Diffeomorphisms Research Articles

Gravitational edge modes, coadjoint orbits, and hydrodynamics

The phase space of general relativity in a finite subregion is characterized by edge modes localized at the codimension-2 boundary, transforming under an infinite-dimensional group of symmetries. The quantization of this symmetry algebra is conjectured to be an important aspect of quantum gravity. As a step towards quantization, we derive a complete classification of the positive-area coadjoint orbits of this group for boundaries that are topologically a 2-sphere. This classification parallels Wigner’s famous classification of representations of the Poincaré group since both groups have the structure of a semidirect product. We find that the total area is a Casimir of the algebra, analogous to mass in the Poincaré group. A further infinite family of Casimirs can be constructed from the curvature of the normal bundle of the boundary surface. These arise as invariants of the little group, which is the group of area-preserving diffeomorphisms, and are the analogues of spin. Additionally, we show that the symmetry group of hydrodynamics appears as a reduction of the corner symmetries of general relativity. Coadjoint orbits of both groups are classified by the same set of invariants, and, in the case of the hydrodynamical group, the invariants are interpreted as the generalized enstrophies of the fluid.

Deformations of circle-valued Morse functions on 2-torus

In this paper we give an algebraic description of fundamental groups of orbits of circle-valued Morse functions on T2 with respect to the action of the group of diffeomorphisms of T2

(Un)distorted diffeomorphisms in different regularities

We build the first examples of diffeomorphisms that are distorted in a group of C1 diffeomorphisms yet undistorted in the corresponding group of C2 diffeomorphisms. This explicit construction is performed for the closed interval.

Homological stability for moduli spaces of disconnected submanifolds, I

A well-known property of unordered configuration spaces of points (in an open, connected manifold) is that their homology stabilises as the number of points increases. We generalise this result to moduli spaces of submanifolds of higher dimension, where stability is with respect to the number of components having a fixed diffeomorphism type and isotopy class. As well as for unparametrised submanifolds, we prove this also for partially-parametrised submanifolds -- where a partial parametrisation may be thought of as a superposition of parametrisations related by a fixed subgroup of the mapping class group. In a companion paper (arXiv:1807.07558), this is further generalised to submanifolds equipped with labels in a bundle over the embedding space, from which we deduce corollaries for the stability of diffeomorphism groups of manifolds with respect to parametrised connected sum and addition of singularities.

On C0-Continuity of the Spectral Norm for Symplectically Non-Aspherical Manifolds

Abstract The purpose of this paper is to study the relation between the $C^0$-topology and the topology induced by the spectral norm on the group of Hamiltonian diffeomorphisms of a closed symplectic manifold. Following the approach of Buhovsky–Humilière–Seyfaddini, we prove the $C^0$-continuity of the spectral norm for complex projective spaces and negative monotone symplectic manifolds. The case of complex projective spaces provides an alternative approach to the $C^0$-continuity of the spectral norm proven by Shelukhin. We also prove a partial $C^0$-continuity of the spectral norm for rational symplectic manifolds. Some applications such as the Arnold conjecture in the context of $C^0$-symplectic topology are also discussed.

On the local well–posedness of the two component b-family of equations

In this paper we consider the two component b-family of equations on \({\mathbb {R}}\). We write the equations on a Sobolev type diffeomorphism group. As an application of this formulation we show that the dependence on the initial data is nowhere locally uniformly continuous. In particular it is nowhere locally Lipschitz and nowhere locally Hölder continuous.

Hofer-Like Geometry and Flux Theory

This paper meticulously revisit and study the flux geometry of any compact connected oriented manifold (M, Ω). We generalize several well- known factorization results, exhibit some orbital conditions under which flux geometry can be studied, give a proof of the discreteness of the flux group for volume-preserving diffeomorphisms, derive that any smooth isotopy in the group of all vanishing-flux volume-preserving diffeomorphisms is a vanishing- flux path, and show that the kernel of flux for volume-preserving diffeomorphisms is C 1−closed inside the group of all volume-preserving diffeomorphisms isotopic to the identity map: We recover several well-known results from symplectic geometry. We use the above studies to construct a right-invariant metric on the group of all volume-preserving diffeomorphisms isotopic to the identity map and study the induced geometry. In the case of a symplectic volume form, the restriction of our metric to the group Ham(N, ω), of all Hamiltonian diffeomorphisms of a closed symplectic manifold (N, ω), is controlled from above by the usual Hofer metric in general, while the Hofer-like metric control our metric in the case where the Riemannian structure is compatible with the symplectic structure (in particular, our construction implies the non-degeneracy of the Hofer and Hofer-like norms).

