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 ${S^n}$, ${n} \geq 5$, where the first eigenvalue of the Laplacian on functions remains bounded above from zero, and the first eigenvalue of the Laplacian on $1$-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 (${S^n}$, some metric) to an (${S^{n - 1}} \times {S^1}$, some metric).