Maximum principle for hypersurfaces
Let \(M\) be a Riemannian or Lorentzian manifold. Firstly, the author gives a short proof of the following interior maximum principle for hypersurfaces; Let \(W_+\) and \(W_-\) be disjoint open domains in \(M\) with spacelike connected \(C^2\)-boundaries having a point in common. If the mean curvatures \(H_+\) of \(\partial W_+\) and \(H_-\) of \(\partial W_-\) satisfy \(H_-\le -a\) and \(H_+\le a\) for some real number \(a\), then \(\partial W_-=\partial M_+\), and \(H_+=-H_-=a\). The boundary maximum principle in a manifold with boundary is shown similarly. Next, the author proves that the maximum principle holds even if \(\partial W_+\) satisfies only that \(\partial W_+\) has generalized mean curvature \(\le a\), that is, for any \(p\in \partial W_+\) there are open domains \(W_{p,j}\), \(j=1,2,\ldots\), whose boundaries are \(C^2\)-hypersurfaces near \(p\) with shape operator \(A_{p,j}\) and mean curvature \(H_{p,j}\) at the point \(p\), with the following properties: (a) \(W_{p,1}\subset W_{p,2}\subset\cdots\subset W\), (b) \(p\in \partial W_{p,j}\), (c) there is a locally uniform upper bound for \(A_{p,1}\), (d) \(H_{p,j}\le a+\varepsilon\) for some sequence \(\varepsilon_ j\to 0\). Many applications of the maximum principle to Riemannian and Lorentzian manifolds with lower Ricci curvature bounds are given. For example he gets: Let \(M\) be a complete connected non-compact manifold with \(\mathrm{Ric}\ge 0\) which contains a compact connected two-sided hypersurface \(S\) (without boundary). Then \(S\) is totally geodesic and bounds a domain \(W\) isometric to \(S\times (0,\infty)\), and \(N\setminus W\) is compact unless \(M\) is isometric to \(S\times R\).
scientific article; zbMATH DE number 4111507
scientific article