Connection on the Group of Diffeomorphisms as a Bundle Over the Space of Functions

Connection on the Group of Diffeomorphisms as a Bundle Over the Space of Functions

Topologie

The Oberwolfach conference “Topologie” is one of only a few opportunities for researchers from many different areas in algebraic and geometric topology to meet and exchange ideas. On this occasion, because of the Corona pandemic, only about 20 participants attended in person, but another \sim 25 attended online. Speakers were selected from both groups. A topic of special interest emphasized at the workshop was the rational homotopy theory of embedding spaces and relations to graph complexes and formality. Two 50 minute lectures on this theme were given by Thomas Willwacher, and one by Victor Turchin. The rest of the program covered a wide range of topics, among them: homotopy properties of diffeomorphism groups of high dimensional manifolds, advances in the classification of high-dimensional highly connected smooth manifolds, parametrized algebraic surgery in relation to hermitian algebraic K-theory, other advances in and geometric applications of algebraic K-theory, stable homotopy interpretation of link invariants, geometry of surface bundles and cohomology of mapping class groups, boundary concepts in geometric group theory, and Koszul duality for operads.

Quasi-morphisms on surface diffeomorphism groups

We show that the identity component of the group of diffeomorphisms of a closed oriented surface of positive genus admits many unbounded quasi-morphisms. As a corollary, we also deduce that this group is not uniformly perfect and its fragmentation norm is unbounded, answering a question of Burago--Ivanov--Polterovich. As a key tool we construct a hyperbolic graph on which these groups act, which is the analog of the curve graph for the mapping class group.

Mather invariant, distortion, and conjugates for diffeomorphisms of the interval

We relate the Mather invariant of diffeomorphisms of the (closed) interval to their asymptotic distortion. For maps with only parabolic fixed points, we show that the former is trivial if and only if the latter vanishes. As a consequence, we obtain that such a diffeomorphism of the interval with no fixed point in the interior contains the identity in the closure of its C1+bv conjugacy class if and only if it is the time-1 map of a C1 vector field. A corollary of this is that diffeomorphisms that do not arise from vector fields are undistorted in the whole group of interval diffeomorphisms. Several related results in other regularity classes are obtained, and many open questions are addressed.

The helicity uniqueness conjecture in 3d hydrodynamics

We prove that the helicity is the only regular Casimir function for the coadjoint action of the volume-preserving diffeomorphism group $\text{SDiff}(M)$ on smooth exact divergence-free vector fields on a closed three-dimensional manifold $M$. More precisely, any regular $C^1$ functional defined on the space of $C^\infty$ (more generally, $C^k$, $k\ge 4$) exact divergence-free vector fields and invariant under arbitrary volume-preserving diffeomorphisms can be expressed as a $C^1$ function of the helicity. This gives a complete description of Casimirs for adjoint and coadjoint actions of $\text{SDiff}(M)$ in 3D and completes the proof of Arnold-Khesin's 1998 conjecture for a manifold $M$ with trivial first homology group. Our proofs make use of different tools from the theory of dynamical systems, including normal forms for divergence-free vector fields, the Poincar\'e-Birkhoff theorem, and a division lemma for vector fields with hyperbolic zeros.

Geometric hydrodynamics and infinite-dimensional Newton’s equations

We revisit the geodesic approach to ideal hydrodynamics and present a related geometric framework for Newton’s equations on groups of diffeomorphisms and spaces of probability densities. The latter setting is sufficiently general to include equations of compressible and incompressible fluid dynamics, magnetohydrodynamics, shallow water systems and equations of relativistic fluids. We illustrate this with a survey of selected examples, as well as with new results, using the tools of infinite-dimensional information geometry, optimal transport, the Madelung transform, and the formalism of symplectic and Poisson reduction.

Kinematics of arbitrary spin matter fields in loop quantum gravity

Loop quantum gravity envisions a small scale structure of spacetime that is markedly different from that of the classical spacetime continuum. This has ramifications for the excitation of matter fields and for their coupling to gravity. There is a general understanding of how to formulate scalar fields, spin $\frac{1}{2}$ fields, and gauge fields in the framework of loop quantum gravity. The goal of the present work is to investigate kinematical aspects of this coupling. We will study implications of the Gau\ss{} and diffeomorphism constraints for the quantum theory: We define and study a less ambiguous variant of the Baez-Krasnov path observables, and we investigate symmetry properties of spin network states imposed by diffeomorphism group averaging. We will do this in a setting which allows for matter excitations of spin $\frac{1}{2}$ and higher. In the case of spin $\frac{1}{2}$, we will also discuss extensions of it by introducing an electromagnetic field and antiparticles. We finally discuss how far the picture with matter excitations of higher spin can be obtained from classical actions for higher spin fields.

