Diffeomorphisms Research Articles

Dispersionless Multi-Dimensional Integrable Systems and Related Conformal Structure Generating Equations of Mathematical Physics

Using diffeomorphism group vector fields on $\mathbb{C}$-multiplied tori and the related Lie-algebraic structures, we study multi-dimensional dispersionless integrable systems that describe conformal structure generating equations of mathematical physics. An interesting modification of the devised Lie-algebraic approach subject to spatial-dimensional invariance and meromorphicity of the related differential-geometric structures is described and applied in proving complete integrability of some conformal structure generating equations. As examples, we analyze the Einstein-Weyl metric equation, the modified Einstein-Weyl metric equation, the Dunajski heavenly equation system, the first and second conformal structure generating equations and the inverse first Shabat reduction heavenly equation. We also analyze the modified Pleba\'nski heavenly equations, the Husain heavenly equation and the general Monge equation along with their multi-dimensional generalizations. In addition, we construct superconformal analogs of the Whitham heavenly equation.

Some finiteness results for groups of automorphisms of manifolds

We prove that in dimensions not equal to 4, 5, or 7, the homology and homotopy groups of the classifying space of the topological group of diffeomorphisms of a disk fixing the boundary are finitely generated in each degree. The proof uses homological stability, embedding calculus and the arithmeticity of mapping class groups. From this we deduce similar results for the homeomorphisms of $R^n$ and various types of automorphisms of 2-connected manifolds.

Vanishing Distance Phenomena and the Geometric Approach to SQG

In this article we study the induced geodesic distance of fractional order Sobolev metrics on the groups of (volume preserving) diffeomorphisms and symplectomorphisms. The interest in these geometries is fueled by the observation that they allow for a geometric interpretation for prominent partial differential equations in the field of fluid dynamics. These include in particular the modified Constantin–Lax–Majda and surface quasi-geostrophic equations. The main result of this article shows that both of these equations stem from a Riemannian metric with vanishing geodesic distance.

Least action principles for incompressible flows and geodesics between shapes

As V. I. Arnold observed in the 1960s, the Euler equations of incompressible fluid flow correspond formally to geodesic equations in a group of volume-preserving diffeomorphisms. Working in an Eulerian framework, we study incompressible flows of shapes as critical paths for action (kinetic energy) along transport paths constrained to have characteristic-function densities. The formal geodesic equations for this problem are Euler equations for incompressible, inviscid potential flow of fluid with zero pressure and surface tension on the free boundary. The problem of minimizing this action exhibits an instability associated with microdroplet formation, with the following outcomes: any two shapes of equal volume can be approximately connected by an Euler spray—a countable superposition of ellipsoidal geodesics. The infimum of the action is the Wasserstein distance squared, and is almost never attained except in dimension 1. Every Wasserstein geodesic between bounded densities of compact support provides a solution of the (compressible) pressureless Euler system that is a weak limit of (incompressible) Euler sprays.

The Λ-BMS4 group of dS4 and new boundary conditions for AdS4

Using the dictionary between Bondi and Fefferman–Graham gauges, we identify the analogues of the Bondi news, Bondi mass and Bondi angular momentum aspects at the boundary of generic asymptotically locally (A)dS4 spacetimes. We introduce the -BMS4 group as the residual symmetry group of the metric in Bondi gauge after boundary gauge fixing. This group consists of infinite-dimensional non-abelian supertranslations and superrotations and it reduces in the asymptotically flat limit to the extended BMS4 group. Furthermore, we present new boundary conditions for asymptotically locally AdS4 spacetimes which admit times the group of area-preserving diffeomorphisms as the asymptotic symmetry group. The boundary conditions amount to fix two components of the holographic stress-tensor while allowing two components of the boundary metric to fluctuate. They correspond to a deformation of a holographic CFT3 which is coupled to a fluctuating spatial metric of fixed area.

The dispersionless completely integrable heavenly type Hamiltonian flows and their differential-geometric structure

There are reviewed modern investigations devoted to studying nonlinear dispersiveless heavenly type integrable evolutions systems on functional spaces within the modern differential-geometric and algebraic tools. Main accent is done on the loop diffeomorphism group vector fields on the complexified torus and the related Lie-algebraic structures, generating dispersionless heavenly type integrable systems.

EQUIVARIANT ALGEBRAIC INDEX THEOREM

AbstractWe prove a $\unicode[STIX]{x1D6E4}$-equivariant version of the algebraic index theorem, where $\unicode[STIX]{x1D6E4}$ is a discrete group of automorphisms of a formal deformation of a symplectic manifold. The particular cases of this result are the algebraic version of the transversal index theorem related to the theorem of A. Connes and H. Moscovici for hypo-elliptic operators and the index theorem for the extension of the algebra of pseudodifferential operators by a group of diffeomorphisms of the underlying manifold due to A. Savin, B. Sternin, E. Schrohe and D. Perrot.

Embedding certain diffeomorphism groups in the quasi-isometry groups of Euclidean spaces

We show that certain groups of diffeomorphisms and PL-homeomorphisms embed in the group of all quasi-isometry classes of the Euclidean spaces.

