Smooth Riemannian Manifold Research Articles

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.

Gradient formulas for jump processes on manifolds

Consider jump processes on a connected compact smooth Riemannian manifold, which are constructed by the canonical projection of the processes on the bundle of orthonormal frames. The condition under which the M-valued process is Markovian will be revisited as seen in Applebaum-Estrade [1]. Moreover, the gradient formula, which will be also called the integration by parts formula, can be also studied. The obtained formula can be regarded as the extended version of the celebrated Bismut formula on the case of diffusion processes.

Emergent dynamics of a thermodynamic Cucker-Smale ensemble on complete Riemannian manifolds

We study emergent collective behaviors of a thermodynamic Cucker-Smale (TCS) ensemble on complete smooth Riemannian manifolds. For this, we extend the TCS model on the Euclidean space to a complete smooth Riemannian manifold by adopting the work [ 30 ] for a CS ensemble, and provide a sufficient framework to achieve velocity alignment and thermal equilibrium. Compared to the model proposed in [ 30 ], our model has an extra thermodynamic observable denoted by temperature, which is assumed to be nonidentical for each particle. However, for isothermal case, our model reduces to the previous CS model in [ 30 ] on a manifold in a small velocity regime. As a concrete example, we study emergent dynamics of the TCS model on the unit \begin{document}$ d $\end{document} -sphere \begin{document}$ \mathbb{S}^d $\end{document} . We show that the asymptotic emergent dynamics of the proposed TCS model on the unit \begin{document}$ d $\end{document} -sphere exhibits a dichotomy, either convergence to zero velocity or asymptotic approach toward a common great circle. We also provide several numerical examples illustrating the aforementioned dichotomy on the asymptotic dynamics of the TCS particles on \begin{document}$ \mathbb{S}^2 $\end{document} .

Kriging Riemannian Data via Random Domain Decompositions

Data taking value on a Riemannian manifold and observed over a complex spatial domain are becoming more frequent in applications, for example, in environmental sciences and in geoscience. The analysis of these data needs to rely on local models to account for the nonstationarity of the generating random process, the nonlinearity of the manifold, and the complex topology of the domain. In this article, we propose to use a random domain decomposition approach to estimate an ensemble of local models and then to aggregate the predictions of the local models through Fréchet averaging. The algorithm is introduced in complete generality and is valid for data belonging to any smooth Riemannian manifold but it is then described in details for the case of the manifold of positive definite matrices, the hypersphere and the Cholesky manifold. The predictive performances of the method are explored via simulation studies for covariance matrices and correlation matrices, where the Cholesky manifold geometry is used. Finally, the method is illustrated on an environmental dataset observed over the Chesapeake Bay (USA). Supplementary materials for this article are available online.

Cross-covariance based affinity for graphs.

The accuracy of graph based learning techniques relies on the underlying topological structure and affinity between data points, which are assumed to lie on a smooth Riemannian manifold. However, the assumption of local linearity in a neighborhood does not always hold true. Hence, the Euclidean distance based affinity that determines the graph edges may fail to represent the true connectivity strength between data points. Moreover, the affinity between data points is influenced by the distribution of the data around them and must be considered in the affinity measure. In this paper, we propose two techniques, CCGAL and CCGAN that use cross-covariance based graph affinity (CCGA) to represent the relation between data points in a local region. CCGAL also explores the additional connectivity between data points which share a common local neighborhood. CCGAN considers the influence of respective neighborhoods of the two immediately connected data points, which further enhance the affinity measure. Experimental results of manifold learning on synthetic datasets show that CCGA is able to represent the affinity measure between data points more accurately. This results in better low dimensional representation. Manifold regularization experiments on standard image dataset further indicate that the proposed CCGA based affinity is able to accurately identify and include the influence of the data points and its common neighborhood that increase the classification accuracy. The proposed method outperforms the existing state-of-the-art manifold regularization methods by a significant margin.

