General Riemannian Manifolds Research Articles

On the stability of constant higher order mean curvature hypersurfaces in a Riemannian manifold

AbstractWe propose a notion of stability for constant ‐mean curvature hypersurfaces in a general Riemannian manifold and we give some applications. When the ambient manifold is a Space Form, our notion coincides with the known one, given by means of the variational problem.

Multi-objective Variational Curves

AbstractRiemannian cubics in tension are critical points of the linear combination of two objective functionals, namely the squared $$L^2$$ L 2 norms of the velocity and acceleration of a curve on a Riemannian manifold. We view this variational problem of finding a curve as a multi-objective optimization problem and construct the Pareto fronts for some given instances where the manifold is a sphere and where the manifold is a torus. The Pareto front for the curves on the torus turns out to be particularly interesting: the front is disconnected and it reveals two distinct Riemannian cubics with the same boundary data, which is the first known nontrivial instance of this kind. We also discuss some convexity conditions involving the Pareto fronts for curves on general Riemannian manifolds.

Unified Framework for Diffusion Generative Models in SO(3): Applications in Computer Vision and Astrophysics

Diffusion-based generative models represent the current state-of-the-art for image generation. However, standard diffusion models are based on Euclidean geometry and do not translate directly to manifold-valued data. In this work, we develop extensions of both score-based generative models (SGMs) and Denoising Diffusion Probabilistic Models (DDPMs) to the Lie group of 3D rotations, SO(3). SO(3) is of particular interest in many disciplines such as robotics, biochemistry and astronomy/cosmology science. Contrary to more general Riemannian manifolds, SO(3) admits a tractable solution to heat diffusion, and allows us to implement efficient training of diffusion models. We apply both SO(3) DDPMs and SGMs to synthetic densities on SO(3) and demonstrate state-of-the-art results. Additionally, we demonstrate the practicality of our model on pose estimation tasks and in predicting correlated galaxy orientations for astrophysics/cosmology.

A Discovery Tour in Random Riemannian Geometry

We study random perturbations of a Riemannian manifold (M,g)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(\ extsf{M},\ extsf{g})$$\\end{document} by means of so-called Fractional Gaussian Fields, which are defined intrinsically by the given manifold. The fields h∙:ω↦hω\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$h^\\bullet : \\omega \\mapsto h^\\omega $$\\end{document} will act on the manifold via the conformal transformation g↦gω:=e2hωg\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extsf{g}\\mapsto \ extsf{g}^\\omega := e^{2h^\\omega }\\,\ extsf{g}$$\\end{document}. Our focus will be on the regular case with Hurst parameter H>0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$H>0$$\\end{document}, the critical case H=0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$H=0$$\\end{document} being the celebrated Liouville geometry in two dimensions. We want to understand how basic geometric and functional-analytic quantities like diameter, volume, heat kernel, Brownian motion, spectral bound, or spectral gap change under the influence of the noise. And if so, is it possible to quantify these dependencies in terms of key parameters of the noise? Another goal is to define and analyze in detail the Fractional Gaussian Fields on a general Riemannian manifold, a fascinating object of independent interest.

Removable sets and [formula omitted]-uniqueness on manifolds and metric measure spaces

We study symmetric diffusion operators on metric measure spaces. Our main question is whether essential self-adjointness or Lp-uniqueness are preserved under the removal of a small closed set from the space. We provide characterizations of the critical size of removed sets in terms of capacities and Hausdorff dimension without any further assumption on removed sets. As a key tool we prove a non-linear truncation result for potentials of nonnegative functions. Our results are robust enough to be applied to Laplace operators on general Riemannian manifolds as well as sub-Riemannian manifolds and metric measure spaces satisfying curvature-dimension conditions. For non-collapsing Ricci limit spaces with two-sided Ricci curvature bounds we observe that the self-adjoint Laplacian is already fully determined by the classical Laplacian on the regular part.

