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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)
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