Eigenforms of the Laplacian for Riemannian V-submersions

Gilkey, Petere; Kim, Hong-Jong; Park, JeongHyeong

doi:10.2748/tmj/1140727070

Cited by 5 publications

(2 citation statements)

References 17 publications

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“…to hold for arbitrary f ∈ C ∞ (Y ) is that the fibers of the submersion π Y are minimal, i.e., they are surfaces of vanishing mean curvature [54][55][56]. Clearly, the latter is not the case given the nonconstant curvature of the streamlines in Fig.…”

Section: Koopman Eigenvalues and Eigenfunctionsmentioning

confidence: 99%

Extraction and prediction of coherent patterns in incompressible flows through space–time Koopman analysis

Giannakis

Das

2020

Physica D: Nonlinear Phenomena

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We develop methods for detecting and predicting the evolution of coherent spatiotemporal patterns in incompressible time-dependent fluid flows driven by ergodic dynamical systems. Our approach is based on representations of the generators of the Koopman and Perron-Frobenius groups of operators governing the evolution of observables and probability measures on Lagrangian tracers, respectively, in a smooth orthonormal basis learned from velocity field snapshots through the diffusion maps algorithm. These operators are defined on the product space between the state space of the fluid flow and the spatial domain in which the flow takes place, and as a result their eigenfunctions correspond to global space-time coherent patterns under a skew-product dynamical system. Moreover, using this data-driven representation of the generators in conjunction with Leja interpolation for matrix exponentiation, we construct model-free prediction schemes for the evolution of observables and probability densities defined on the tracers. We present applications to periodic Gaussian vortex flows and aperiodic flows generated by Lorenz 96 systems.

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Section: Koopman Eigenvalues and Eigenfunctionsmentioning

confidence: 99%

Extraction and prediction of coherent patterns in incompressible flows through space–time Koopman analysis

Giannakis

Das

2020

Physica D: Nonlinear Phenomena

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“…Many researchers have studied the spectrum of the Laplacian and Dirac-type operators on families of manifolds where the metric is collapsed. We point out in particular the references [9], [12], [18], where the behavior of the spectrum of Laplacians on Riemannian submersions are noted under collapse of the fiber metrics. In [22], R. R. Mazzeo and R. B. Melrose related the properties of the Laplace eigenvalues under adiabatic limits in a Riemannian fiber bundle to the Leray spectral sequence, and J. A.Álvarez-López and Y. Kordyukov extended this analysis in [2] to the more general case of Riemannian foliations; see [20] for an exposition and further references.…”

Section: Introductionmentioning

confidence: 99%

Riemannian flows and adiabatic limits

Habib

Richardson

2018

Int. J. Math.

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The Regularized Mean Curvature Flow for Invariant Hypersurfaces in a Hilbert Space

Koike

2014

Springer Proceedings in Mathematics &Amp; Statistics

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In this paper, we investigate the regularized mean curvature flow starting from an invariant hypersurface in a Hilbert space equipped with an isometric and almost free action of a Hilbert Lie group whose orbits are regularized minimal. We prove that, if the invariant hypersurface satisfies a certain kind of horizontally convexity condition and its image by the orbit map of the Hilbert Lie group action is included by the geodesic ball of some radius, then it collapses to an orbit of the Hilbert Lie group action along the regularized mean curvature flow. As an application of this result to the gauge theory, we derive a result for the behaviour of the holonomy elements (along a fixed loop) of connections belonging to some based gauge-invariant hypersurface in the space of connections of the principal bundle having a compact Lie group as the structure group over a compact Riemannian manifold along the regularized mean curvature flow starting from the hypersurface. almost all relevant cases, these regularized traces coincide. In this paper, we adopt the regularized trace defined in [HLO]. Let M be a proper Fredholm submanifold in V immersed by f . If, for each normal vector ξ of f , the regularized trace Tr r A ξ of the shape operator A ξ of f and the trace Tr A 2 ξ of A 2 ξ exist, then M (or f ) is said to be regularizable. See Section 3 about the definition of the regularized trace Tr r A ξ . Let M be a Hilbert manifold and f t (0 ≤ t < T ) be a C ∞ -family of regularizable immersions of codimension one of M into V which admit a unit normal vector field ξ t . The regularized mean curvature vector H t is defined by. We call f t 's (0 ≤ t < T ) the regularized mean curvature flow if the following evolution equation holds:R. S. Hamilton ([Ha]) proved the existenceness and the uniqueness (in short time) of solutions of a weakly parabolic equation for sections of a finite dimensional vector bundle. The evolution equation (1.1) is regarded as the evolution equation for sections of the infinite dimensional trivial vector bundle M × V over M . Also, M is not compact. Hence we cannot apply the Hamilton's result to this evolution equation (1. 1). Also, we must impose certain kind of infinite dimensional invariantness for f because M is not compact. Thus, we cannot show the existenceness and the uniqueness (in short time) of solutions of (1.1) in general. However we ([Koi]) showed the existenceness and the uniqueness (in short time) of solutions of (1.1) in the following special case. We consider a isometric almost free action of a Hilbert Lie group G on a Hilbert space V whose orbits are regularized minimal, that is, they are regularizable submanifold and their regularized mean curvature vectors vanish, where "almost free" means that the isotropy group of the action at each point is finite. Denote by N the orbit space V /G, which is an orbifold. Give N the Riemannian orbimetric g N such that the orbit map φ : V → N is a Riemannian orbisubmersion. Let M (⊂ V ) be a G-invariant submanifold in V . Assume that M := φ(M ) is compa...

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Eigenforms of the Laplacian for Riemannian V-submersions

Abstract: Let π : Z → Y be a Riemannian V -submersion of compact Vmanifolds. We study when the pull-back of an eigenform of the Laplacian on Y is an eigenform of the Laplacian on Z, and when the associated eigenvalue can change.2000 Mathematics Subject Classification. Primary 58J50.

Cited by 5 publications

References 17 publications

Extraction and prediction of coherent patterns in incompressible flows through space–time Koopman analysis

Extraction and prediction of coherent patterns in incompressible flows through space–time Koopman analysis

Riemannian flows and adiabatic limits

The Regularized Mean Curvature Flow for Invariant Hypersurfaces in a Hilbert Space

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