riemannian manifold.

Topological Analysis of Vector-Field Guided Path Following on Manifolds

A path-following control algorithm enables a system’s trajectories under its guidance to converge to and evolve along a given geometric desired path. There exist various such algorithms, but many of them can only guarantee local convergence to the desired path in its neighborhood. In contrast, the control algorithms using a well-designed guiding vector field can ensure almost global convergence of trajectories to the desired path; here, “almost” means that in some cases, a measure-zero set of trajectories converge to the singular set where the vector field becomes zero (with all other trajectories converging to the desired path). In this article, we first generalize the guiding vector field from the Euclidean space to a general smooth Riemannian manifold. This generalization can deal with path-following in some abstract configuration space (such as robot arm joint space). Then, we show several theoretical results from a topological viewpoint. Specifically, we are motivated by the observation that singular points of the guiding vector field exist in many examples where the desired path is homeomorphic to the unit circle, but it is unknown whether the existence of singular points always holds in general (i.e., is inherent in the topology of the desired path). In the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$n$</tex-math></inline-formula> -dimensional Euclidean space, we provide an affirmative answer, and conclude that it is not possible to guarantee global convergence to desired paths that are homeomorphic to the unit circle. Furthermore, we show that there always exist nonpath-converging trajectories (i.e., trajectories that do not converge to the desired path) starting from the boundary of a ball containing the desired path in an <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$n$</tex-math></inline-formula> -dimensional Euclidean space where <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$n \geq 3$</tex-math></inline-formula> . Examples are provided to illustrate the theoretical results.

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.