Deforming an \(\varepsilon\)-close to hyperbolic metric to a warped product
Let \((M,g)\) be a Riemannian manifold with center \(o\). Let \(g\) be of the form \(g = g_r +dr^2\), where \(r\) is the distance from \(o\), and consider the warped product \(\mathcal{W}_rg = \sinh^2(r)g'+dr^2\) (\(g'\) does not depend on \(r\)) for \(r\) less than some fixed \(r_0\). The purpose of this paper is to answer the question: If \(g\) is \(\epsilon\)-close to a hyperbolic metric, then to what extent is the warp forced metric \(\mathcal{W}_{r}g\) close to hyperbolic? The main result is the following theorem: Let \((M, g)\) be a Riemannian manifold with center \(o\) and let \(S\subset M\). If \(g\) is radially \(\epsilon\)-close to hyperbolic on \(S\), with charts of excess \(\xi>1\), then \(\mathcal{W}_{r_0}g\) is radially \(\eta\)-close to the hyperbolic metric on \(S-\overline{B}_{r_0-1-\xi}\) with charts of excess \(\xi-1\), provided that \(\eta\geq e^{27+12\xi}(e^{-2r_0} +\epsilon)\) and \(r_0> 3+ 2\epsilon\).
radially \(\epsilon\)-close to hyperbolic
scientific article; zbMATH DE number 6694752