The index of constant mean curvature surfaces in hyperbolic 3-space
Let \(M\) be a complete surface with constant mean curvature \(|H|<1\) immersed into hyperbolic 3-space with second fundamental form \(A\). Recall that the Morse index of the immersion is the maximal number of negative eigenvalues of the stability operator \(L:=\Delta+2-|A|^2\) in compact subdomains. Let \(\phi\) be the traceless fundamental form of \(M\), \(\phi:= A-H\text{Id}\). In this paper we show, under the assumption \(\int_M|\phi|^2 dv_M<\infty\) (\(M\) has ``finite total curvature''), that the Morse index of the immersion is finite. The proof makes use of two main ingredients. One ingredient is the fact that the tensor \(\phi\) satisfies a nonlinear elliptic equation (this idea goes back to J. Simons); another ingredient is that \(M\) satisfies a nice Sobolev inequality (D. Hoffman and J. Spruck). These ingredients, together with the fact that the surface \(M\) has finite total curvature imply that the operator \(L\) is bounded from below and essentially selfadjoint. One can then prove that the essential spectrum of \(L\) is bounded from below by \(2(1-H^2)\). Because the discrete spectrum of \(L\) can only accumulate at \(-\infty\) or at the infimum of the essential spectrum, one can conclude from the assumption \(H^2<1\) that the index of the stability operator \(L\) is finite. A similar statement for \(n\)-dimensional immersions with constant mean curvature (or with parallel mean curvature vector) into hyperbolic space \((n\geq 3)\) is also true. An interesting aspect of the proofs is that they use, in a crucial way, the essential spectrum of (the selfadjoint extension of) \(L\).
scientific article; zbMATH DE number 1028163