Time-dependent Diffeomorphisms Research Articles

The Laplacian Flow of Locally Conformal Calibrated G2-Structures

We consider the Laplacian flow of locally conformal calibrated G 2 -structures as a natural extension to these structures of the well-known Laplacian flow of calibrated G 2 -structures. We study the Laplacian flow for two explicit examples of locally conformal calibrated G 2 manifolds and, in both cases, we obtain a flow of locally conformal calibrated G 2 -structures, which are ancient solutions, that is they are defined on a time interval of the form ( − ∞ , T ) , where T > 0 is a real number. Moreover, for each of these examples, we prove that the underlying metrics g ( t ) of the solution converge smoothly, up to pull-back by time-dependent diffeomorphisms, to a flat metric as t goes to − ∞ , and they blow-up at a finite-time singularity.

Laplacian Flow of Closed G $$_2$$ 2 -Structures Inducing Nilsolitons

We study the existence of left invariant closed \(G_2\)-structures defining a Ricci soliton metric on simply connected nonabelian nilpotent Lie groups. For each one of these \(G_2\)-structures, we show long time existence and uniqueness of solution for the Laplacian flow on the noncompact manifold. Moreover, considering the Laplacian flow on the associated Lie algebra as a bracket flow on \({\mathbb {R}}^7\) in a similar way as in Lauret (Commun Anal Geom 19(5):831–854, 2011) we prove that the underlying metrics \(g(t)\) of the solution converge smoothly, up to pull-back by time-dependent diffeomorphisms, to a flat metric, uniformly on compact sets in the nilpotent Lie group, as \(t\) goes to infinity.

Curvature flows for almost-hermitian Lie groups

We study curvature flows in the locally homogeneous case (e.g. compact quotients of Lie groups, solvmanifolds, nilmanifolds) in a unified way by considering a generic flow under just a few natural conditions on the broad class of almost-hermitian structures. As a main tool, we use an ODE system defined on the variety of 2 n 2n -dimensional Lie algebras, called the bracket flow, whose solutions differ from those to the original curvature flow by only pull-back by time-dependent diffeomorphisms. The approach, which has already been used to study the Ricci flow on homogeneous manifolds, is useful to better visualize the possible pointed limits of solutions, under diverse rescalings, as well as to address regularity issues. Immortal, ancient and self-similar solutions arise naturally from the qualitative analysis of the bracket flow. The Chern-Ricci flow and the symplectic curvature flow are considered in more detail.

The Ricci flow for simply connected nilmanifolds

We prove that the Ricci flow g(t) starting at any metric on R that is invariant by a transitive nilpotent Lie group N can be obtained by solving an ODE for a curve of nilpotent Lie brackets on R. By using that this ODE is the negative gradient flow of a homogeneous polynomial, we obtain that g(t) is type-III, and, up to pull-back by time-dependent diffeomorphisms, that g(t) converges to the flat metric, and the rescaling |R(g(t))| g(t) converges to a Ricci soliton in C∞, uniformly on compact sets in R. The Ricci soliton limit is also invariant by some transitive nilpotent Lie group, though possibly nonisomorphic to N .

A Note on Deformations of 2D Fluid Motions Using 3D Born-Infeld Equations

Classical fluid motions can be described either by time dependent diffeomorphisms (Lagrangian description) or by the corresponding generating vector fields (Eulerian description). We are interested in interpolating two such fluid motions. For both analytic and numerical purposes, we would like the corresponding interpolating vector fields to evolve according to some hyperbolic evolution PDEs. In the 2D case, it turns out that a convenient set of such PDEs can be deduced from the BornInfeld (BI) theory of Electromagnetism. The 3D BI equations were originally designed as a nonlinear correction to the linear Maxwell equations allowing finite electric fields for point charges. They depend on a parameter λ. As λ goes to infinity, the classical Maxwell equations are recovered. It turns out that in the opposite case, as λ goes to zero, the BI equations provide a solution to our problem. These equations, as λ = 0, also describe classical strings (extremal surfaces) evolving in the 4D Minkowski space-time. In their static version, they can be interpreted in the framework of optimal transportation theory.

Time-dependent diffeomorphisms as quantum canonical transformations and the time-dependent harmonic oscillator

Quantum canonical transformations corresponding to time-dependent diffeomorphisms of the configuration space are studied. A special class of these transformations which correspond to time-dependent dilatations is used to identify a previously unknown class of exactly solvable time-dependent harmonic oscillators. The Caldirola-Kanai oscillator belongs to this class. For a general time-dependent harmonic oscillator, it is shown that choosing the dilatation parameter to satisfy the classical equation of motion, one obtains the solution of the Schrödinger equation. A simple generalization of this result leads to the reduction of the Schrödinger equation to a second-order ordinary differential equation whose special case is the auxiliary equation of the Lewis-Riesenfeld invariant theory. The time-evolution operator is expressed in terms of a positive real solution of this equation in a closed form, and the time-dependent position and momentum operators are calculated.