Eigenvalue pinching and application to the stability and the almost umbilicity of hypersurfaces. (English) Zbl 1243.53071

Grosjean, J.-F.; Roth, J.

Math. Z. 271, No. 1-2, 469-488 (2012).

Summary: We give pinching theorems for the first nonzero eigenvalue of the Laplacian on compact hypersurfaces of ambient spaces with bounded sectional curvature. As an application we deduce a rigidity result for stable constant mean curvature hypersurfaces \(M\) of these spaces \(N\). Indeed, we prove that if \(M\) is included in a ball of radius small enough then the Hausdorff-distance between \(M\) and a geodesic sphere \(S\) of \(N\) is small. Moreover \(M\) is diffeomorphic and quasi-isometric to \(S\). As other application, we obtain rigidity results for almost umbilic hypersurfaces.

MSC:

53C21	Methods of global Riemannian geometry, including PDE methods; curvature restrictions
53C42	Differential geometry of immersions (minimal, prescribed curvature, tight, etc.)
58J50	Spectral problems; spectral geometry; scattering theory on manifolds
53C24	Rigidity results

Keywords:

spectrum; Laplacian; pinching results; hypersurfaces

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