Riemannian metrics with large first eigenvalue on forms of degree $p$

Abstract:

Let (M, g) be a compact, connected, ${C^\infty }$ Riemannian manifold of n dimensions. Denote by ${\lambda _{1,p}}(M,g)$ the first nonzero eigenvalue of the Laplace operator acting on differential forms of degree p. We prove that for $n \geq 4$ and $2 \leq p \leq n - 2$, there exists a family of metrics ${g_t}$ of volume one, such that ${\lambda _{1,p}}(M,{g_t}) \to \infty$ as $t \to \infty$.