Diffeomorphisms Research Articles

Ping-pong partitions and locally discrete groups of real-analytic circle diffeomorphisms, I: Construction

Following the recent advances in the study of groups of circle diffeomorphisms, we describe an efficient way of classifying the topological dynamics of locally discrete, finitely generated, virtually free subgroups of the group \mathsf{Diff}^\omega_+(\mathbb{S}^1) of orientation-preserving real-analytic circle diffeomorphisms, which include all subgroups of \mathsf{Diff}^\omega_+(\mathbb{S}^1) acting with an invariant Cantor set. An important tool that we develop, of independent interest, is the extension of classical ping-pong lemma to actions of fundamental groups of graphs of groups. Our main motivation is an old conjecture by Dippolito [Ann. of Math. (2) 107 (1978), 403–453] from foliation theory, which we solve in this restricted but significant setting: this and other consequences of the classification will be treated in more detail in a companion work (by a slightly different list of authors).

Ping-pong partitions and locally discrete groups of real-analytic circle diffeomorphisms, II: Applications

In the first part of this work we have established an efficient method to obtain a topological classification of locally discrete, finitely generated, virtually free subgroups of real-analytic circle diffeomorphisms. In this second part we describe several consequences, among which the solution (within this setting) to an old conjecture by P. R. Dippolito [Ann. Math. 107 (1978), 403–453] that actions with invariant Cantor sets must be semi-conjugate to piecewise linear actions. In addition, we exhibit examples of locally discrete, minimal actions which are not of Fuchsian type.

Extended phase space in general gauge theories

In a recent paper, it was shown that in diffeomorphism-invariant theories, the symplectic vector fields induced by spacetime diffeomorphisms are integrable if one introduces an extended phase space. In this paper we extend the notion of extended phase space to all gauge theories with arbitrary combinations of internal and spacetime local symmetries. We formulate this in terms of a corresponding Atiyah Lie algebroid, a geometric object derived from a principal bundle which features internal symmetries and diffeomorphisms on an equal footing. In this language, gauge transformations are understood as morphisms between Atiyah Lie algebroids that preserve the geometric structures encoded therein. The extended configuration space of a gauge theory can subsequently be understood as the space of pairs (φ,Φ), where φ is a Lie algebroid morphism and Φ is a field configuration in the non-extended sense. Starting from this data, we outline a very powerful, manifestly geometric approach to the extended phase space. Using this approach, we find that the action of the group of gauge transformations and diffeomorphisms on the symplectic geometry of any covariant theory is integrable. We motivate our construction by carefully examining the need for extended phase space in Chern-Simons gauge theories and display its usefulness by re-computing the charge algebra. We also describe the implementation of the configuration algebroid in Einstein-Yang-Mills theories.

On the Diffeomorphism Groups of Foliated Manifolds

On the Diffeomorphism Groups of Foliated Manifolds

Diffeomorphism groups of prime 3-manifolds

Abstract Let X be a compact orientable non-Haken 3-manifold modeled on the Thurston geometry Nil {\operatorname{Nil}} . We show that the diffeomorphism group Diff ⁡ ( X ) {\operatorname{Diff}(X)} deformation retracts to the isometry group Isom ⁡ ( X ) {\operatorname{Isom}(X)} . Combining this with earlier work by many authors, this completes the determination the homotopy type of Diff ⁡ ( X ) {\operatorname{Diff}(X)} for any compact, orientable, prime 3-manifold X.

On the Lie-Algebraic Integrability of the Calogero-Degasperis Dynamical System and Its Generalizations

We studied the Lax type integrability of the Calogero-Degasperis nonlinear dynamical system, possessing only one local conserved quantity. Based on the gradientholonomic integrability approach there are stated tboth the bi-Hamiltonian structure of the Calogero-Degasperis dynamical system and isomorphism of its symmetries group to the semidirect product of the diffeomorphism group of the circle and the abelian group of functions on it. We also constructed a rich algebra of non-Hamiltonian symmetries, related to the Bäcklund transformed general symmetries of the corresponding linearization of the Calogero-Degasperis dynamical system. There is also analyzed in detail the inverse problem of classifying integrable generalized Calogero-Degasperis type dynamical systems a priori possessing a finite number of conserved quantities.

A nonperturbative approach to Hawking radiation and black hole quantum hair

We present a nonperturbative derivation of the subleading order in Hawking radiation based on diffeomorphism symmetry breaking during black hole evaporation. The diffeomorphism group of horizon admits a nontrivial phase factor which encodes information about infalling matter during formation. This nonintegrable phase represents the black hole quantum hair as it arises from the diffeomorphisms that change the physical state of the black hole. During evaporation, the decrease in total area breaks the diffeomorphism symmetry and leads to a dynamical shift in that phase factor. This shift affects the usual Hawking spectrum via dispersion relation and results in the subleading term in Hawking radiation. The higher order terms are locally insensitive to the Unruh radiation due to the lack of diffeomorphism groups on the local Rindler horizon at the low energy scale. This explains the generic difference between Hawking radiation and Unruh radiation. In addition, this phase shift indicates the decrease of the total number of degrees of freedom in horizon phase space during evaporation as past Page time. This enables us to escape from the firewall paradox and provide an account for the resolution to the information paradox.

