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Nonlinear phenomena in the spectral theory of geometric linear differential operators. (English) Zbl 0857.58042
Arveson, William (ed.) et al., Quantization, nonlinear partial differential equations, and operator algebra. 1994 John von Neumann symposium on quantization and nonlinear wave equations, June 7-11, 1994, MIT, Cambridge, MA, USA. Providence, RI: American Mathematical Society. Proc. Symp. Pure Math. 59, 27-65 (1996).
The extremal problem for the functional determinant of a natural linear elliptic operator on a Riemannian manifold is studied. Viewing the determinant as a function of the Riemannian metric, the author proves sharp inequalities comparing nonlinear functionals of the metric and its derivatives. The derivation and use of such inequalities in new situations, especially essentially tensor-valued inequalities, leads back to linear theory and the classification of conformally covariant differential operators.
For the entire collection see [Zbl 0840.00054].

58J52 Determinants and determinant bundles, analytic torsion
58J50 Spectral problems; spectral geometry; scattering theory on manifolds
58C40 Spectral theory; eigenvalue problems on manifolds
53A30 Conformal differential geometry (MSC2010)