Analogous sharp bounds also hold for other Geometries and for certain degenerate Laplacians associated with these geometries like the after on a compact | Applications there are to the global embedding of such CR manifolds in C n |
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Conversely, 2 characterizes the Laplace—Beltrami operator completely, in the sense that it is the only operator with this property | More precisely if we multiply the eigenvalue eqn |
It is convenient to regard the sphere as isometrically embedded into R n as the unit sphere centred at the origin | "Embeddability for 3-dimensional CR manifolds and CR Yamabe Invariants" |
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The resulting operator is called the Laplace—de Rham operator named after | Flanders, Harley 1989 , Differential forms with applications to the physical sciences, Dover,• The sign is merely a convention, and both are common in the literature |
Chavel, Isaac 1984 , Eigenvalues in Riemannian Geometry, Pure and Applied Mathematics, 115 2nd ed.
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