Weighted nonlinear flag manifolds as coadjoint orbits

Abstract A weighted nonlinear flag is a nested set of closed submanifolds, each submanifold endowed with a volume density. We study the geometry of Fréchet manifolds of weighted nonlinear flags, in this way generalizing the weighted nonlinear Grassmannians. When the ambient manifold is symplectic, we use these nonlinear flags to describe a class of coadjoint orbits of the group of Hamiltonian diffeomorphisms, orbits that consist of weighted isotropic nonlinear flags.

Deep neural networks on diffeomorphism groups for optimal shape reparametrization

Deep neural networks on diffeomorphism groups for optimal shape reparametrization

Diffeomorphism groups of Morse-Bott foliation on the solid Klein bottle by Klein bottles parallel to the boundary

Let $\mathcal{G}$ be a Morse-Bott foliation on the solid Klein bottle $\mathbf{K}$ into $2$-dimensional Klein bottles parallel to the boundary and one singular circle $S^1$. Let also $S^1\widetilde{\times}S^2$ be the twisted bundle over $S^1$ which is a union of two solid Klein bottles $\mathbf{K}_0$ and $\mathbf{K}_1$ with common boundary $K$. Then the above foliation $\mathcal{G}$ on both $\mathbf{K}_0$ and $\mathbf{K}_1$ gives a foliation $\mathcal{G}'$ on $S^1\widetilde{\times}S^2$ into parallel Klein bottles and two singluar circles. The paper computes the homotopy types of groups of foliated (sending leaves to leaves) and leaf preserving diffeomorphisms for foliations $\mathcal{G}$ and $\mathcal{G}'$.

Homotopy types of diffeomorphism groups of polar Morse–Bott foliations on lens spaces, 1

Homotopy types of diffeomorphism groups of polar Morse–Bott foliations on lens spaces, 1

On the local well‐posedness of the 1D Green–Naghdi system on Sobolev spaces

AbstractIn this paper, we consider the local well‐posedness of the 1D Green–Naghdi system. This system describes the evolution of water waves over an uneven bottom in the shallow water regime in terms of the water depth h and the horizontal velocity u. Using a Lagrangian formulation of this system on a Sobolev‐type diffeomorphism group, we prove local well‐posedness for in the Sobolev space , where is the parameterization of the bottom and where we assume that the water surface has an equilibrium at height 1. This improves the present local well‐posedness range by one degree.

A note on the Nielsen realization problem for 𝐾3 surfaces

We will show the following three theorems on the diffeomorphism and homeomorphism groups of a K 3 K3 surface. The first theorem is that the natural map π 0 ( D i f f ( K 3 ) ) → A u t ( H 2 ( K 3 ; Z ) ) \pi _{0}(Diff(K3)) \to Aut(H^{2}(K3;\mathbb {Z})) has a section over its image. The second is that there exists a subgroup G G of π 0 ( D i f f ( K 3 ) ) \pi _{0}(Diff(K3)) of order two over which there is no splitting of the map D i f f ( K 3 ) → π 0 ( D i f f ( K 3 ) ) Diff(K3) \to \pi _{0}(Diff(K3)) , but there is a splitting of H o m e o ( K 3 ) → π 0 ( H o m e o ( K 3 ) ) Homeo(K3) \to \pi _{0}(Homeo(K3)) over the image of G G in π 0 ( H o m e o ( K 3 ) ) \pi _{0}(Homeo(K3)) , which is non-trivial. The third is that the map π 1 ( D i f f ( K 3 ) ) → π 1 ( H o m e o ( K 3 ) ) \pi _{1}(Diff(K3)) \to \pi _{1}(Homeo(K3)) is not surjective. Our proof of these results is based on Seiberg-Witten theory and the global Torelli theorem for K 3 K3 surfaces.

PFH spectral invariants on the two-sphere and the large scale geometry of Hofer’s metric

We resolve three longstanding questions related to the large scale geometry of the group of Hamiltonian diffeomorphisms of the two-sphere, equipped with Hofer’s metric. Namely: (1) we resolve the Kapovich–Polterovich question by showing that this group is not quasi-isometric to the real line; (2) more generally, we show that the kernel of Calabi over any proper open subset is unbounded; and (3) we show that the group of area and orientation preserving homeomorphisms of the two-sphere is not a simple group. We also find, as a corollary, that the group of area-preserving diffeomorphisms of the open disc, equipped with an area form of finite area, is not perfect. Central to all of our proofs are new sequences of spectral invariants over the two-sphere, defined via periodic Floer homology.