Topologically Stable Chain Recurrence Classes for Diffeomorphisms

Let f:M→M be a diffeomorphism of a finite dimension, smooth compact Riemannian manifold M. In this paper, we demonstrate that if a diffeomorphism f lies within the C1 interior of the set of all chain recurrence class-topologically stable diffeomorphisms, then the chain recurrence class is hyperbolic.

A Class of Fully Nonlinear Equations Arising in Conformal Geometry

Abstract In this paper, we consider the equations of Krylov type in conformal geometry on closed smooth Riemannian manifolds, which can be viewed as an extension of $\sigma _k$-Yamabe equation. Moreover, we prove local gradient and 2nd-derivative estimates for solutions to these equations and establish an existence result.

Barriers of the McKean–Vlasov energy via a mountain pass theorem in the space of probability measures

We show that the empirical process associated with a system of weakly interacting diffusion processes exhibits a form of noise-induced metastability. The result is based on an analysis of the associated McKean–Vlasov free energy, which, for suitable attractive interaction potentials, has at least two distinct global minimisers at the critical parameter value β=βc. On the torus, one of these states is the spatially homogeneous constant state, and the other is a clustered state. We show that a third critical point exists at this value. As a result, we obtain that the probability of transition of the empirical process from the constant state scales like exp⁡(−NΔ), with Δ the energy gap at β=βc. The proof is based on a version of the mountain pass theorem for lower semicontinuous and λ-geodesically convex functionals on the space of probability measures P2(M) equipped with the 2-Wasserstein metric, where M is a complete, connected, and smooth Riemannian manifold.

Emergent Behaviors of Cucker–Smale Flocks on Riemannian Manifolds

In this article, we present a new Cucker-Smale model on smooth Riemannian manifolds using the concepts of covariant derivative and parallel transport, and we also study its emergent dynamics under an a priori assumption on the energy functional. For Euclidean space, our proposed model coincides with the original Cucker-Smale model. As concrete examples, we consider three Riemannian manifolds: the unit 2-sphere, the unit circle, and the Poincaré half-plane, and provide explicit reductions from the proposed general model to aforementioned manifolds via explicit formulas for the covariant derivative and parallel transport.

Nodal solutions for a Paneitz-Branson type equation

Let (M,g) be a smooth compact Riemannian manifold of dimension n≥5. Denote Lg the Paneitz-Branson type operator. In this paper, we show that there exists a nodal solution (solution with changing sign) of the nonlinear Paneitz-Branson type equation Lgv=ϵ|v|N−2v where ϵ<0. At the end, we give a geometric application of the above equation.

Mollifier smoothing of C0-Finsler structures

A $C^0$-Finsler structure is a continuous function $F:TM \rightarrow [0,\infty)$ defined on the tangent bundle of a differentiable manifold $M$ such that its restriction to each tangent space is an asymmetric norm. We use the convolution of $F$ with the standard mollifier in order to construct a mollifier smoothing of $F$, which is a one parameter family of Finsler structures $F_\varepsilon$ (of class $\mathit{C}^\infty$ on $TM\backslash 0$) that converges uniformly to $F$ on compact subsets of $TM$. We prove that when $F$ is a Finsler structure, then the Chern connection, the Cartan connection, the Hashiguchi connection, the Berwald connection and the flag curvature of $F_\varepsilon$ converges uniformly on compact subsets to the corresponding objects of $F$. As an application of this mollifier smoothing, we study examples of two-dimensional piecewise smooth Riemannian manifolds with nonzero total curvature on a line segment. We also indicate how to extend this study to the correspondent piecewise smooth Finsler manifolds.

Bubble tree convergence for conformal metrics with [formula omitted] bounds

Given a sequence of conformal metrics {gk=uk4n−2g0} on a smooth compact boundaryless Riemannian manifold (Mn,g0). Assume the volume of gk and Ln2 norm of scalar curvatures both are bounded. We prove that, after passing to a subsequence, uk weakly converges to the bubble tree limit (u,u1,1,…,ui,α,…,ul,αl),1≤i≤l<∞,1≤α≤αi<∞ in W2,p, for some p<n2, where u∈W2,p(M,g0) and ui,α∈W2,p(Rn,gRn). Moreover, after passing to a subsequence,the sequence of metric spaces (M,dk) defined by gk converges to a connected metric space (M∞,d∞) in the Gromov-Hausdorff topology sense and limk→∞Vol(M,gk)=Hn(M∞,d∞), where Hn is the n dimensional Hausdorff measure defined by d∞.

