General Riemannian 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.

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.

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.

Metric-Driven 3D Frame Field Generation.

Controlling the size and shear of elements is crucial in pure hex or hex-dominant meshing. To this end, non-orthonormal frame fields that are almost everywhere integrable (except for the singularities) can play a key role. However, it is often challenging or impossible to generate such a frame field under the tight control of a general Riemannian metric field. Therefore, we propose to solve a relatively weaker problem, i.e., generating such a frame field for a Riemannian metric field that is flat away from singularities. Such a metric field admits a local isometry to 3D Euclidean space. Applying Cartans first structural equation to the associated rotation field, i.e., the rotation part of the frame field, we show that the rotation field must have zero covariant derivatives under the 3D connection induced by the metric field. This observation leads to a metric-aware smoothness measure, equivalent to local integrability. The use of such a measure can be justified on meshes associated with locally flat metric fields. We also propose a method to generate smooth metric fields under a few intuitive constraints. On cuboid shapes, our method generates singularities aware of the metric fields, which makes the parameterization match the input metric fields better than the conventional methods. For generic shapes, while our method generates visually similar results to those using boundary frame fields to guide the metric field generation, the integrability and consistency of the metric fields are still improved, as reflected by the statistics.

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.

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.

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.

Well-defined forward operators in dynamic diffractive tensor tomography using viscosity solutions of transport equations

We consider a general setting for dynamic tensor field tomography in an inhomogeneous refracting and absorbing medium as an inverse source problem for the associated transport equation. Following Fermat's principle, the Riemannian metric in the considered domain is generated by the refractive index of the medium. There is a wealth of results for the inverse problem of recovering a tensor field from its longitudinal ray transform in a static Euclidean setting, whereas there are only a few inversion formulas and algorithms existing for general Riemannian metrics and time-dependent tensor fields. It is a well-known fact that tensor field tomography is equivalent to an inverse source problem for a transport equation where the ray transform serves as given boundary data. We prove that this result extends to the dynamic case. Interpreting dynamic tensor tomography as an inverse source problem represents a holistic approach in this field. To guarantee that the forward mappings are well defined, it is necessary to prove existence and uniqueness for the underlying transport equations. Unfortunately, the bilinear forms of the associated weak formulations do not satisfy the coercivity condition. To this end we transfer to viscosity solutions and prove their unique existence in appropriate Sobolev (static case) and Sobolev–Bochner (dynamic case) spaces under a certain assumption that allows only small variations of the refractive index. Numerical evidence is given that the viscosity solution solves the original transport equation if the viscosity term turns to zero.

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 .

Parseval Wavelet Frames on Riemannian Manifold

We construct Parseval wavelet frames in $$L^2(M)$$ for a general Riemannian manifold M and we show the existence of wavelet unconditional frames in $$L^p(M)$$ for $$1< p <\infty $$ . This is made possible thanks to smooth orthogonal projection decomposition of the identity operator on $$L^2(M)$$ , which was recently proven by Bownik et al. (Potential Anal 54:41–94, 2021). We also show a characterization of Triebel–Lizorkin $${\mathbf {F}}_{p,q}^s(M)$$ and Besov $${\mathbf {B}}_{p,q}^s(M)$$ spaces on compact manifolds in terms of magnitudes of coefficients of Parseval wavelet frames. We achieve this by showing that Hestenes operators are bounded on $${\mathbf {F}}_{p,q}^s(M)$$ and $${\mathbf {B}}_{p,q}^s(M)$$ spaces on manifolds M with bounded geometry.

Hydrodynamic derivation of the Gross–Pitaevskii equation in general Riemannian metric

Here we show that the Gross–Pitaevskii equation (GPE) for Bose–Einstein condensates (BECs) admits hydrodynamic interpretation in a general Riemannian metric, and show that in this metric the momentum equation has a new term that is associated with local curvature and density distribution profile. In particular conditions of steady state a new Einstein’s field equation is determined in presence of negative curvature. Since GPE governs BECs defects that are useful, analogue models in cosmology, a relativistic form of GPE is also considered to show connection with models of analogue gravity, thus providing further grounds for future investigations of black hole dynamics in cosmology.