The deformation space of Delaunay triangulations of the sphere

In this paper, we determine the topology of the spaces of convex polyhedra inscribed in the unit $2$-sphere and the spaces of strictly Delaunay geodesic triangulations of the unit $2$-sphere. These spaces can be regarded as discretized groups of diffeomorphisms of the unit $2$-sphere. Hence, it is natural to conjecture that these spaces have the same homotopy types as those of their smooth counterparts. The main result of this paper confirms this conjecture for the unit $2$-sphere. It follows from an observation on the variational principles on triangulated surfaces developed by I. Rivin. On the contrary, the similar conjecture does not hold in the cases of flat tori and convex polygons. We will construct simple examples of flat tori and convex polygons such that the corresponding spaces of Delaunay geodesic triangulations are not connected.

Hofer's metric in compact Lie groups

In this article, we study the Hofer geometry of a compact Lie group K which acts by Hamiltonian diffeomorphisms on a symplectic manifold M . Generalized Hofer norms on the Lie algebra of K are introduced and analyzed with tools from group invariant convex geometry, functional and matrix analysis. Several global results on the existence of geodesics and their characterization in finite-dimensional Lie groups K endowed with bi-invariant Finsler metrics are proved. We relate the conditions for being a geodesic in the group K and in the group of Hamiltonian diffeomorphisms. These results are applied to obtain necessary and sufficient conditions on the moment polytope of the momentum map, for the commutativity of the Hamiltonians of geodesics. Particular cases are studied, where a generalized non-crossing of eigenvalues property of the Hamiltonians hold.

Pointed vortex loops in ideal 2D fluids

We study a special kind of singular vorticities in ideal 2D fluids that combine features of point vortices and vortex sheets, namely pointed vortex loops. We focus on the coadjoint orbits of the area-preserving diffeomorphism group of determined by them. We show that a polarization subgroup consists of diffeomorphisms that preserve the loop as a set, thus the configuration space is the space of loops that enclose a fixed area, without information on vorticity distribution and attached points.

The energy–momentum complex in non-local gravity

In General Relativity, the issue of defining the gravitational energy contained in a given spatial region is still unresolved, except for particular cases of localized objects where the asymptotic flatness holds for a given spacetime. In principle, a theory of gravity is not self-consistent, if the whole energy content is not uniquely defined in a specific volume. Here, we generalize the Einstein gravitational energy–momentum pseudotensor to non-local theories of gravity where analytic functions of the non-local integral operator [Formula: see text] are taken into account. We apply the Noether theorem to a gravitational Lagrangian, supposed invariant under the one-parameter group of diffeomorphisms, that is, the infinitesimal rigid translations. The invariance of non-local gravitational action under global translations leads to a locally conserved Noether current, and thus, to the definition of a gravitational energy–momentum pseudotensor, which is an affine object transforming like a tensor under affine transformations. Furthermore, the energy–momentum complex remains locally conserved, thanks to the non-local contracted Bianchi identities. The continuity equations for the gravitational pseudotensor and the energy–momentum complex, taking into account both gravitational and matter components, can be derived. Finally, the weak field limit of pseudotensor is performed to lowest order in metric perturbation in view of astrophysical applications.

Proper extensions of the 2-sphere’s conformal group present entropy and are 4-transitive

AbstractIn this paper, we prove using elementary techniques that any group of diffeomorphisms acting on the 2-sphere and properly extending the conformal group of Möbius transformations must be at least 4-transitive or, more precisely, arc 4-transitive. As an important consequence, we derive that any such group must always contain an element of positive topological entropy. We also provide a self-contained characterization, in terms of transitivity, of the Möbius transformations within the full group of sphere diffeomorphisms.

Determining the invariant of inter-frame processing for constructing the image similarity metric

The relevance of modeling digital images is determined by the need to implement approaches in the study of localization and identification of objects in order to reduce the amount of data. In this paper, the study object is the topology of a discrete two-dimensional image within the framework of the problem of determining the invariants of diffeomorphic transformations. Geographic information objects (GIOs) refer to objects that are on a given surface or objects that locally change the surface. With regard to objects, it is assumed that the change in their geolocation in the process of forming both single images and an extended series of frames obtained in the process of continuous monitoring is insignificant. In the process of scanning the surface, possible changes in the position of the image source are taken into account, for example, such as yawing, rolling, and pitch in the case of unmanned aerial vehicles (UAVs). These maneuvers are represented as a group of diffeomorphisms that are controlled by the internal gyroscopes of the carrier and the external navigation system. Based on the studies reported here, the initial ontology of digital images (ODI) has been determined by using the model of color spaces and functions of a special kind. The presence of an ontology makes it possible to build an adequate topology of color distribution in the image and take into account the specificity of the distribution of different colors in a digital image. The study results indicate that a promising method is to determine the similarity by constructing a color atlas structure graph (CASG) based on ODI and by determining invariants as a fragment of CASG inherited by all images in the sequence. The scope and conditions for the practical use of the result include its application to the analysis of images by methods of artificial intelligence