Sobolev Metrics on the Manifold of All Riemannian Metrics

Abstract

On the manifold M(M ) of all Riemannian metrics on a compact manifold M one can consider the natural L 2 -metric as described first by [11]. In this paper we consider variants of this metric which in general are of higher order. We derive the geodesic equations, we show that they are well-posed under some conditions and induce a locally diffeomorphic geodesic exponential mapping. We give a condition when Ricci flow is a gradient flow for one of these metrics.

Overview of the Geometries of Shape Spaces and Diffeomorphism Groups

“…Nevertheless, it has been shown that the geodesic equations are (locally) well-posed, assuming certain conditions on the operator field L defining the metric; see [15]. These conditions are satisfied by the class of Sobolev type metrics and conformal metrics but not by the scalar curvature weighted metrics.…”

“…Subsequently Freed and Groisser [49] and Michor and Gil-Medrano [52] computed the geodesic equation and found explicit solution formulas. The geodesic equation for higher order Sobolev type metrics and Scalar curvature metrics can be found in [15] and for volume weighted metrics in [15,33].…”

“…The resulting metric G L , which is a the Sobolev metric of order l, has been introduced in [15]. Other metrics, that have been studied include conformal transformations of the L 2 -metric [15,32],…”

Geodesic completeness of the $$H^{3/2}$$ metric on $$\mathrm {Diff}(S^{1})$$

“…Since then, an analogous result has been found for a whole variety of PDEs, including the inviscid Burgers equation, the Hunter-Saxton equation, the Camassa-Holm equation [6,19], and the modified Constantin-Lax-Majda (mCLM) equation [10,14]. Building on the pioneering work of Ebin and Marsden [11], these geometric interpretations have been used to obtain rigourous well-posedness and stability results for the corresponding PDEs [9,26,25,23,3,15,21].…”

“…This result is not true anymore for the critical index s = 3 2 , as D s (S 1 ) is only a topological group for s > 3 2 ; the metric extends only to a smooth, weak Riemannian metric on the Sobolev completion D q (S 1 ), for high enough q > 3 2 . For the metric corresponding to the Euler-Weil-Petersson equation, which is of critical order 3 2 , Gay-Balmaz and Ratiu [16] found an interpretation as a strong Riemannian metric on a certain subgroup of all quasisymmetric homeomorphisms of the circle and used this to conclude that the velocity field remains in H 3/2 . Although we believe that a similar approach would be feasible for general metrics of order 3 2 both the results and methods are entirely distinct from the approach in this paper.…”