Geometric and energy-aware decomposition of the Navier–Stokes equations: A port-Hamiltonian approach

A port-Hamiltonian model for compressible Newtonian fluid dynamics is presented in entirely coordinate-independent geometric fashion. This is achieved by the use of tensor-valued differential forms that allow us to describe the interconnection of the power preserving structure which underlies the motion of perfect fluids to a dissipative port which encodes Newtonian constitutive relations of shear and bulk stresses. The relevant diffusion and the boundary terms characterizing the Navier–Stokes equations on a general Riemannian manifold arise naturally from the proposed construction.

Bounds for discrepancies in the Hamming space

We derive bounds for the ball Lp-discrepancies in the Hamming space for 0<p<∞ and p=∞. Sharp estimates of discrepancies have been obtained for many spaces such as the Euclidean spheres and more general compact Riemannian manifolds. In the present paper, we show that the behavior of discrepancies in the Hamming space differs fundamentally because the volume of the ball in this space depends on its radius exponentially while such a dependence for the Riemannian manifolds is polynomial.

Einstein field equations for Bose-Einstein condensates in cosmology

In this work we consider the Gross-Pitaevskii equation for Bose-Einstein condensates (BECs) in a general Riemannian metric. Given initial conditions dictated by an external potential, we consider the free expansion of the condensate when the external potential is turned off. Focusing on the forces associated with the geometry of the initial configuration, we show how these are related to the Ricci curvature tensor and the Ricci scalar and we find an Einstein field equation governing the steady flow. Some important correlations between the study of defects in BECs and the appearance of cosmological singularities will be addressed, in particular the emergence of an effective Lorentzian spacetime geometry, which is what is needed to obtain Hawking radiation effects.

New formulas for the Laplacian of distance functions and applications

The goal of the paper is to prove an exact representation formula for the Laplacian of the distance (and more generally for an arbitrary 1-Lipschitz function) in the framework of metric measure spaces satisfying Ricci curvature lower bounds in a synthetic sense (more precisely in essentially non-branching MCP(K,N)-spaces). Such a representation formula makes apparent the classical upper bounds and also some new lower bounds, together with a precise description of the singular part. The exact representation formula for the Laplacian of 1-Lipschitz functions (in particular for distance functions) holds also (and seems new) in a general complete Riemannian manifold. We apply these results to prove the equivalence of CD(K,N) and a dimensional Bochner inequality on signed distance functions. Moreover we obtain a measure-theoretic Splitting Theorem for infinitesimally Hilbertian essentially non-branching spaces verifying MCP(0,N).

Stochastic heat equations for infinite strings with values in a manifold

In the paper, we construct conservative Markov processes corresponding to the martingale solutions to the stochastic heat equation on R + \mathbb {R}^+ or R \mathbb {R} with values in a general Riemannian manifold, which is only assumed to be complete and stochastic complete. This work is an extension of the previous paper of Röckner and the second, third, and fourth authors [SIAM J. Math. Anal. 52 (2020), pp. 2237–2274] on finite volume case. Moveover, we also obtain some functional inequalities associated to these Markov processes. This implies that on infinite volume case, the exponential ergodicity of the solution of the Ricci curvature is strictly positive and the non-ergodicity of the process if the sectional curvature is negative.

Riemannian Proximal Policy Optimization

In this paper, we propose a general Riemannian proximal optimization algorithm with guaranteed convergence to solve Markov decision process (MDP) problems. To model policy functions in MDP, we employ Gaussian mixture model (GMM) and formulate it as a non-convex optimization problem in the Riemannian space of positive semidefinite matrices. For two given policy functions, we also provide its lower bound on policy improvement by using bounds derived from the Wasserstein distance of GMMs. Preliminary experiments show the efficacy of our proposed Riemannian proximal policy optimization algorithm.

Small-time expansion of the Fokker–Planck kernel for space and time dependent diffusion and drift coefficients

We study the general solution of the Fokker–Planck equation in d dimensions with arbitrary space and time dependent diffusion matrix and drift terms. We show how to construct the solution for arbitrary initial distributions as an asymptotic expansion for small time. This generalizes the well-known asymptotic expansion of the heat kernel for the Laplace operator on a general Riemannian manifold. We explicitly work out the general solution to leading and next-to-leading order in this small-time expansion, as well as to next-to-next-to-leading order for vanishing drift. We illustrate our results on a several examples.