Abstract
For any (not necessarily complete) Riemannian manifold, we construct a larger Riemannian metric which is complete and with bounded sectional curvatures. As an application, log-Sobolev inequalities are established on arbitrary Riemannian manifolds with reference measures having smooth and strictly positive densities. In particular, a conjecture of the second-named author (see [J. Math. Anal. Appl. 300 (2004) 426–435]) is solved.
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