Monodromy of isometric deformations of CMC surfaces (Q5952194)

It is well known that an arbitrary simply connected domain \(D\) of CMC-\(H\) (i.e. Constant Mean Curvature \(H)\) surfaces in the space form \(M^3(c)\) of constant curvature \(c\) has the following two special properties:NEWLINENEWLINENEWLINE(i) it admits a real analytic isometric deformation preserving the mean curvature function in \(M^3(c)\). This is called a \(H\)-deformation of the surface; NEWLINENEWLINENEWLINE(ii) it can be isometrically immersed in another space form \(M^3(t)\) as a CMC-\(\sqrt{H^2+ c-t}\) surface. This is called a \(t\)-deformation of the surface.NEWLINENEWLINENEWLINEIn the present paper the authors investigate the monodromy of \(H\)- and \(t\)-deformations. They prove: NEWLINENEWLINENEWLINETheorem 1. Let \((\Sigma^2,ds^2)\) be a Riemannian 2-manifold and \(F:\Sigma^2\to M^3(c)\) with \(c\in\mathbb{R}\) an isometric immersion of constant mean curvature \(H\). If the \(t\)-deformation is single-valued on \(\Sigma^2\), so is the \(H\)-deformation, too. Moreover the converse is also true unless the original surface is minimal in \((\mathbb{R}^3,\sum^3_{j=1} (dx_j)^2)\).NEWLINENEWLINENEWLINEIn the case of \(c\leq 0\) and \(|H|= \sqrt{|c|}\) this result has been obtained by \textit{M. Umehara} and \textit{K. Yamada} in Math. Ann. 304, 203-224 (1996; Zbl 0841.53050). Finally, the authors derive the following: NEWLINENEWLINENEWLINECorollary. Let \((\Sigma^2,ds^2)\) be a closed Riemannian 2-manifold and \(x: \Sigma^2 \to M^3(c)\) with \(c\in\mathbb{R}\) an isometric immersion of constant mean curvature \(H\), then the number of congruent classes \(N_x:=\#\{x: (\Sigma^2, ds^2) \to M^3(c)\); isometric \(\text{CMC-}H\) immersion\} is finite. In particular, there do not exist any global non-trivial isometric deformations of CMC surfaces preserving the mean curvature.