On the gap between the first eigenvalues of the Laplacian on functions and \(p\)-forms. (English) Zbl 1021.58025

Takahashi, Junya

Ann. Global Anal. Geom. 23, No. 1, 13-27 (2003).

Let \((M,g)\) be a connected closed oriented \(m\)-dimensional Riemannian manifold, let \(\lambda_{k,p}(M,g)\) be the \(k^{th}\) positive eigenvalue of the \(p\)-form valued Laplacian, counted with multiplicity.
The author shows that there is no a priori relationship between \(\lambda_{1,p}\) and \(\lambda_{1,0}\) for \(2\leq p\leq m-2\) and \(m\geq 4\) showing:
Theorem: Let \(M\) be a closed connected oriented smooth manifold of dimension \(m\geq 4\). Suppose that \(2\leq p\leq m-2\). Then there exist Riemannian metrics \(g_i\) on \(M\) so that \(\lambda_{1,p}(g_1)>\lambda_{1,0}(g_1)\), \(\lambda_{1,p}(g_2)=\lambda_{1,0}(g_2)\), and \(\lambda_{1,p}(g_3)<\lambda_{1,0}(g_3)\).
The author also studies the situation when an additional geometric hypothesis is imposed.
Theorem: Let \((M,g)\) be a closed connected oriented smooth manifold of dimension \(m\). If \((M,g)\) has a non-trivial parallel \(p\)-form, then \(\lambda_{1,p}(g)\leq\lambda_{1,0}(M,g)\).

Reviewer: Peter B.Gilkey (Eugene)

MSC:

58J50	Spectral problems; spectral geometry; scattering theory on manifolds
35P15	Estimates of eigenvalues in context of PDEs
53C25	Special Riemannian manifolds (Einstein, Sasakian, etc.)
53C43	Differential geometric aspects of harmonic maps

Keywords:

Laplacian on forms; spectrum; comparison of eigenvalues; collapsing Riemannian manifolds; parallel forms