The Plateau-Douglas Problem for Singular Configurations and in General Metric Spaces

Assume you are given a finite configuration Gamma of disjoint rectifiable Jordan curves in {mathbb {R}}^n. The Plateau-Douglas problem asks whether there exists a minimizer of area among all compact surfaces of genus of at most p which span Gamma . While the solution to this problem is well-known, the classical approaches break down if one allows for singular configurations Gamma , where the curves are potentially non-disjoint or self-intersecting. Our main result solves the Plateau-Douglas problem for such potentially singular configurations. Moreover, our proof works not only in {mathbb {R}}^n but in general proper metric spaces. In particular, the existence of an area minimizer is new for disjoint configurations of Jordan curves in general complete Riemannian manifolds. A minimal surface of fixed genus p bounding a given configuration Gamma need not always exist, even in the most regular settings. Concerning this problem, we also generalize the approach for singular configurations via minimal sequences satisfying conditions of cohesion and adhesion to the setting of metric spaces.

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.

The Dirichlet problem for fully nonlinear elliptic equations on Riemannian manifolds

We study the Dirichlet problem for a broad class of fully nonlinear elliptic equations in Euclidean space as well as on general Riemannian manifolds. Under a set of fundamental structure conditions which have become standard since the pioneering work of Cafferalli, Nirenberg and Spruck, we prove that the Dirichlet problem admits a (unique) smooth solution provided that there exists a subsolution. The conditions are essentially optimal, especially with no geometric restrictions to the boundary of the underlying manifold, which is important in applications. We search new ideas and techniques to make use of the concavity condition and subsolution in order to overcome difficulties in deriving key a priori estimates. Along the way we discover some interesting properties of concave functions which should be useful in other fields. Our methods can be adopted to treat other types of fully nonlinear elliptic and parabolic equations on real or complex manifolds. We shall also solve some new equations, which were not covered by previous results even in Rn, with interesting properties.

Almost-sure asymptotics for Riemannian random waves

We consider the Riemannian random wave model of Gaussian linear combinations of Laplace eigenfunctions on a general compact Riemannian manifold. With probability one with respect to the Gaussian coefficients, we establish that, both for large and small band models, the process properly rescaled and evaluated at an independently and uniformly chosen point X on the manifold, converges in distribution under the sole randomness of X towards an universal Gaussian field as the frequency tends to infinity. This result extends the celebrated central limit Theorem of Salem–Zygmund for trigonometric polynomials series to the more general framework of compact Riemannian manifolds. We then deduce from the above convergence the almost-sure asymptotics of the nodal volume associated with the random wave. To the best of our knowledge, in the real Riemannian case, these asymptotics were only known in expectation and not in the almost sure sense due to the lack of sufficiently accurate variance estimates. This in particular addresses a question of S. Zelditch regarding the almost sure equidistribution of nodal volume.

Black Holes have Intrinsic Scalar Curvature

The scalar curvature [Formula: see text] is invariant under isometric symmetries (distance invariance) associated with metric spaces. Gravitational Riemannian manifolds are metric spaces. For Minkowski Space, the distance invariant is [Formula: see text], where [Formula: see text], [Formula: see text] are arbitrary 4-vectors. Thus the isometry symmetry associated with Minkowski Space is the Poincaré Group. The Standard Model Lagrangian density [Formula: see text] is also invariant under the Poincaré Group, so for Minkowski Space, the scalar curvature and the Standard Model Lagrangian density are proportional to each other. We show that this proportionality extends to general gravitational Riemannian manifolds, not just for Minkowski Space. This predicts that Black Holes have non-zero scalar curvatures [Formula: see text]. For Schwarzschild Black Holes, [Formula: see text] is predicted to be [Formula: see text], where [Formula: see text] is the Schwarzschild radius. The existence of [Formula: see text] means that Black Holes cannot evaporate.