Geodesic motion on groups of diffeomorphisms with H1 metric as geometric generalised Lagrangian mean theory

ABSTRACTGeneralized Lagrangian mean theories are used to analyse the interactions between mean flows and fluctuations, where the decomposition is based on a Lagrangian description of the flow. A systematic geometric framework was recently developed by Gilbert and Vanneste who cast the decomposition in terms of intrinsic operations on the group of volume preserving diffeomorphisms or on the full diffeomorphism group. In this setting, the mean of an ensemble of maps can be defined as the Riemannian centre of mass on either of these groups. We apply this decomposition in the context of Lagrangian averaging where equations of motion for the mean flow arise via a variational principle from a mean Lagrangian, obtained from the kinetic energy Lagrangian of ideal fluid flow via a small amplitude expansion for the fluctuations. We show that the Euler-α equations arise as Lagrangian averaged Euler equations when using the -geodesic mean on the volume preserving diffeomorphism group of a manifold without boundaries, imposing a “Taylor hypothesis”, which states that first order fluctuations are transported as a vector field by the mean flow, and assuming that fluctuations are statistically nearly isotropic. Similarly, the EPDiff equations arise as the Lagrangian averaged Burgers' equations using the same argument on the full diffeomorphism group. A serious drawback of this construction is that the assumptions of Lie transport of the fluctuation vector field and isotropy of fluctuations cannot persist except for an asymptotically vanishing interval of time. To remedy the problem of persistence of isotropy, we suggest adding strong mean-reverting stochastic term to the Taylor hypothesis and identify a scaling regime in which the inclusion of the stochastic term leads to the same averaged equations up to a constant.

Global fixed points for nilpotent actions on the torus

An isotopic to the identity map of the 2-torus, that has zero rotation vector with respect to an invariant ergodic probability measure, has a fixed point by a theorem of Franks. We give a version of this result for nilpotent subgroups of isotopic to the identity diffeomorphisms of the 2-torus. In such a context we guarantee the existence of global fixed points for nilpotent groups of irrotational diffeomorphisms. In particular, we show that the derived group of a nilpotent group of isotopic to the identity diffeomorphisms of the 2-torus has a global fixed point.

On a class of immersions of spheres into space forms of nonpositive curvature

Let \( M^{n+1} \) (\( n \ge 2 \)) be a simply-connected space form of sectional curvature \( -\kappa ^2 \) for some \( \kappa \ge 0 \), and I an interval not containing \( [-\kappa ,\kappa ] \) in its interior. It is known that the domain of a closed immersed hypersurface of M whose principal curvatures lie in I must be diffeomorphic to the n-sphere \( \mathbb {S}^n \). These hypersurfaces are thus topologically rigid. The purpose of this paper is to show that they are also homotopically rigid. More precisely, for fixed I, the space \( \mathcal {F} \) of all such hypersurfaces is either empty or weakly homotopy equivalent to the group of orientation-preserving diffeomorphisms of \( \mathbb {S}^n \). An equivalence assigns to each element of \( \mathcal {F} \) a suitable modification of its Gauss map. For M not simply-connected, \( \mathcal {F} \) is the quotient of the corresponding space of hypersurfaces of the universal cover of M by a natural free proper action of the fundamental group of M.

Groups with infinitely many ends acting analytically on the circle

This article takes the inspiration from two milestones in the study of non minimal actions of groups on the circle: Duminy's theorem about the number of ends of semi-exceptional leaves and Ghys' freeness result in analytic regularity. Our first result concerns groups of analytic diffeomorphisms with infinitely many ends: if the action is non expanding, then the group is virtually free. The second result is a Duminy's theorem for minimal codimension one foliations: either non expandable leaves have infinitely many ends, or the holonomy pseudogroup preserves a projective structure.

Geodesic distance for right-invariant metrics on diffeomorphism groups: critical Sobolev exponents

We study the geodesic distance induced by right-invariant metrics on the group $\operatorname{Diff}_c(M)$ of compactly supported diffeomorphisms of a manifold $M$, and show that it vanishes for the critical Sobolev norms $W^{s,n/s}$, where $n$ is the dimension of $M$ and $s\in(0,1)$. This completes the proof that the geodesic distance induced by $W^{s,p}$ vanishes if $sp\le n$ and $s<1$, and is positive otherwise. The proof is achieved by combining the techniques of two recent papers --- [JM19] by the authors, which treated the sub-critical case, and [BHP18] of Bauer, Harms and Preston, which treated the critical 1-dimensional case.

Examples of diffeomorphism group cocycles with no periodic approximation

We construct a finitely generated subgroup of Diff ∞ ⁡ ( S 3 × S 1 ) \operatorname {Diff}^{\infty }(\mathbb {S}^3 \times \mathbb {S}^1) where every element is conjugate to an isometry but such that the group action itself is far from isometric (the group has “exponential growth of derivatives”). As a corollary, one obtains a locally constant Diff ∞ ⁡ ( S 3 × S 1 ) \operatorname {Diff}^{\infty }(\mathbb {S}^3 \times \mathbb {S}^1) valued cocycle over a hyperbolic dynamical system which has elliptic behavior over its periodic orbits but which preserves a measure with non-zero top fiber-Lyapunov exponent. Additionally, we provide new examples of Banach cocycles not satisfying the periodic approximation property as first shown in Ergodic Theory Dynam. Systems 39 (2019), pp. 689–706.

