Parabolicity criteria and characterization results for submanifoldsof bounded mean curvature in model manifolds with weights

Let P be a submanifold properly immersed in a rotationally symmetric manifold having a pole and endowed with a weight e h. The aim of this paper is twofold. First, by assuming certain control on the h-mean curvature of P, we establish comparisons for the h-capacity of extrinsic balls in P, from which we deduce criteria ensuring the h-parabolicity or h-hyperbolicity of P. Second, we employ functions with geometric meaning to describe submanifolds of bounded h-mean curvature which are confined into some regions of the ambient manifold. As a consequence, we derive half-space and Bernstein-type theorems generalizing previous ones. Our results apply for some relevant h-minimal submanifolds appearing in the singularity theory of the mean curvature flow

Intrinsic and extrinsic comparison results for isoperimetric quotients and capacities in weighted manifolds

Let (M, g)be a complete non-compact Riemannian manifold together with a function eh, which weights the Hausdorff measures associated to the Riemannian metric. In this work we assume lower or upper radial bounds on some weighted or unweighted curvatures of Mto deduce comparisons for the weighted isoperimetric quotient and the weighted capacity of metric balls in Mcentered at a point o ∈M. As a consequence, we obtain parabolicity and hyperbolicity criteria for weighted manifolds generalizing previous ones. A basic tool in our study is the analysis of the weighted Laplacian of the distance function from o. The technique extends to non-compact submanifolds properly immersed in Munder certain control on their weighted mean curvature

A note on the p-parabolicity of submanifolds

We give a geometric criterion which shows p-parabolicity of a class of submanifolds in a Riemannian manifold, with controlled second fundamental form, for p ≥ 2

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

Geometric evolution equations and p-harmonic theory with applications in differential geometry.

In this dissertation, we consider parabolic (e.g. Ricci flow) and elliptic (e.g. p-harmonic equations) partial differential equations on Riemannian manifolds and use them to study geometric and topological problems. More specifically, to classify a special class of Ricci flow equations, we constructed a family of new entropy functionals in the sense of Perelman. We study the monotonicity of these functionals and use this property to prove that a compact steady gradient Ricci breather is necessarily Ricci-flat. We introduce a new approach to prove the monotonicity formula of Perelman's W -entropy functional and we construct similar entropy functionals on expanders from this new viewpoint. We prove that a large family of complete non-compact Riemannian manifolds cannot be stably minimally immersed into Euclidean space as a hypersurface which serves as a non-existence theorem considering the Generalized Bernstein Conjecture. We give another yet simpler proof for a theorem of do Carmo and Peng, concerning stable minimal hypersurfaces in Euclidean space with certain integral curvature condition. In the study of p-harmonic geometry, we develop a classification theory of Riemannian manifolds by using p-superharmonic functions in the weak sense. We gave sharp estimates as sufficient conditions for a p-parabolic manifold. By developing a Generalized Uniformization Theorem, a Generalized Bochner's Method, and an iterative method, we approach various geometric and variational problems in complete noncompact manifolds of general dimensions