An inverse problem for a semi-linear elliptic equation in Riemannian geometries

We study the inverse problem of unique recovery of a complex-valued scalar function V:M×C→C, defined over a smooth compact Riemannian manifold (M,g) with smooth boundary, given the Dirichlet-to-Neumann map, in a suitable sense, for the elliptic semi-linear equation −Δgu+V(x,u)=0. We show that uniqueness holds for a large class of non-linearities when the manifold is conformally transversally anisotropic. The proof is constructive and is based on a multiple-fold linearization of the semi-linear equation near complex geometric optic solutions for the linearized operator and the resulting non-linear interactions. These interactions result in the study of a weighted integral transform along geodesics, that we call the Jacobi weighted ray transform.

On the singular second-order elliptic equation

Let (M,g) be a smooth and compact Riemannian manifold with dimensions n≥3. This paper deals with the existence of nontrivial solutions for a nonlinear second-order elliptic equation with a singular term and critical Sobolev growth. We provide a geometric application of this singular equation.

An existence theorem on the isoperimetric ratio over scalar-flat conformal classes

Let (M,g) be a smooth compact Riemannian manifold of dimension n with smooth boundary ∂M, admitting a scalar-flat conformal metric. We prove that the supremum of the isoperimetric ratio over the scalar-flat conformal class is strictly larger than the best constant of the isoperimetric inequality in the Euclidean ball, and consequently is achieved, if either (i) 9≤n≤11 and ∂M has a nonumbilic point; or (ii) 7≤n≤9, ∂M is umbilic and the Weyl tensor does not vanish identically on the boundary. This is a continuation of the work [12] by the second named author and Xiong.

Sharp Cheeger\u2013Buser Type Inequalities in mathsf {RCD}(K,infty ) Spaces

The goal of the paper is to sharpen and generalise bounds involving Cheeger’s isoperimetric constant h and the first eigenvalue lambda _{1} of the Laplacian. A celebrated lower bound of lambda _{1} in terms of h, lambda _{1}ge h^{2}/4, was proved by Cheeger in 1970 for smooth Riemannian manifolds. An upper bound on lambda _{1} in terms of h was established by Buser in 1982 (with dimensional constants) and improved (to a dimension-free estimate) by Ledoux in 2004 for smooth Riemannian manifolds with Ricci curvature bounded below. The goal of the paper is twofold. First: we sharpen the inequalities obtained by Buser and Ledoux obtaining a dimension-free sharp Buser inequality for spaces with (Bakry–Émery weighted) Ricci curvature bounded below by Kin mathbb R (the inequality is sharp for K>0 as equality is obtained on the Gaussian space). Second: all of our results hold in the higher generality of (possibly non-smooth) metric measure spaces with Ricci curvature bounded below in synthetic sense, the so-called mathsf {RCD}(K,infty ) spaces.

The algebra of Wick polynomials of a scalar field on a Riemannian manifold

On a connected, oriented, smooth Riemannian manifold without boundary we consider a real scalar field whose dynamics is ruled by [Formula: see text], a second-order elliptic partial differential operator of Laplace type. Using the functional formalism and working within the framework of algebraic quantum field theory and of the principle of general local covariance, first we construct the algebra of locally covariant observables in terms of equivariant sections of a bundle of smooth, regular polynomial functionals over the affine space of the parametrices associated to [Formula: see text]. Subsequently, adapting to the case in hand a strategy first introduced by Hollands and Wald in a Lorentzian setting, we prove the existence of Wick powers of the underlying field, extending the procedure to smooth, local and polynomial functionals and discussing in the process the regularization ambiguities of such procedure. Subsequently we endow the space of Wick powers with an algebra structure, dubbed E-product, which plays in a Riemannian setting the same role of the time-ordered product for field theories on globally hyperbolic spacetimes. In particular, we prove the existence of the E-product and we discuss both its properties and the renormalization ambiguities in the underlying procedure. As the last step, we extend the whole analysis to observables admitting derivatives of the field configurations and we discuss the quantum Møller operator which is used to investigate interacting models at a perturbative level.

