Marco Rigoli

Maximum principles on Riemannian manifolds and applications

Preliminaries and some geometric motivations Further typical applications of Yaus technique Stochastic completeness and the weak maximum principle The weak maximum principle for the

A remark on the maximum principle and stochastic completeness

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Maximum principles and geometric applications

-Laplacian

Average decay of Fourier transforms and geometry of convex sets

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Some non-linear function theoretic properties of Riemannian manifolds

-parabolicity and some further remarks Curvature and the maximum principle for the

A general form of the weak maximum principle and some applications.

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Yamabe-type equations on complete, noncompact manifolds

-Laplacian Bibliography.

Some remarks on the weak maximum principle

We prove that the stochastic completeness of a Riemannian manifold (M, ) is equivalent to the validity of a weak form of the Omori-Yau maximum principle. Some geometric applications of this result are also presented.

Lichnerowicz-type equations on complete manifolds

A crash course in Riemannian geometry.- The Omori-Yau maximum principle.- New forms of the maximum principle.- Sufficient conditions for the validity of the weak maximum principle.- Miscellany results for submanifolds.- Applications to hypersurfaces.- Hypersurfaces in warped products.- Applications to Ricci Solitons.- Spacelike hypersurfaces in Lorentzian spacetimes.

A finiteness theorem for the space of

Let B be a convex body in R2, with piecewise smooth boundary and let ^?B denote the Fourier transform of its characteristic function. In this paper we determine the admissible decays of the spherical Lp averages of ^?B and we relate our analysis to a problem in the geometry of convex sets. As an application we obtain sharp results on the average number of integer lattice points in large bodies randomly positioned in the plane.