Alberto G. Setti

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

Ricci almost solitons

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A remark on the maximum principle and stochastic completeness

-Laplacian

Some characterizations of space-forms

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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

Yamabe-type equations on complete, noncompact manifolds

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Estimates for functions of the Laplace operator on homogeneous trees

-Laplacian Bibliography.

ENERGY ESTIMATES AND LIOUVILLE THEOREMS FOR HARMONIC MAPS

We introduce a natural extension of the concept of gradient Ricci soliton: the Ricci almost soliton. We provide existence and rigidity results, we deduce a-priori curvature estimates and isolation phenomena, and we investigate some topological properties. A number of differential identities involving the relevant geometric quantities are derived. Some basic tools from the weighted manifold theory such as general weighted volume comparisons and maximum principles at infinity for diffusion operators are discussed.

The range of the Helgason-Fourier transformation on homogeneous trees

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.

The connectivity at infinity of a manifold and Lq,p-Sobolev inequalities

Integral conditions on the traceless Ricci tensor are used to characterize Euclidean and hyperbolic spaces among complete, locally conformally flat manifolds of constant scalar curvature. The main tools are vanishing-type results for L P -solutions of a large class of differential inequalities. Further applications of the technique are also given.