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Essential spectrum and heat kernel. (English) Zbl 0634.58031

Let M be a complete Riemannian manifold and \({}^ p\Lambda (M)\) its vector space of exterior differential p-forms. The Laplacian \(\Delta\) acts on the space of compactly supported exterior differentiable p-forms. If \(\Delta \phi =\lambda \phi\), \(\lambda\in {\mathbb{R}}\), then \(\phi\) is called an eigen p-form. If \(\lambda =0\), that means \(\Delta\phi =0\), then \(\phi\) is called harmonic p-form. We denote by Sp \(\Delta\) the spectrum of \(\Delta\), that means Sp \(\Delta\) \(=\{\lambda /\Delta\phi= \lambda\phi\}\). Since \(\Delta\) is positive semidefinite Sp \(\Delta\) is a subset of \([0,\infty)\). Let \(\rho\) be the infimum of Sp\(\Delta\setminus 0\). The essential spectrum of \(\Delta\) denoted Ess Sp \(\Delta\), consists of the cluster points in the spectrum of \(\Delta\) and the eigenvalues with infinite multiplicity. If \(\alpha\) is the infimum of Ess Sp \(\Delta\), then we assume that \(\alpha >0\). It is obvious that Sp \(\Delta\setminus\) Ess Sp \(\Delta\) consists of isolated eigenvalues having finite multiplicity. The hypothesis \(\alpha >0\) therefore implies \(0<\rho \leq \alpha.\)
Let \(u\in L^ 2(\Lambda^ p(M))\) be a solution of the heat equation \((\partial /\partial t+\Delta)u=0\). The aim of the present paper is to study the essential spectrum and heat kernel. Some of the results can be stated as follows:
I. For \(t\geq t_ 0\), \(\int_{M}| u|^ 2e^{2\sqrt{\beta r}}\leq C_ 3e^{-2\lambda t}+C_ 4e^{-2\gamma t}.\)
II. For \(t\geq t_ 0>0\), \(\int_{M}| \bar K(t,x,x_ 0)|^ 2 e^{2\sqrt{\beta r}}\leq B_ 1e^{-2\lambda t}+B_ 2e^{-2\gamma t}.\)
The integral runs over the variable x. \(\bar K(t,x,x_ 0)=K(t,x,x_ 0)- H(x,x_ 0|\), where \(K(t,x,x_ 0)\) is the fundamental solution of the heat equation. \(H(x,x_ 0)\) is the kernel for projection onto the space of \(L^ 2(\Lambda^ p(M))\) harmonic forms and \(x_ 0\) a basic point on M.
Reviewer: G.Tsagas

MSC:

58J35 Heat and other parabolic equation methods for PDEs on manifolds
53C20 Global Riemannian geometry, including pinching
58J50 Spectral problems; spectral geometry; scattering theory on manifolds
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