Asymptotic distributions associated with the Laplacian for forms

✍ Scribed by Matthew P. Gaffney

• Publisher: John Wiley and Sons
• Year: 1958
• Tongue: English
• Weight: 496 KB
• Volume: 11
• Category: Article
• ISSN: 0010-3640
• DOI: 10.1002/cpa.3160110405

✦ Synopsis

In 1953 Minakshisundarum [6] published a new and simpler proofbased on the heat equation-of results concerning the characteristic functions and values of the Laplace-Beltrami operator on a closed Riemannian manifold which he had obtained earlier in collaboration with Pleijel.

Shortly afterward Professor M. H. Stone suggested that I carry out a similar program for differential forms-with the emphasis on finding the asymptotic distribution of the characteristic forms and values of the Laplacian.

The method consists of deriving the expansion for the fundamental solution of the heat equation "

,(P)w,(Q)e-A't

(Sections 1 and 2), finding the asymptotic behaviour of the fundamental solution (Section 3), and then applying Karamata's Tauberian theorem to these results, thus obtaining the desired asymptotic distributions (Section 3).

We consider an oriented, compact, C" Riemannian manifold M of dimension n. For differential forms on M the Laplacian is given by d = dd+dd (see [l], [2], [4]; A for functions is the negative of that used by Minalcshisundarum), and the heat equation is (A+a/at)tl, = 0 with t > 0.

A fundamental solution of the heat equation is a continuous double $-form 6 ( P, Q ; t ) , 0 5 $ 5 n, t > 0, C2 in Q and C1 in t, which for fixed P satisfies the heat equation i=l 3 1. dQ88 sz A Q + -6(P, Q ; t) = 0 ( aai ) and which also satisfies the relation for all continuous $-forms a.