scholarly journals Finite propagation speed, kernel estimates for functions of the Laplace operator, and the geometry of complete Riemannian manifolds

1982 ◽  
Vol 17 (1) ◽  
pp. 15-53 ◽  
Author(s):  
Jeff Cheeger ◽  
Mikhail Gromov ◽  
Michael Taylor
Keyword(s):  
Laplace Operator ◽  
Riemannian Manifolds ◽  
Propagation Speed ◽  
Kernel Estimates ◽  
Finite Propagation ◽  
The Laplace Operator ◽  
Finite Propagation Speed ◽  
Complete Riemannian Manifolds

scholarly journals On compactness of isospectral conformal, metrics of 4-manifolds

1995 ◽  
Vol 140 ◽  
pp. 77-99 ◽  
Author(s):  
Xingwang Xu
Keyword(s):  
Laplace Operator ◽  
Riemannian Manifolds ◽  
Compact Manifold ◽  
Riemannian Metrics ◽  
Conformal Class ◽  
Conformal Metrics ◽  
Laplace Operators ◽  
Definition Of ◽  
The Laplace Operator ◽  
Isometry Class

In this paper, we are interested in the compactness of isospectral conformal metrics in dimension 4.Let us recall the definition of the isospectral metrics. Two Riemannian metrics g, g′ on a compact manifold are said to be isospectral if their associated Laplace operators on functions have identical spectrum. There are now numeruos examples of compact Riemannian manifolds which admit more than two metrics such that they are isospectral but not isometric. That is to say that the eigenvalues of the Laplace operator Δ on the functions do not necessarily determine the isometry class of (M, g). If we further require the metrics stay in the same conformal class, the spectrum of Laplace operator still does not determine the metric uniquely ([BG], [BPY]).

scholarly journals Finite propagation speed and kernels of strictly elliptic operators

1985 ◽  
Vol 8 (1) ◽  
pp. 75-91 ◽  
Author(s):  
David Gurarie
Keyword(s):  
Singular Perturbation ◽  
Schrödinger Operators ◽  
Propagation Speed ◽  
Elliptic Operators ◽  
Schrodinger Operators ◽  
Finite Propagation ◽  
Finite Propagation Speed

We establish estimates of the resolvent and other related kernels and discussLp-theory for a class of strictly elliptic operators onRn. The class of operators considered in the paper is of the formA0+Bwith the leading elliptic partA0and a “singular” perturbationB, whose coefficients haveLp-type and are modeled after Schrödinger operators.

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