Diffeomorphisms Research Articles

Structure theorems for actions of homeomorphism groups

We give general classification and structure theorems for actions of groups of homeomorphisms and diffeomorphisms on manifolds, reminiscent of classical results for actions of (locally) compact groups. This gives a negative answer to Ghys’s extension problem for diffeomorphisms of manifolds with boundary, as well as a classification of all homomorphisms Homeo0(M)→Homeo0(N) when dim(M)=dim(N) (and related results for diffeomorphisms), and a complete classification of actions of Homeo0(S1) on surfaces. This resolves many problems in a program initiated by Ghys, and gives definitive answers to conjectures of Militon and Hurtado and a question of Rubin.

Constructing the Virasoro Groups Using Differential Cohomology

Abstract The Virasoro groups are a family of central extensions of $\textrm{Diff}^+(S^1)$, the group of orientation-preserving diffeomorphisms of $S^1$, by the circle group $\mathbb T$. We give a novel, geometric construction of these central extensions using “off-diagonal” differential lifts of the first Pontryagin class, thus affirmatively answering a question of Freed–Hopkins.

The Toda Flow as a Porous Medium Equation

We describe the geometry of the incompressible porous medium (IPM) equation: we prove that it is a gradient dynamical system on the group of area-preserving diffeomorphisms and has a special double-bracket form. Furthermore, we show its similarities and differences with the dispersionless Toda system. The Toda flow describes an integrable interaction of several particles on a line with an exponential potential between neighbours, while its continuous version is an integrable PDE, whose physical meaning was obscure. Here we show that this continuous Toda flow can be naturally regarded as a special IPM equation, while the key double-bracket property of Toda is shared by all equations of the IPM type, thus manifesting their gradient and non-autonomous Hamiltonian origin. Finally, we comment on Toda and IPM modifications of the QR diagonalization algorithm, as well as describe double-bracket flows in an invariant setting of general Lie groups with arbitrary inertia operators.

The diffeomorphism group of the solid closed torus and Hochschild homology

We prove that for a self-injective ribbon Grothendieck-Verdier category C \mathcal {C} in the sense of Boyarchenko-Drinfeld the cyclic action on the Hochschild complex of C \mathcal {C} extends to an action of the diffeomorphism group of the solid closed torus S 1 × D 2 \mathbb {S}^1 \times \mathbb {D}^2 .

Geometric Hydrodynamics in Open Problems

Geometric Hydrodynamics has flourished ever since the celebrated 1966 paper of V. Arnold. In this paper we present a collection of open problems along with several new constructions in fluid dynamics and a concise survey of recent developments and achievements in this area. The topics discussed include variational settings for different types of fluids, models for invariant metrics, the Cauchy and boundary value problems, partial analyticity of solutions to the Euler equations, their steady and singular vorticity solutions, differential and Hamiltonian geometry of diffeomorphism groups, long-time behaviour of fluids, as well as mechanical models of direct and inverse cascades.

Stable cohomology of graph complexes

We study three graph complexes related to the higher genus Grothendieck–Teichmüller Lie algebra and diffeomorphism groups of manifolds. We show how the cohomology of these graph complexes is related, and we compute the cohomology as the genus g tends to infty . As a byproduct, we find that the Malcev completion of the genus g mapping class group relative to the symplectic group is Koszul in the stable limit, partially answering a question of Hain.

Bounded and unbounded cohomology of homeomorphism and diffeomorphism groups

We determine the bounded cohomology of the group of homeomorphisms of certain low-dimensional manifolds. In particular, for the group of orientation-preserving homeomorphisms of the circle and of the closed 2-disc, it is isomorphic to the polynomial ring generated by the bounded Euler class. These seem to be the first examples of groups for which the entire bounded cohomology can be described without being trivial. We further prove that the $$C^r$$ -diffeomorphisms groups of the circle and of the closed 2-disc have the same bounded cohomology as their homeomorphism groups, so that both differ from the ordinary cohomology of $$C^r$$ -diffeomorphisms when $$r>1$$ . Finally, we determine the low-dimensional bounded cohomology of homeo- and diffeomorphism of the spheres $$S^n$$ and of certain 3-manifolds. In particular, we answer a question of Ghys by showing that the Euler class in $$H^4(\textrm{Homeo}_\circ (S^3))$$ is unbounded.

Geometric Structures on the Orbits of Loop Diffeomorphism Groups and Related Heavenly-Type Hamiltonian Systems. II

Geometric Structures on the Orbits of Loop Diffeomorphism Groups and Related Heavenly-Type Hamiltonian Systems. II

Models of Viscous Fluids Generated by Martingales on the Groups of Diffeomorphisms

Models of Viscous Fluids Generated by Martingales on the Groups of Diffeomorphisms

Geometric Structures on the Orbits of Loop Diffeomorphism Groups and Related Heavenly-Type Hamiltonian Systems. I

Geometric Structures on the Orbits of Loop Diffeomorphism Groups and Related Heavenly-Type Hamiltonian Systems. I

There are no exotic actions of diffeomorphism groups on 1-manifolds

Let M be a manifold and N a 1 -dimensional manifold. Assuming that r\neq\dim(M)+1 , we show that any nontrivial homomorphism \rho:\mathsf{Diff}^r_c(M)\rightarrow\mathsf{Homeo}(N) has a standard form: necessarily M is 1 -dimensional, and there are countably many embeddings \phi_i: M\rightarrow N with disjoint images such that the action of \rho is conjugate (via the product of the \phi_i ) to the diagonal action of \mathsf{Diff}^r_c(M) on M\times M\times\cdots on \bigcup_i\phi_i(M) , and trivial elsewhere. This solves a conjecture of Matsumoto. We also show that the groups \mathsf{Diff}^r_c(M) have no countable index subgroups.