Certain Geometric Aspects of a Class of Almost Contact Structures on a Smooth Metric Manifold

The classication of Smooth Geometrical Manifolds still remains an open problem. The concept of almost contact Riemannian manifolds provides neat descriptions and distinctions between classes of odd and even dimensional manifolds and their geometries. We construct an almost contact structure which is related to almost contact 3-structure carried on a smooth Riemannian manifold (M, gM) of dimension (5n + 4) such that gcd(2, n) = 1. Starting with the almost contact metric manifolds (N4n+3, gN) endowed with structure tensors (ϕi, ξj , ηk) such that 1 ≤ i, j, k ≤ 3 of types (1, 1), (1, 0), (0, 1) respectively, we establish that there exists a structure (ϕ4, ξ4, η4) on (N4n+3 ⊗ Rd) ≈ M; gcd(4, d) = 1, d|2n + 1, constructed as linear combinations of the three structures on (N4n+3, gN) . We study some algebraic properties of the tensors of the constructed almost contact structure and further explore the Geometry of the two manifolds (N4n+3⊗Rd) ≈ M and N4n+3 via a !submersion F : (N4n+3 ⊗Rd) ↩→ (N4n+3) and the metrics gM respective gN between them. This provides new forms of Gauss-Weigarten's equations, Gauss-Codazzi equations and the Ricci equations incorporating the submersion other than the First and second Fundamental coecients only. Fundamentally, this research has revealed that the structure (ϕ4, ξ4, η4) is constructible and it is carried on the hidden compartment of the manifold M∼=(N4n+3 ⊗ Rd) (d|2n + 1) which is related to the manifold (N4n+3).

On the geometry of the tangent bundle with vertical rescaled generalized Cheeger- Gromoll metric

Let (M, g) be an n-dimensional smooth Riemannian manifold. In the present paper, we introduce a new class of natural metrics denoted by G f and called the vertical rescaled generalized Cheeger-Gromoll metric on the tangent bundle TM. We calculate its Levi-Civita connection and Riemannian curvature tensor. We study the geometry of (TM, G f ).

Dynamical aspects of the generalized Schrödinger problem via Otto calculus – A heuristic point of view

The defining equation (\ast)\qquad \dot \omega_t=-F'(\omega_t) of a gradient flow is kinetic in essence. This article explores some dynamical (rather than kinetic) features of gradient flows (i) by embedding equation (\ast) into the family of slowed down gradient flow equations: \dot \omega^{\varepsilon}_t=- \varepsilon F'( \omega ^{ \varepsilon}_t) , where \varepsilon &gt; 0 , and (ii) by considering the accelerations \ddot \omega ^{ \varepsilon}_t . We shall focus on Wasserstein gradient flows. Our approach is mainly heuristic. It relies on Otto calculus. A special formulation of the Schrödinger problem consists in minimizing some action on the Wasserstein space of probability measures on a Riemannian manifold subject to fixed initial and final data. We extend this action minimization problem by replacing the usual entropy, underlying the Schrödinger problem, with a general function on the Wasserstein space. The corresponding minimal cost approaches the squared Wasserstein distance when the fluctuation parameter \varepsilon tends to zero. We show heuristically that the solutions satisfy some Newton equation, extending a recent result of Conforti. The connection with Wasserstein gradient flows is established and various inequalities, including evolutional variational inequalities and contraction inequalities under a curvature-dimension condition, are derived with a heuristic point of view. As a rigorous result we prove a new and general contraction inequality for the Schrödinger problem under a Ricci lower bound on a smooth and compact Riemannian manifold.