Geometric stochastic heat equations

We consider a natural class of $\mathbf{R}^d$-valued one-dimensional stochastic PDEs driven by space-time white noise that is formally invariant under the action of the diffeomorphism group on $\mathbf{R}^d$. This class contains in particular the KPZ equation, the multiplicative stochastic heat equation, the additive stochastic heat equation, and rough Burgers-type equations. We exhibit a one-parameter family of solution theories with the following properties: - For all SPDEs in our class for which a solution was previously available, every solution in our family coincides with the previously constructed solution, whether that was obtained using It\^o calculus (additive and multiplicative stochastic heat equation), rough path theory (rough Burgers-type equations), or the Hopf-Cole transform (KPZ equation). - Every solution theory is equivariant under the action of the diffeomorphism group, i.e. identities obtained by formal calculations treating the noise as a smooth function are valid. - Every solution theory satisfies an analogue of It\^o's isometry. - The counterterms leading to our solution theories vanish at points where the equation agrees to leading order with the additive stochastic heat equation. In particular, points 2 and 3 show that, surprisingly, our solution theories enjoy properties analogous to those holding for both the Stratonovich and It\^o interpretations of SDEs simultaneously. For the natural noisy perturbation of the harmonic map flow with values in an arbitrary Riemannian manifold, we show that all these solution theories coincide. In particular, this allows us to conjecturally identify the process associated to the Markov extension of the Dirichlet form corresponding to the $L^2$-gradient flow for the Brownian loop measure.

The action spectrum and $$C^0$$ symplectic topology

Our first main result states that the spectral norm on the group of Hamiltonian diffeomorphisms, introduced in the works of Viterbo, Schwarz and Oh, is continuous with respect to the C^0 topology, when M is symplectically aspherical. This statement was previously proven only in the case of closed surfaces. As a corollary, using a recent result of Kislev and Shelukhin, we obtain C^0 continuity of barcodes on aspherical symplectic manifolds, and furthermore define barcodes for Hamiltonian homeomorphisms. We also present several applications to Hofer geometry and dynamics of Hamiltonian homeomorphisms. Our second main result is related to the Arnold conjecture about fixed points of Hamiltonian diffeomorphisms. The recent example of a Hamiltonian homeomorphism, on any closed symplectic manifold of dimension greater than 2, having only one fixed point, shows that the conjecture does not admit a direct generalization to the C^0 setting. However, in this paper we demonstrate that a reformulation of the conjecture in terms of fixed points as well as spectral invariants still holds for Hamiltonian homeomorphisms on symplectically aspherical manifolds.

Positivity for the curvature of the diffeomorphism group corresponding to the incompressible Euler equation with Coriolis force

Abstract We investigate the geometry of the central extension $\widehat{\mathcal D}_{\mu}(S^{2})$ of the group of volume-preserving diffeomorphisms of the 2-sphere equipped with an $L^{2}$-metric, for which geodesics correspond to solutions of the incompressible Euler equation with Coriolis force. In particular, we calculate the Misiołek curvature of this group. This value is related to the existence of a conjugate point and its positivity directly implies the positivity of the sectional curvature.

The Structure of Homeomorphism and Diffeomorphism Groups

The Structure of Homeomorphism and Diffeomorphism Groups

THE FLUX HOMOMORPHISM AND CENTRAL EXTENSIONS OF DIFFEOMORPHISM GROUPS

Let $D$ be a closed unit disk in dimension two and $G_{\rm rel}$ the group of symplectomorphisms on $D$ preserving the origin and the boundary $\partial D$ pointwise. We consider the flux homomorphism on $G_{\rm rel}$ and construct a central $\mathbb{R}$-extension called the flux extension. We determine the Euler class of this extension and investigate the relation among the extension, the group $2$-cocycle defined by Ismagilov, Losik, and Michor, and the Calabi invariant of $D$.

Port-Hamiltonian modeling of ideal fluid flow: Part II. Compressible and incompressible flow

Part I of this paper presented a systematic derivation of the Stokes–Dirac structure underlying the port-Hamiltonian model of ideal fluid flow on Riemannian manifolds. Starting from the group of diffeomorphisms as a configuration space for the fluid, the Stokes–Dirac structure is derived by Poisson reduction and then augmented by boundary ports and distributed ports. The additional boundary ports have been shown to appear naturally as surface terms in the pairings of dual maps, always neglected in standard Hamiltonian theory. The port-Hamiltonian model presented in Part I corresponded only to the kinetic energy of the fluid and how its energy variables evolve such that the energy is conserved.In Part II, we utilize the distributed port of the kinetic energy port-Hamiltonian system for representing a number of fluid-dynamical systems. By adding internal energy we model compressible flow, both adiabatic and isentropic, and by adding constraint forces we model incompressible flow. The key tools used are the interconnection maps relating the dynamics of fluid motion to the dynamics of advected quantities.