The index of constant mean curvature surfaces in hyperbolic 3-space
type: Item
instance of: scholarly article
title: The index of constant mean curvature surfaces in hyperbolic 3-space
zbMATH Open document ID: 0868.53010
DOI: 10.1007/PL00004288
published in: Mathematische Zeitschrift
publication date: 1997-07-03T00:00:00Z
review text: 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) \(L\).
Mathematics Subject Classification ID: 53A10
Mathematics Subject Classification ID: 53C42
zbMATH DE Number: 1028163
zbMATH Keywords: constant mean curvature surface
zbMATH Keywords: hyperbolic 3-space
zbMATH Keywords: Morse index
zbMATH Keywords: stability operator
zbMATH Keywords: essential spectrum
author: Pierre Bérard
author: Manfredo P. do Carmo
author: Walcy Santos
MaRDI profile type: Publication
full work available at URL: pl00004288
OpenAlex ID: W2073871803
description: scientific article; zbMATH DE number 1028163