Pointwise Weyl laws for Schrödinger operators with singular potentials

We consider the Schrödinger operators HV=−Δg+V with singular potentials V on general n-dimensional Riemannian manifolds and study the eigenvalues and eigenfunctions under this perturbation. These singular potentials appear naturally in physics, most notably the Coulomb potential |x|−1. Sogge and the first author [14] proved the sharp Weyl laws for these HV with potentials in the Kato class, which is the minimal assumption to ensure that HV is essentially self-adjoint and bounded from below and the eigenfunctions of HV are bounded. Later, Frank-Sabin [9] studied the problem on the pointwise Weyl laws for these HV in three dimensions by extending the method of Avakumović [2], while it is unknown how to reconstruct this argument in other dimensions. In this paper, we completely solve this problem in any dimensions by using a different argument. First, we establish the pointwise Weyl law for potentials in the Kato class on any n-dimensional manifolds. This extends the 3-dimensional results of Frank-Sabin [9] by a different method. Second, we prove that the pointwise Weyl law with the standard sharp error term O(λn−1) holds for potentials in Ln(M). This extends the classical results for smooth potentials by Avakumović [2], Levitan [18] and Hörmander [12] to critically singular potentials. In three dimensions, this L3 condition also naturally appears in Boccato-Brennecke-Cenatiempo-Schlein [6] on the ground state energy of the Hamilton operator in the Gross-Pitaevskii regime. These two results are sharp, and our proof exploits Li-Yau's heat kernel bounds and Blair-Sire-Sogge's eigenfunction estimates.

Additive models for symmetric positive-definite matrices and Lie groups

Summary We propose and investigate an additive regression model for symmetric positive-definite matrix-valued responses and multiple scalar predictors. The model exploits the Abelian group structure inherited from either of the log-Cholesky and log-Euclidean frameworks for symmetric positive-definite matrices and naturally extends to general Abelian Lie groups. The proposed additive model is shown to connect to an additive model on a tangent space. This connection not only entails an efficient algorithm to estimate the component functions, but also allows one to generalize the proposed additive model to general Riemannian manifolds. Optimal asymptotic convergence rates and normality of the estimated component functions are established, and numerical studies show that the proposed model enjoys good numerical performance, and is not subject to the curse of dimensionality when there are multiple predictors. The practical merits of the proposed model are demonstrated through an analysis of brain diffusion tensor imaging data.

Positive scalar curvature on foliations: The noncompact case

Let (M,gTM) be a noncompact (not necessarily complete) enlargeable Riemannian manifold in the sense of Gromov-Lawson and F an integrable subbundle of TM. Let kF be the leafwise scalar curvature associated to gF=gTM|F. We show that if either TM or F is spin, then inf(kF)≤0. This generalizes the famous result of Gromov-Lawson on enlargeable manifolds to the case of foliations. It also extends an ansatz of Gromov on hyper-Euclidean spaces to general enlargeable Riemannian manifolds, as well as recent results on compact enlargeable foliated manifolds due to Benameur-Heitsch et al. to the noncompact situation.

Hardy-Poincaré-Sobolev type inequalities on hyperbolic spaces and related Riemannian manifolds

We establish sharpened forms of the Hardy type identities and inequalities which are substantial improvements of Hardy inequalities for the operators −ΔH−(N−1)24 on the hyperbolic spaces HN. More precisely, we study the refined Hardy-Poincaré-Sobolev type inequalities in the spirit of Brezis-Vázquez [17] and Brezis-Marcus [15] on hyperbolic spaces, spherically symmetric Riemannian manifolds and more general Riemannian manifolds. Spherically symmetric manifolds are also of significant importance in physics and general relativity. Our approaches are to first establish the Hardy-Poincaré-Sobolev type identities using the notion of Bessel pairs on hyperbolic spaces and related Riemannian manifolds. Using a Hardy identity on upper half space, we also establish a sharp Sobolev inequality on a novel example of noncomplete and non-extendable Riemannian manifold with positive Ricci curvature. In particular, in dimension 3, our example shows that the completeness assumption of the manifolds in the rigidity result of Ledoux [47] cannot be removed.

Large-scale geometry obstructs localization

We explain the coarse geometric origin of the fact that certain spectral subspaces of topological insulator Hamiltonians are delocalized, in the sense that they cannot admit an orthonormal basis of localized wavefunctions, with respect to any uniformly discrete set of localization centers. This is a robust result requiring neither spatial homogeneity nor symmetries and applies to Landau levels of disordered quantum Hall systems on general Riemannian manifolds.

