On Cheng’s Eigenvalue Comparison Theorems

We prove Cheng’s eigenvalue comparison theorems [4] for geodesic balls within the cut locus under weaker geometric hypothesis, Theorems (1.1, 3.1, 3.2) and we also show that there are certain geometric rigidity in case of equality of the eigenvalues. This rigidity becomes isometric rigidity under upper sectional curvature bounds or lower Ricci curvature bounds. We construct examples of smooth metrics showing that our results are true extensions of Chengs theorem. We also construct a family of complete smooth metrics on Rn non-isometric to the constant sectional curvature κ metrics of the simply connected space forms M(κ) such that the geodesic balls BRn(r), BMn(κ)(r) have the same first eigenvalue and the geodesic spheres ∂BRn(s) and ∂BMn(κ)(s), 0 < s ≤ r, have the same mean curvatures. In the end we construct examples of Riemannian manifolds M with arbitrary topology with positive fundamental tone λ > 0 that generalize Veeravalli’s examples, [6]. Mathematics Subject Classification: (2000): 53C40, 53C42, 58C40