# zbMATH — the first resource for mathematics

Sobolev metrics on the manifold of all Riemannian metrics. (English) Zbl 1275.58007
Let $$\mathrm{Met}(M)$$ be the space of all smooth Riemannian metrics on a compact manifold $$M$$. The space $$\mathrm{Met}(M)$$ is an open, convex, positive cone in the Frechet space $$S^2(M)$$ of all smooth, symmetric 2-forms on $$M$$. Therefore, $$\mathrm{Met}(M)$$ is a Frechet manifold and for any $$g\in \mathrm{Met}(M)$$, the tangent space $$T_g\mathrm{Met}(M)$$ can be naturally identified with the space $$S^2(M)$$. The manifold $$\mathrm{Met}(M)$$ possesses the canonical weak $$L^2$$-Riemannian metric as described first by D. Ebin: if $$h,k\in T_g\mathrm{Met}(M)=S^2(M)$$, then $$(h,k)_g=\int_M g(h,k) \mathrm{vol}(g)=\int_M g^{ik}g^{jl}h_{ij}k_{kl} vol(g)$$.
In this paper, the authors investigate stronger metrics on $$\mathrm{Met}(M)$$ than the $$L^2$$-metric. These are metrics of the following form: $$G_g(h,k)=\Phi(\mathrm{Vol})\int_M g(h,k) \mathrm{vol}(g)$$, or $$=\int_M \Phi(\mathrm{Scal}).g(h,k) \mathrm{vol}(g)$$, or $$=\int_M g((1+\Delta)^ph,k)\mathrm{vol}(g)$$, where $$\Phi$$ is a suitable real-valued function, $$\mathrm{Vol}$$ is the total volume of $$(M,g)$$, $$\mathrm{Scal}$$ is the scalar curvature of $$(M,g)$$. The authors describe all these metrics uniformly as $$G^P_g(h,k)=\int_M g(P_gh,k) \mathrm{vol}(g)$$, where $$P_g:S^2(M)\to S^2(M)$$ is a positive, symmetric, bijective pseudo-differential operator of order $$2p, p\geq 0,$$ depending smoothly on the metric $$g$$. The authors derive the geodesic equations, show that they are well-posed under some conditions and induce a locally diffeomorphic geodesic exponential mapping. The authors give a condition when the Ricci flow is a gradient flow for one of these metrics.

##### MSC:
 58D17 Manifolds of metrics (especially Riemannian) 58E30 Variational principles in infinite-dimensional spaces 58B20 Riemannian, Finsler and other geometric structures on infinite-dimensional manifolds 35A01 Existence problems for PDEs: global existence, local existence, non-existence 53C44 Geometric evolution equations (mean curvature flow, Ricci flow, etc.) (MSC2010)
Full Text: