Keller--Osserman conditions for diffusion-type operators on Riemannian Manifolds

In this paper we obtain generalized Keller-Osserman conditions for wide classes of differential inequalities on weighted Riemannian manifolds of the form $L u\geq b(x) f(u) \ell(|\nabla u|)$ and $L u\geq b(x) f(u) \ell(|\nabla u|) - g(u) h(|\nabla u|)$, where $L$ is a non-linear diffusion-type operator. Prototypical examples of these operators are the $p$-Laplacian and the mean curvature operator. While we concentrate on non-existence results, in many instances the conditions we describe are in fact necessary for non-existence. The geometry of the underlying manifold does not affect the form of the Keller-Osserman conditions, but is reflected, via bounds for the modified Bakry-Emery Ricci curvature, by growth conditions for the functions $b$ and $\ell$. We also describe a weak maximum principle related to inequalities of the above form which extends and improves previous results valid for the \vp-Laplacian

Dirichlet parabolicity and L1L^1-Liouville property under localized geometric conditions

We shed a new light on the $L^1$-Liouville property for positive, superharmonic functions by providing many evidences that its validity relies on geometric conditions localized on large enough portions of the space. We also present examples in any dimension showing that the $L^1$-Liouville property is strictly weaker than the stochastic completeness of the manifold. The main tool in our investigations is represented by the potential theory of a manifold with boundary subject to Dirichlet boundary conditions. The paper incorporates, under a unifying viewpoint, some old and new aspects of the theory, with a special emphasis on global maximum principles and on the role of the Dirichlet Green's kernel

Ricci almost solitons

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 L1 L^1 -Liouville property on graphs

In this paper we investigate the $L^1$-Liouville property, underlining its connection with stochastic completeness and other structural features of the graph. We give a characterization of the $L^1$-Liouville property in terms of the Green function of the graph and use it to prove its equivalence with stochastic completeness on model graphs. Moreover, we show that there exist stochastically incomplete graphs which satisfy the $L^1$-Liouville property and prove some comparison theorems for general graphs based on inner-outer curvatures. We also introduce the Dirichlet $L^1$-Liouville property of subgraphs and prove that if a graph has a Dirichlet $L^1$-Liouville subgraph, then it is $L^1$-Liouville itself. As a consequence, we obtain that the $L^1$-Liouville property is not affected by a finite perturbation of the graph and, just as in the continuous setting, a graph is $L^1$-Liouville provided that at least one of its ends is Dirichlet $L^1$-Liouville

Spectral and stochastic properties of the f-Laplacian, solutions of PDE's at infinity and geometric applications

The aim of this paper is to suggest a new viewpoint to study qualitative properties of solutions of semilinear elliptic PDE's defined outside a compact set. The relevant tools come from spectral theory and from a combination of stochastic properties of the relevant differential operators. Possible links between spectral and stochastic properties are analyzed in detail

Remarks on non-compact gradient Ricci solitons

In this paper we show how techniques coming from stochastic analysis, such as stochastic completeness (in the form of the weak maximum principle at infinity), parabolicity and $L^p$-Liouville type results for the weighted Laplacian associated to the potential may be used to obtain triviality, rigidity results, and scalar curvature estimates for gradient Ricci solitons under $L^p$ conditions on the relevant quantities.Comment: The main changes over the previous version are that Theorem 2 has been improved by removing the pointwise growth assumption on $|\nabla f|$ in the case $p=1$, and that in Theorem 3, a shrinking gradient Ricci soliton is shown to be isometric to $\R^m$ in the endpoint case $S_*=0