A roughness penalty approach to estimate densities over two-dimensional manifolds

An innovative nonparametric method for density estimation over general two-dimensional Riemannian manifolds is proposed. The method follows a functional data analysis approach, combining maximum likelihood estimation with a roughness penalty that involves a differential operator appropriately defined over the manifold domain, thus controlling the smoothness of the estimate. The proposed method can accurately handle point pattern data over complicated curved domains. Moreover, it is able to capture complex multimodal signals, with strongly localized and highly skewed modes, with varying directions and intensity of anisotropy. The estimation procedure exploits a discretization in finite element bases, enabling great flexibility on the spatial domain. The method is tested through simulation studies, showing the strengths of the proposed approach. Finally, the density estimation method is illustrated with an application to the distribution of earthquakes in the world.

Oracle inequalities and upper bounds for kernel density estimators on manifolds and more general metric spaces

We prove oracle inequalities and upper bounds for kernel density estimators on a very broad class of metric spaces. Precisely we consider the setting of a doubling measure metric space in the presence of a non-negative self-adjoint operator whose heat kernel enjoys Gaussian regularity. Many classical settings like Euclidean spaces, spheres, balls, cubes as well as general Riemannian manifolds, are contained in our framework. Moreover the rate of convergence we achieve is the optimal one in these special cases. Finally we provide the general methodology of constructing the proper kernels when the manifold under study is given and we give precise examples for the case of the sphere.

A mixed elliptic-parabolic boundary value problem coupling a harmonic-like map with a nonlinear spinor

Abstract In this paper, we solve a new elliptic-parabolic system arising in geometric analysis that is motivated by the nonlinear supersymmetric sigma model of quantum field theory. The corresponding action functional involves two fields, a map from a Riemann surface into a Riemannian manifold and a spinor coupled to the map. The first field has to satisfy a second-order elliptic system, which we turn into a parabolic system so as to apply heat flow techniques. The spinor, however, satisfies a first-order Dirac-type equation. We carry that equation as a nonlinear constraint along the flow. With this novel scheme, in more technical terms, we can show the existence of Dirac-harmonic maps from a compact spin Riemann surface with smooth boundary to a general compact Riemannian manifold via a heat flow method when a Dirichlet boundary condition is imposed on the map and a chiral boundary condition on the spinor.

Reflecting time-Space Gaussian random field on compact Riemannian manifold and excursion probability

Motivated from measuring the performance of quantum computing and storage systems together with nanorheology over different shapes of devices, we introduce a reflecting time-space Gaussian random field (RGRF) on a general -parameter compact Riemannian manifold to model the quantum particle movement dynamics. The main task in studying the RGRF is to estimate an asymptotic upper bound of its excursion probability, which asymptotically converges to zero. In doing so, we establish a relationship between the RGRF and its netput Gaussian random field (GRF) via a Skorohod mapping. Then, as an intermediate step, we approximate the excursion probability of the GRF by constructing a chart oriented homeomorphism mapping between a flat manifold and a general compact Riemannian manifold. The netput GRF can be either isotropic or anisotropic. Case studies in terms of anisotropic GRFs over a flat manifold such as normalized fractional Brownian sheets (FBSs) and those over a sphere such as normalized anisotropic fractional spherical Brownian motions (FSBMs) are also presented, where an α-FSBM is introduced and its existence is proved for some constant .

Intrinsic Cramér–Rao bounds for distributed Bayesian estimator

This paper addresses the intrinsic Cramér–Rao bounds (CRBs) for a distributed Bayesian estimator whose states and measurements are on Riemannian manifolds. As Euclidean-based CRBs for recursive Bayesian estimator are no longer applicable to general Riemannian manifolds, the bounds need redesigning to accommodate the non-zero Riemannian curvature. To derive the intrinsic CRBs, we append a coordination step to the recursive Bayesian procedure, where the proposed sequential steps are prediction, measurement and coordination updates. In the coordination step, the estimator minimises the Kullback–Liebler divergence to obtain the consensus of multiple probability density functions (PDFs). Employing the PDFs from those steps together with the affine connection on manifolds the Fisher Information Matrix (FIM) and the curvature terms of the corresponding intrinsic bounds are derived. Subsequently, the design of a distributed estimator for Riemannian information manifold with Gaussian distribution – referred to as distributed Riemannian Kalman filter – is also presented to exemplify the application of the proposed intrinsic bounds. Finally, simulations utilising the designed filter for a distributed quaternionic estimation problem verifies that the covariance matrices of the filter are never below the formulated intrinsic CRBs.