# Can we run to infinity? The diameter of the diffeomorphism group with respect to right-invariant Sobolev metrics

@article{Bauer2019CanWR, title={Can we run to infinity? The diameter of the diffeomorphism group with respect to right-invariant Sobolev metrics}, author={Martin Bauer and Cy Maor}, journal={Calculus of Variations and Partial Differential Equations}, year={2019}, volume={60}, pages={1-35} }

The group $${\text {Diff}}({\mathcal {M}})$$ Diff ( M ) of diffeomorphisms of a closed manifold $${\mathcal {M}}$$ M is naturally equipped with various right-invariant Sobolev norms $$W^{s,p}$$ W s , p . Recent work showed that for sufficiently weak norms, the geodesic distance collapses completely (namely, when $$sp\le \dim {\mathcal {M}}$$ s p ≤ dim M and $$s<1$$ s < 1 ). But when there is no collapse, what kind of metric space is obtained? In particular, does it have a finite or infinite…

## 2 Citations

### A Geometric View on the Generalized Proudman-Johnson and r-Hunter-Saxton Equations

- MathematicsJ. Nonlinear Sci.
- 2022

It is shown that two families of equations, the generalized inviscid Proudman–Johnson equation, and the r-Hunter–Saxton equation coincide for a certain range of parameters, and a new geometric interpretation of these Proudman- Johnson equations as geodesic equations of right invariant homogeneous W -Finsler metrics on the diffeomorphism group is given.

### Singular solutions of the r-Camassa-Holm equation

- Mathematics
- 2022

This paper introduces the r-CH equation, which describes a geodesic flow on the manifold of diffeomorphisms acting on the real line induced by the W 1 r metric. The conserved energy is ‖u‖rW1 r for…

## References

SHOWING 1-10 OF 70 REFERENCES

### Vanishing geodesic distance for right-invariant Sobolev metrics on diffeomorphism groups

- MathematicsAnnals of Global Analysis and Geometry
- 2019

We study the geodesic distance induced by right-invariant metrics on the group $${\text {Diff}}_\text {c}({\mathcal {M}})$$Diffc(M) of compactly supported diffeomorphisms, for various Sobolev norms…

### Geodesic distance for right invariant Sobolev metrics of fractional order on the diffeomorphism group

- Mathematics
- 2011

We study Sobolev-type metrics of fractional order s ≥ 0 on the group Diffc(M) of compactly supported diffeomorphisms of a manifold M. We show that for the important special case M = S1, the geodesic…

### The $L^p$-diameter of the group of area-preserving diffeomorphisms of $S^2$

- Mathematics
- 2016

We show that for each $p \geq 1,$ the $L^p$-metric on the group of area-preserving diffeomorphisms of the two-sphere has infinite diameter. This solves the last open case of a conjecture of…

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

- MathematicsAnnals of Global Analysis and Geometry
- 2019

We study the geodesic distance induced by right-invariant metrics on the group \({\text {Diff}}_\text {c}(\mathcal {M})\) of compactly supported diffeomorphisms of a manifold \(\mathcal {M}\) and…

### Computing Large Deformation Metric Mappings via Geodesic Flows of Diffeomorphisms

- MathematicsInternational Journal of Computer Vision
- 2004

The Euler-Lagrange equations characterizing the minimizing vector fields vt, t∈[0, 1] assuming sufficient smoothness of the norm to guarantee existence of solutions in the space of diffeomorphisms are derived.

### Geodesic distance for right invariant Sobolev metrics of fractional order on the diffeomorphism group. II

- Mathematics
- 2013

The geodesic distance vanishes on the group $$\text{ Diff }_c(M)$$ of compactly supported diffeomorphisms of a Riemannian manifold $$M$$ of bounded geometry, for the right invariant weak Riemannian…

### The geometry of a vorticity model equation

- Mathematics
- 2010

We show that the modified Constantin-Lax-Majda equation modeling vortex and quasi-geostrophic dynamics [27] can be recast as the geodesic flow on the subgroup $\mathrm{Diff}_{1}^{\infty}(\mathbb{S})$…

### On Completeness of Groups of Diffeomorphisms

- Mathematics
- 2014

We study completeness properties of the Sobolev diffeomorphism groups $\mathcal D^s(M)$ endowed with strong right-invariant Riemannian metrics when the underlying manifold $M$ is $\mathbb R^d$ or…

### The Hofer norm of a contactomorphism

- Mathematics
- 2014

We show that the $L^{\infty}$-norm of the contact Hamiltonian induces a non-degenerate right-invariant metric on the group of contactomorphisms of any closed contact manifold. This contact Hofer…

### Large-scale geometry of homeomorphism groups

- MathematicsErgodic Theory and Dynamical Systems
- 2017

Let $M$ be a compact manifold. We show that the identity component $\operatorname{Homeo}_{0}(M)$ of the group of self-homeomorphisms of $M$ has a well-defined quasi-isometry type, and study its…