Smooth Perturbations of the Functional Calculus and Applications to Riemannian Geometry on Spaces of Metrics

Bauer, Martin; Bruveris, Martins; Harms, Philipp; Michor, Peter W.

doi:10.1007/s00220-021-04264-y

Cited by 7 publications

(4 citation statements)

References 66 publications

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“…Thus, there is no loss or gain of regularity along geodesics. Similar results have been shown in many specific cases [20,18,14,10]. We will formulate the following theorem for general right invariant flows; the result for the geodesic equation follows by interpreting this equation as a flow equation (with respect to the geodesic spray) and noting that the geodesic spray is a right-invariant vector field.…”

Section: Riemannian Geometry On Half-lie Groupssupporting

confidence: 53%

Regularity and completeness of half-Lie groups

Bauer¹,

Harms²,

Michor³

2023

Preprint

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Half Lie groups exist only in infinite dimensions: They are smooth manifolds and topological groups such that right translations are smooth, but left translations are merely required to be continuous. The main examples are groups of H s or C k diffeomorphisms and semidirect products of a Lie group with kernel an infinite dimensional representation space. Here, we investigate mainly Banach half-Lie groups, the groups of their C k -elements, extensions, and right invariant strong Riemannian metrics on them: surprisingly the full Hopf-Rinow theorem holds, which is wrong in general even for Hilbert manifolds.

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Section: Riemannian Geometry On Half-lie Groupssupporting

confidence: 53%

Regularity and completeness of half-Lie groups

Bauer¹,

Harms²,

Michor³

2023

Preprint

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“…The local well-posedness of the geodesic equation when the inertia operator A is a differential operator has been implicitly solved in the seminal article of Ebin and Marsden [24], see also [52,53,20,58,32,47,44,38,39], and hence for H k -metrics on diffeomorphism groups, where k is an integer. This result has been extended to invariant metrics on several related spaces of mappings, such as spaces of immersions, Riemannian metrics and the Virasoro-Bott group, see [39,6,7,3,11,4]. In a series of papers [29,28,5,41], the local and global well-posedness problem for the general EPDiff equation on Diff ∞ (T d ) or Diff H ∞ (R d ) when the inertia operator is a non-local Fourier multiplier was solved.…”

Section: Introductionmentioning

confidence: 99%

Well-posedness of the EPDiff equation with a pseudo-differential inertia operator

Bauer

Bruveris²,

Cismas

et al. 2020

Journal of Differential Equations

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In this article we study the class of right-invariant, fractional order Sobolev-type metrics on groups of diffeomorphisms of a compact manifold M . Our main result concerns well-posedness properties for the corresponding Euler-Arnold equations, also called the EPDiff equations, which are of importance in mathematical physics and in the field of shape analysis and template registration. Depending on the order of the metric, we will prove both local and global well-posedness results for these equations. As a result of our analysis we will also obtain new commutator estimates for elliptic pseudo-differential operators.

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“…\end{equation*}$$Thus, all of these matrices are positive definite. Taking the square root of a symmetric positive definite matrix is smooth by the implicit function theorem or, more generally, because the functional calculus is real analytic, see [3]. In particular, the matrix square root is Lipschitz continuous on compacts.…”

Section: Brownian Motion Of Two Landmarksmentioning

confidence: 99%

Long‐time existence of Brownian motion on configurations of two landmarks

Habermann,

Harms,

Sommer

2024

Bulletin of London Math Soc

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We study Brownian motion on the space of distinct landmarks in , considered as a homogeneous space with a Riemannian metric inherited from a right‐invariant metric on the diffeomorphism group. As of yet, there is no proof of long‐time existence of this process, despite its fundamental importance in statistical shape analysis, where it is used to model stochastic shape evolutions. We make some first progress in this direction by providing a full classification of long‐time existence for configurations of exactly two landmarks, governed by a radial kernel. For low‐order Sobolev kernels, we show that the landmarks collide with positive probability in finite time, whilst for higher‐order Sobolev and Gaussian kernels, the landmark Brownian motion exists for all times. We illustrate our theoretical results by numerical simulations.

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Smooth Perturbations of the Functional Calculus and Applications to Riemannian Geometry on Spaces of Metrics

Cited by 7 publications

References 66 publications

Regularity and completeness of half-Lie groups

Regularity and completeness of half-Lie groups

Well-posedness of the EPDiff equation with a pseudo-differential inertia operator

Long‐time existence of Brownian motion on configurations of two landmarks

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