Abstract
Smooth valuations on manifolds are studied by establishing a link with the Rumin-de Rham complex of the co-sphere bundle. Several operations on differential forms induce operations on smooth valuations: signature operator, Rumin-Laplace operator, Euler-Verdier involution and derivation operator. As an application, Alesker’s Hard Lefschetz Theorem for even translation invariant valuations on a finite-dimensional Euclidean space is generalized to all translation invariant valuations. The proof uses Kähler identities, the Rumin-de Rham complex and spectral geometry.
Citation
A. Bernig. L. Bröcker. "Valuations on manifolds and Rumin cohomology." J. Differential Geom. 75 (3) 433 - 457, March 2007. https://doi.org/10.4310/jdg/1175266280
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