Abstract
We prove existence of instantaneously complete Yamabe flows on hyperbolic space of arbitrary dimension mge 3. The initial metric is assumed to be conformally hyperbolic with conformal factor and scalar curvature bounded from above. We do not require initial completeness or bounds on the Ricci curvature. If the initial data are rotationally symmetric, the solution is proven to be unique in the class of instantaneously complete, rotationally symmetric Yamabe flows.
Highlights
The Yamabe flow was introduced by Richard Hamilton [11]. It describes a family of Riemannian metrics g(t) subject to the equation ∂t g = −Rg and tends to evolve a given initial metric towards a metric of vanishing scalar curvature
Ma and An [13] proved short-time existence of Yamabe flows on noncompact, locally conformally flat manifolds M under the assumption that the initial manifold (M, g0) is complete with Ricci tensor bounded from below
Peter Topping and Gregor Giesen [7,16,17] introduced the notion of instantaneous completeness and obtained existence and uniqueness of instantaneously complete Ricci/Yamabe flows on arbitrary surfaces
Summary
Short-time existence of a solution u to Eq (1) for given u(·, 0) = u0 > 0 on convex, bounded domains ⊂ H with suitable boundary data is proven by applying the inverse function theorem on Banach spaces. Richard Hamilton [9, § IV.11] uses the same technique to prove existence of solutions to the heat equation for manifolds. Local Hölder estimates lead to a uniform existence time for all domains. In a second step we derive uniform gradient estimates, which do not depend on the domain. By considering an exhaustion of H with convex, bounded domains, we obtain a locally uniformly bounded sequence of solutions which allows a subsequence converging to a solution of (1) on all of H
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