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Last multipliers for Riemannian geometries, Dirichlet forms and Markov diffusion semigroups. (English) Zbl 1390.53026

Summary: We start this study with last multipliers and the Liouville equation for a symmetric and non-degenerate tensor field \(Z\) of \((0, 2)\)-type on a given Riemannian geometry \((M, g)\) as a measure of how far away is \(Z\) from being divergence-free (and hence \(g^C\)-harmonic) with respect to \(g\). The some topics are studied also for the Riemannian curvature tensor of \((M, g)\) and finally for a general tensor field of \((1, k)\)-type. Several examples are provided, some of them in relationship with Ricci solitons. Inspired by the Riemannian setting, we introduce last multipliers in the abstract framework of Dirichlet forms and symmetric Markov diffusion semigroups. For the last framework, we use the Bakry-Emery carré du champ associated to the infinitesimal generator of the semigroup.

MSC:

53C21 Methods of global Riemannian geometry, including PDE methods; curvature restrictions
53C25 Special Riemannian manifolds (Einstein, Sasakian, etc.)
35Q75 PDEs in connection with relativity and gravitational theory
53C99 Global differential geometry
53B20 Local Riemannian geometry
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