A general halfspace theorem for constant mean curvature surfaces (Q2851618)

A surface \(S\) of constant mean curvature in a 3-dimensional Riemannian manifold \(M\) is called stable, if for every smooth function \(u\) with compact support on \(S\), NEWLINE\[NEWLINE \int_S \|\nabla u\|^2-[2\text{Ric}(\vec{n},\vec{n})+|A|^2]u^2\geq 0,NEWLINE\]NEWLINE where Ric denotes the Ricci tensor of \(M\), \(\vec{n}\) a unit normal vector field to \(S\), and \(|A|\) the norm of the second fundamental form of \(S\). NEWLINENEWLINENEWLINEA continuous function \(u\) on a domain \(\Omega\subseteq M\) is called superharmonic, if for every domain \(U\subset\Omega\) which is relatively compact in \(\Omega\), and for every harmonic function \(v\in C^2(U)\cap C^0(\overline{U})\), \(v|_{\partial U}\leq u|_{\partial U}\) implies \(v\leq u\).NEWLINENEWLINEThe manifold \(M\) is called parabolic, if every bounded superharmonic function on \(M\) is constant. If \(N\) is a manifold with boundary, \(N\) is called parabolic at infinity, if every bounded non-positive superharmonic function \(u\) on \(N\) with \(u|_{\partial N}=0\) is constant. NEWLINENEWLINENEWLINELet \((\Sigma, \text{d}\sigma_0^2)\) be a 2-dimensional complete Riemannian manifold. The 3-dimensional manifold with boundary \(M_+(\varepsilon):=\Sigma\times[0, \varepsilon]\) \((\varepsilon>0)\) with a Riemannian metric \(\text{d}s^2=\text{d}\sigma_t^2+\text{d}t^2\) \((0\leq t\leq\varepsilon)\), \(t\mapsto\text{d}\sigma_t^2\) being a smooth family of Riemannian metrics such that \(\text{d}s^2\) is complete, is called outside \(\varepsilon\)-half neighborhood of \(\Sigma\). Analogously, \(M_-(\varepsilon):=\Sigma\times [-\varepsilon,0]\) is called inside \(\varepsilon\)-half neighborhood of \(\Sigma\).NEWLINENEWLINENEWLINELet \(\Sigma_t:=\Sigma\times\{t\}\). \(M_+(\varepsilon)\) is said to satisfy the \(H\leq H_0\) hypothesis \((H_0\geq 0)\), if for every \(t\in [0,\varepsilon]\) the mean curvature of \(\Sigma_t\) is everywhere at most \(H_0\). \(M_-(\varepsilon)\) is said to satisfy the \(H\geq H_0\) hypothesis, if for every \(t\in [-\varepsilon, 0]\) the mean curvature of \(\Sigma_t\) is everywhere at least \(H_0\).NEWLINENEWLINELet \(\pi_t:\Sigma_t\rightarrow\Sigma\) be the projection onto the first component. An (inside or outside) \(\varepsilon\)-half neighborhood \(M_\pm(\varepsilon)\) of \(\Sigma\) is called regular, if the following conditions are satisfied: NEWLINENEWLINENEWLINE(1) There exists \(k\geq 1\) such that for every \(t\), \(\pi_t\) is \(k\)-quasi-isometric (which means that the norms of \(\text{d}\pi_t\) and its inverse are everywhere bounded by \(k\)). NEWLINENEWLINENEWLINE(2) There exists \(C>0\) such that for every \(t\), the norm of the second fundamental form of \(\Sigma_t\) is bounded by \(C\).NEWLINENEWLINENEWLINE(3) \(M_\pm(\varepsilon)\) is geometrically bounded. NEWLINENEWLINENEWLINEFinally, let \(S\) be a surface of constant mean curvature \(H_0\) \((H_0\geq 0)\) in \(M_\pm(\varepsilon)\setminus\Sigma_0\) (more generally, \(S\) can be the image of an immersion into \(M\), a proper one as a mapping), \(D\) the connected component of \(M_\pm(\varepsilon)\setminus S\) that contains \(\Sigma_0\). \(S\) is said to be well oriented, if for every point in \(S\cap\partial D\) the mean curvature vector of \(S\) points into \(D\) (resp. not into \(D\)) in the case \(S\subset M_+(\varepsilon)\) (resp. \(S\subset M_-(\varepsilon)\)).NEWLINENEWLINENEWLINEAfter these preparations, the main theorem of the author is stated as follows.NEWLINENEWLINENEWLINELet \((\Sigma,\text{d}\sigma_0^2)\) be a complete orientable parabolic 2-dimensional Riemannian manifold, \(\varepsilon>0\) and \(H_0\geq 0\). Let \(S\subset M_\pm(\varepsilon)\setminus\Sigma_0\) be a surface of constant mean curvature \(H_0\) in the more general sense described above, which may have a boundary in \(\Sigma_\varepsilon\). It is assumed that \(M_\pm(\varepsilon)\) is regular.NEWLINENEWLINENEWLINE(1) If \(S\subset M_+(\varepsilon)\), \(S\) is well oriented and \(M_+(\varepsilon)\) satisfies the \(H\leq H_0\) hypothesis, then there exists \(t\in\,]0,\varepsilon]\) such that \(S\subseteq\Sigma_t\).NEWLINENEWLINENEWLINE(2) If \(S\subset M_-(\varepsilon)\) and \(M_-(\varepsilon)\) satisfies the \(H\geq H_0\) hypothesis, then there exists \(t\in [-\varepsilon,0[\) such that \(S\subseteq\Sigma_t\).NEWLINENEWLINENEWLINEIn the last section of the article the author establishes an analogous result in the case where the ambient manifold is a Lie group with a left invariant Riemannian metric. He also presents some applications of it.