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In this paper, we investigate the validity of first and second order Lp estimates for the solutions of the Poisson equation depending on the geometry of the underlying manifold. We first present Lp estimates of the gradient under the assumption that the Ricci tensor is lower bounded in a local integral sense, and construct the first counterexample showing that they are false, in general, without curvature restrictions. Next, we obtain Lp estimates for the second order Riesz transform (or, equivalently, the validity of Lp Calderón–Zygmund inequalities) on the whole scale 1 < p < ∞ by assuming that the injectivity radius is positive and that the Ricci tensor is either pointwise lower bounded, or non-negative in a global integral sense. When 1 < p ≤ 2, analogous Lp bounds on higher even order Riesz transforms are obtained provided that also the derivatives of Ricci are controlled up to a suitable order. [ABSTRACT FROM AUTHOR]

We study decay and compact support properties of positive and bounded solutions of Δ p u ≥ Λ (u) on the exterior of a compact set of a complete manifold with rotational symmetry. In the same setting, we also give a new characterization of stochastic completeness for the p -Laplacian in terms of a global W 1 , p -regularity of such solutions. One of the tools we use is a nonlinear version of the Feller property which we investigate on general Riemannian manifolds and which we establish under integral Ricci curvature conditions. [ABSTRACT FROM AUTHOR]

In this paper we characterize the Neumann-parabolicity of manifolds with boundary in terms of a new form of the classical Ahlfors maximum principle and of a version of the so-called Kelvin--Nevanlinna--Royden criterion. The motivation underlying this study is to obtain new information on the geometry of graphs with prescribed mean curvature inside a Riemannian product of the type N x R. In this direction two kind of results will be presented: height estimates for constant mean curvature graphs parametrized over unbounded domains in a complete manifold, which extend results by A. Ros and H. Rosenberg valid for domains of R², and slice-type results for graphs whose superlevel sets have finite volume. Finally, the use of the Ahlfors maximum principle allows us to establish a connection between the Neumann-parabolicity and the Dirichlet-parabolicity commonly used in minimal surface theory. In particular, we will be able to give a deterministic proof of special cases of a result by R. Neel. [ABSTRACT FROM AUTHOR]

We obtain a comparison principle on annuli, with “catenoid-like” functions, for supersolutions of non-linear elliptic PDEs over exterior domains in a non-positively curved manifold with a pole. This result is applied to get an upper estimate on the growth of such supersolutions and, in particular, of exterior graphs of non-negative mean curvature. [ABSTRACT FROM AUTHOR]

We prove mean and sectional curvature estimates for submanifolds confined into either a horocylinder of N × L or a horoball of N , where N is a Cartan–Hadamard manifold with pinched curvature. Thus, these submanifolds behave in many respects like submanifolds immersed into compact balls and into cylinders over compact balls. The proofs rely on the Hessian comparison theorem for the Busemann function. [ABSTRACT FROM AUTHOR]