Hypersurfaces with prescribed scalar curvature in geometric flows

Abstract

We consider translators to the extrinsic fl ow defi nedby S α in R × hPn or in Pn × χR , where S is the extrinsic scalar curvature, α ∈ {1/2,1}, and n ≥ 3. We show that there exist rotational bowl-type and translating catenoid-type translators in P × χR . In our main existence results we exhibit a one-parameter family of explicit solutions when α = 1/2 in P × χR when P is Hadamard complete manifold with a rotationally symmetric metric. We discuss the variational nature of solitons, we fi nd a one-parameter family of null scalar curvature hypersurfaces when P × χR is Einstein, we use maximum principle to show that if a translating soliton is contained in a slab in I × hP and it is parabolic with respect to the L map then it is contained in a leaf P s = {s} × P and that if P × χR has constant sectional curvature and a translating soliton is parabolic with respect to L ζ map then it is not bounded from above.