Open Collections

Maximum principles at infinity and the Ahlfors-Khas'minskii duality

Banff International Research Station for Mathematical Innovation and Discovery

Maximum principles at infinity (or ``almost maximum principles") are a powerful tool to investigate the geometry of Riemannian manifolds. In this talk I will focus on the the Ekeland, the Omori-Yau principles and their weak versions, in the sense of Pigola-Rigoli-Setti. These last have probabilistic interpretations in terms of stochastic and martingale completeness, that is, the non-explosion of (respectively) the Brownian motion and each martingale on $M$. After an overview to show the usefulness of these principles in geometry, I will move to discuss necessary and sufficient conditions for their validity. In particular, I will focus on an underlying duality that allows to discover new relations between them. Indeed, duality holds for a broad class of fully-nonlinear operators of geometric interest. Our methods use the approach to nonlinear PDEs pioneered by Krylov ('95) and Harvey-Lawson ('09 - ). This is based on joint works with B. Bianchini, M. Rigoli, P. Pucci and L. F. Pessoa.

Mathematics; Partial differential equations; Integral equations

video/mp4

Author affiliation: Scuola Normale Superiore

2017-10-01

Attribution-NonCommercial-NoDerivatives 4.0 International

http://hdl.handle.net/2429/63162

Unreviewed

http://creativecommons.org/licenses/by-nc-nd/4.0/