Spectrum of the Jacobi operator on a minimal hypersurface of the Euclidean space (Q2780524)

This mainly expository paper motivates the research done in other works by the author and S. Nayatani. On a complete minimal hypersurface \((M,g)\) of \(\mathbb{R}^{n+1}\) one considers the Jacobi operator \(A:=\Delta+\tau=\Delta-|B|^2\) arising from the second variation of the volume functional (here \(\tau\) is the scalar curvature and \(B\) the shape operator of the minimal immersion). The author first reviews in detail some of the main known results (for example, that the number of negative eigenvalues of \(A\), called the index of \((M,g)\), is estimated from above by the total scalar curvature \(\int_M |B|^n\): Bérard-Besson, Cheng-Tysk, Ejiri) and states some open problems.NEWLINENEWLINENEWLINEThen, he addresses the problem of determining when the essential spectrum of \(A\) is contained in \([0,\infty)\); he does so by applying the simple, general result proved in [\textit{S. Nayatani} and \textit{H. Urakawa}, J. Funct. Anal. 112, 459-479 (1993; Zbl 0779.58041)]: any Schrödinger operator \(\Delta+V\) on a complete Riemannian manifold has essential spectrum contained in \([0,\infty)\) provided that the negative part \(V_-\) of its potential tends to zero uniformly as the distance from any given point tends to infinity. Also, he gives a sufficient condition for having finite index, based on the asymptotic behavior at infinity of the scalar curvature. Finally, he gives some counterexamples and shows that the index of the catenoid is one. Several open problems are stated.NEWLINENEWLINEFor the entire collection see [Zbl 0981.00018].