A warped product splitting theorem (Q1196422)

Let \(M\) be a Riemannian \(n\)-manifold with \(\text{Ric} \geq k(n-1)\), and \(N \subset M\) an oriented hypersurface. To fix signs, let \(d = d(. ,N)\) be the distance from \(N\) and call \(H(t) = {1\over n-1}\text{div }\nabla d|_{N_ t}\) the mean curvature of the equidistant hypersurface \(N_ t = \{d=t\}\) with respect to the normal vector \(\nabla d\). This satisfies the Riccati inequality \(dH/dt + H^ 2 + k \leq 0\) with equality iff all \(N_ t\) are umbilic and the sectional curvature equals \(k\) on all planes containing the normal vector \(\nabla d\). Consequently, \(H\) is bounded from above by the corresponding mean curvature \(H_ k\) in a space \(M_ k\) of constant curvature \(k\), with equality only in the above mentioned case. If \(k = -\delta^ 2 \leq 0\), the horospheres in \(M_ k\) form such a family with \(H_ k = \delta\). Thus if \(H(0) \leq \delta\) on \(M\), we must have \(H(t) \leq \delta\) for all \(t\). This is used in the present paper to show the following theorem: Suppose that \(M\) has two boundary components \(N\) and \(N'\) where \(N\) is compact with mean curvature \(H \geq \delta\) with respect to the inward normal while the mean curvature of \(N'\) is \(H' \leq \delta\) with respect to the outward normal. Then \(M\) is isometric to a warped product: \(M = [0,L] \times N\) with metric \(ds^ 2 = dt^ 2 + e^{2\delta t}\cdot g_ N\). (Note the different sign convention for \(H\) in the paper.) This result is obtained similarly as the Cheeger-Gromoll splitting theorem by applying the maximum principle to the sum of the two distance functions \(d + d'\). Replacing \(d'\) by a Busemann function, the authors get an analogous result when \(N'\) is removed to \(\infty\). A similar theorem also holds for any \(k\) on regions between two concentric balls if again the mean curvatures are optimal.