A Teichmüller space for negatively curved surfaces

AbstractWe first describe the action of the fundamental group of a closed surface of variable negative curvature on the oriented geodesics in its universal covering in terms of a naturally defined flat connection whose holonomy lies in the group of Hamiltonian diffeomorphisms of . Consideration of the holonomy necessitates an extension from Riemannian to Finsler metrics. The second part of the paper follows the Higgs bundle approach to flat connections adapted to this infinite‐dimensional group and focuses on a family of metrics, relying on a construction of O. Biquard, which is parametrised by the infinite‐dimensional space of CR functions on the unit circle bundle of a hyperbolic surface. This generates an alternative approach to defining a connection and offers the possibility of this vector space representing a moduli space which generalizes and includes the classical Teichmüller space.

Shape analysis via gradient flows on diffeomorphism groups

We study a Riemannian gradient flow on Sobolev diffeomorphisms for the problem of image registration. The energy functional quantifies the effect of transforming a template to a target, while also penalizing non-isometric deformations. The main result is well-posedness of the flow. We also give a geometric description of the gradient in terms of the momentum map.

Asymptotic representations of Hamiltonian diffeomorphisms and quantization

We show that for a special class of geometric quantizations with “small” quantum errors, the quantum classical correspondence gives rise to an asymptotic projective unitary representation of the group of Hamiltonian diffeomorphisms. As an application, we get an obstruction to Hamiltonian actions of finitely presented groups.

Shape Analysis of Functional Data With Elastic Partial Matching.

Elastic Riemannian metrics have been used successfully for statistical treatments of functional and curve shape data. However, this usage suffers from a significant restriction: the function boundaries are assumed to be fixed and matched. In practice, functional data often comes with unmatched boundaries. It happens, for example, in dynamical systems with variable evolution rates, such as COVID-19 infection rate curves associated with different geographical regions. Here, we develop a Riemannian framework that allows for partial matching, comparing, and clustering of functions with phase variability and uncertain boundaries. We extend past work by (1) Defining a new diffeomorphism group G over the positive reals that is the semidirect product of a time-warping group and a time-scaling group; (2) Introducing a metric that is invariant to the action of G; (3) Imposing a Riemannian Lie group structure on G to allow for an efficient gradient-based optimization for elastic partial matching; and (4) Presenting a modification that, while losing the metric property, allows one to control the amount of boundary disparity in the registration. We illustrate this framework by registering and clustering shapes of COVID-19 rate curves, identifying basic patterns, minimizing mismatch errors, and reducing variability within clusters compared to previous methods.

The Flux Homomorphism on a Surface with Boundary and Central Extensions of Diffeomorphism Groups

Let Σg,11 be a genus g compact oriented surface with one boundary component and one marked point x. Let G be the identity component of the group of symplectomorphisms that preserve the marked point x. By using the flux homomorphism and the short exact sequence 1→Grel→G→Diff+(S1)→1, we construct a central ℝ-extension of Diff+(S1). We also show that the second cohomology class corresponding to the central ℝ-extension is equal to the Euler class of Diff+(S1) up to non-zero constant multiple.

Non-trivial smooth families of K3 surfaces

Let X be a complex K3 surface, $$\textrm{Diff}(X)$$ the group of diffeomorphisms of X and $$\textrm{Diff}_0(X)$$ the identity component. We prove that the fundamental group of $$\textrm{Diff}_0(X)$$ contains a free abelian group of countably infinite rank as a direct summand. The summand is detected using families Seiberg–Witten invariants. The moduli space of Einstein metrics on X is used as a key ingredient in the proof.

Stochastic Variational Principles for Dissipative Equations with Advected Quantities

This paper presents symmetry reduction for material stochastic Lagrangian systems with advected quantities whose configuration space is a Lie group. Such variational principles yield deterministic as well as stochastic constrained variational principles for dissipative equations of motion in spatial representation. The general theory is presented for the finite-dimensional situation. In infinite dimensions we obtain partial differential equations and stochastic partial differential equations. When the Lie group is, for example, a diffeomorphism group, the general result is not directly applicable but the setup and method suggest rigorous proofs valid in infinite dimensions which lead to similar results. We apply this technique to the compressible Navier–Stokes equation and to magnetohydrodynamics for charged viscous compressible fluids. A stochastic Kelvin–Noether theorem is presented. We derive, among others, the classical deterministic dissipative equations from purely variational and stochastic principles, without any appeal to thermodynamics.

Two-Point Boundary Value Problems on Diffeomorphism Groups

Abstract We consider a variety of geodesic equations on Sobolev diffeomorphism groups, including the equations of ideal hydrodynamics. We prove that solutions of the corresponding two-point boundary value problems are precisely as smooth as their boundary conditions. We further utilise this regularity property to construct continuously differentiable exponential maps in the Frechét setting.

Conjugate point criteria on the area-preserving diffeomorphism group

This paper answers some questions about conjugate points along the geodesics corresponding to steady 2D Euler flows, posed by a paper of Drivas-Misiołek-Shi-Yoneda. We present a new sufficient criterion for the existence of conjugate points, which improves on the criterion of Misiołek. It applies in any rotational cell of a steady 2D Euler flow, and in case of rotational symmetry it captures all known conjugate points. We give a general construction of the surfaces that admit a steady fluid with given area form, velocity profile, and vorticity profile, and from this we show how to detect conjugate points in a single rotational cell of a steady flow. When the velocity profile has a local extremum, the criterion becomes particularly simple. Several examples are provided, and in an appendix we use the Misiołek criterion to give some new examples of conjugate points along Kolmogorov flows on the torus.