A new optimal estimate for the first stability eigenvalue of closed hypersurfaces in Riemannian space forms
Suppose that \(\Sigma^n\) is a closed \(n\)-dimensional hypersurface of constant mean curvature \(H\) in a complete, simply connected \((n+1)\)-dimensional space form \(Q_c^{n+1}\) of constant curvature \(c \in \{ -1, 0, 1\}\). The stability operator \(J\), which acts on smooth functions on \(\Sigma\), is defined by the formula \[ J = \Delta + |\Phi| + c(H^2+c), \] where \(\Delta\) is the Laplace operator on \(\Sigma\) with its induced metric, and \(|\Phi|\) is the norm of the trace-free part \(\Phi\) of the shape operator \(A\) of \(\Sigma\) in \(Q_c\). The operator \(\Phi\) vanishes at the umbilic points of \(\Sigma\). Under the additional hypothesis that \(\Phi\) satisfies a certain Okumura-type inequality, which is automatically satisfied whenever \(\Sigma\) has exactly two principal curvatures of constant multiplicity, the authors find an upper bound on the first eigenvalue of \(J\) in terms of \(H\), \(c\), \(n\) and \(\Phi\). This upper bound is attained for Clifford tori in the sphere \(S^{n+1}\), but it is not attained for any \(\Sigma\) in Euclidean (\(c=0\)) or hyperbolic (\(c=-1\)) space. The examples of hyperbolic cylinders in hyperbolic space show that the upper bound is not valid in general when \(\Sigma\) is no longer assumed to be compact.
Clifford torus, circular and hyperbolic cylinders
scientific article; zbMATH DE number 6992787