A metric deformation and the first eigenvalue of Laplacian on 1-forms

Abstract

We search for a higher-dimensional analogue of Calabi’s example of a metric deformation, quoted by Cheeger, which inspired him to prove an inequality between the first eigenvalue of the Laplacian on functions and an isoperimetric constant. We construct an example of a metric deformation on S n {S^n} , n ≥ 5 {n} \geq 5 , where the first eigenvalue of the Laplacian on functions remains bounded above from zero, and the first eigenvalue of the Laplacian on 1 1 -forms tends to zero. This metric deformation makes the sphere in the limit into a manifold with a cone singularity, which is an intermediate point on a path of deformation from an ( S n {S^n} , some metric) to an ( S n − 1 × S 1 {S^{n - 1}} \times {S^1} , some metric).