Longitudinal Image Analysis via Path Regression on the Image Manifold

Longitudinal image analysis plays an important role in depicting the development of the brain structure, where image regression and interpolation are two commonly used techniques. In this paper, we develop an efficient model and approach based on a path regression on the image manifold instead of the geodesic regression to avoid the complexity of the geodesic computation. Concretely, first we model the deformation by diffeomorphism; then, a large deformation is represented by a path on the orbit of the diffeomorphism group action. This path is obtained by compositing several small deformations, which can be well approximated by its linearization. Second, we introduce some intermediate images as constraints to the model, which guides to form the best-fitting path. Thirdly, we propose an approximated quadratic model by local linearization method, where a closed form is deduced for the solution. It actually speeds up the algorithm. Finally, we evaluate the proposed model and algorithm on a synthetic data and a real longitudinal MRI data. The results show that our proposed method outperforms several state-of-the-art methods.

The universal n-pointed surface bundle only has n sections

The classifying space BDiff[Formula: see text] of the orientation-preserving diffeomorphism group of a surface [Formula: see text] of genus [Formula: see text] fixing [Formula: see text] points pointwise has a universal bundle [Formula: see text] The [Formula: see text] fixed points provide [Formula: see text] sections [Formula: see text] of [Formula: see text]. In this paper we prove a conjecture of R. Hain that any section of [Formula: see text] is homotopic to some [Formula: see text]. Let [Formula: see text] be the space of ordered [Formula: see text]-tuple of distinct points on [Formula: see text]. As part of the proof of Hain’s conjecture, we prove a result of independent interest: any surjective homomorphism [Formula: see text] is equal to one of the forgetful homomorphisms [Formula: see text], possibly post-composed with an automorphism of [Formula: see text]. We also classify sections of the universal hyperelliptic surface bundle.

Corrigendum: The diffeomorphism group of a K3 surface and Nielsen realization

Corrigendum: The diffeomorphism group of a K3 surface and Nielsen realization

Edge Wave and Boundary Layer of Vortex Matter.

We show that vortex matter, that is, a dense assembly of vortices in an incompressible two-dimensional flow, such as a fast rotating superfluid or turbulent flows with signlike eddies, exhibits (i)a boundary layer of vorticity (vorticity layer) and (ii)a nonlinear wave localized within the vorticity layer, the edge wave. Both are solely an effect of the topological nature of vortices. Both are lost if vortex matter is approximated as a continuous vorticity patch. The edge wave is governed by the integrable Benjamin-Davis-Ono equation, exhibiting solitons with a quantized total vorticity. Quantized solitons reveal the topological nature of the vortices through their dynamics. The edge wave and the vorticity layer are due to the odd viscosity of vortex matter. We also identify the dynamics with the action of the Virasoro-Bott group of diffeomorphisms of the circle, where odd viscosity parametrizes the central extension. Our edge wave is a hydrodynamic analog of the edge states of the fractional quantum Hall effect.

ALMOST FIXED POINTS OF FINITE GROUP ACTIONS ON MANIFOLDS WITHOUT ODD COHOMOLOGY

If X is a smooth manifold and $$ \mathcal{G} $$ is a subgroup of Diff(X) we say that (X, $$ \mathcal{G} $$ ) has the almost fixed point property if there exists a number C such that for any finite subgroup G ≤ $$ \mathcal{G} $$ there is some x ∈ X whose stabilizer Gx ≤ $$ \mathcal{G} $$ satisfies [G : Gx] ≤ C. We say that X has no odd cohomology if its integral cohomology is torsion free and supported in even degrees. We prove that if X is compact and possibly with boundary and has no odd cohomology then (X, Diff(X)) has the almost fixed point property. Combining this with a result of Petrie and Randall we conclude that if Z is a non-necessarily compact smooth real affine variety, and Z has no odd cohomology, then (Z, Aut(Z)) has the almost fixed point property, where Aut(Z) is the group of algebraic automorphisms of Z lifting the identity on Spec ℝ. The main ingredients in the proof are: (1) the Jordan property for diffeomorphism groups of compact manifolds with nonzero Euler characteristic, and (2) the study of λ-stability, a condition on actions of finite abelian groups on manifolds that we introduce in this paper.

Applications of Square Roots of Diffeomorphisms

In this paper, we prove that on any contact manifold ( M , ξ ) there exists an arbitrary C ∞ -small contactomorphism which does not admit a square root. In particular, there exists an arbitrary C ∞ -small contactomorphism which is not “autonomous”. This paper is the first step to study the topology of C o n t 0 ( M , ξ ) ∖ Aut ( M , ξ ) . As an application, we also prove a similar result for the diffeomorphism group Diff ( M ) for any smooth manifold M.