A note on eigenvalues
Let M be a simply connected complete Riemannian manifold with sectional curvature \(K_ M\). Yau conjectured that if \(-k^ 2_ 2\leq K_ M\leq - k^ 2_ 1<0,\) then M has no \(L^ 2\) eigenvalues. The authors construct a counter example to this conjecture. Let \(0\leq k_ 1<k_ 2<\infty\). The authors construct a spherically symmetric metric \(ds^ 2=dr^ 2+f(r)^ 2d\theta^ 2\) on \(M=R^ n\) such that (1) \(-k^ 2_ 2\leq K_ M\leq -k^ 2_ 1\), (2) M has \(L^ 2\) eigenvalues.
scientific article; zbMATH DE number 4153383