##
**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.

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 |

### Keywords:

Laplace operator; heat kernel; complete Riemannian manifold; exterior differential p-forms; essential spectrum; heat equation
Full Text:
DOI

### References:

[1] | Atiyah, M.F; Bott, R; Patodi, V.K, On the heat equation and index theorem, Invent. math., 19, 279-330, (1973) · Zbl 0257.58008 |

[2] | Cheeger, J; Gromov, M, On the characteristic numbers of complete manifolds of bounded curvature and finite volume, (), 115-154 |

[3] | Cheeger, J; Gromov, M; Taylor, M, Finite propagation speed, kernel estimates for functions of the Laplace operator, and the geometry of complete Riemannian manifolds, J. differential geom., 17, 15-53, (1982) · Zbl 0493.53035 |

[4] | Dodziuk, J, Maximum principle for parabolic inequalities and the heat flow on open manifolds, Indiana univ. math. J., 32, 703-716, (1983) · Zbl 0526.58047 |

[5] | Donnelly, H, Eigenforms of the Laplacian on complete Riemannian manifolds, Comm. partial differential equations, 9, 1299-1321, (1984) · Zbl 0552.58027 |

[6] | Donnelly, H, On the essential spectrum of a complete Riemannian manifold, Topology, 20, 1-14, (1981) · Zbl 0463.53027 |

[7] | Donnelly, H; Fefferman, C, Fixed point formula for the Bergman kernel, Amer. J. math., 108, 1241-1257, (1986) · Zbl 0603.32018 |

[8] | Gromov, M; Lawson, H.B, Positive scalar curvature and the Dirac operator on complete Riemannian manifolds, Inst. hautes études sci. publ. math., 58, 295-408, (1983) |

[9] | Li, P; Yau, S.T, On the parabolic kernel of the Schrödinger operator, Acta math., 156, 153-202, (1986) |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.