260 research outputs found
On the equivalence of stochastic completeness, Liouville and Khas'minskii condition in linear and nonlinear setting
Set in Riemannian enviroment, the aim of this paper is to present and discuss
some equivalent characterizations of the Liouville property relative to special
operators, in some sense modeled after the p-Laplacian with potential. In
particular, we discuss the equivalence between the Lioville property and the
Khas'minskii condition, i.e. the existence of an exhaustion functions which is
also a supersolution for the operator outside a compact set. This generalizes a
previous result obtained by one of the authors and answers to a question in
"Aspects of potential theory, linear and nonlinear" by Pigola, Rigoli and
Setti.Comment: 34 pages. The pasting lemma has been improved to fix a technical
problem in the main theorem. Final version, to appear on Trans. Amer. Math.
So
Maps from Riemannian manifolds into non-degenerate Euclidean cones
Let be a connected, non-compact -dimensional Riemannian manifold. In
this paper we consider smooth maps with images
inside a non-degenerate cone. Under quite general assumptions on , we
provide a lower bound for the width of the cone in terms of the energy and the
tension of and a metric parameter. As a side product, we recover some
well known results concerning harmonic maps, minimal immersions and K\"ahler
submanifolds. In case is an isometric immersion, we also show that, if
is sufficiently well-behaved and has non-positive sectional curvature,
cannot be contained into a non-degenerate cone of
.Comment: 19 pages, to appea
Duality between Ahlfors-Liouville and Khas'minskii properties for nonlinear equations
In recent years, the study of the interplay between (fully) non-linear
potential theory and geometry received important new impulse. The purpose of
this work is to move a step further in this direction by investigating
appropriate versions of parabolicity and maximum principles at infinity for
large classes of non-linear (sub)equations on manifolds. The main goal is
to show a unifying duality between such properties and the existence of
suitable -subharmonic exhaustions, called Khas'minskii potentials, which is
new even for most of the "standard" operators arising from geometry, and
improves on partial results in the literature. Applications include new
characterizations of the classical maximum principles at infinity (Ekeland,
Omori-Yau and their weak versions by Pigola-Rigoli-Setti) and of conservation
properties for stochastic processes (martingale completeness). Applications to
the theory of submanifolds and Riemannian submersions are also discussed.Comment: 67 pages. Final versio
Some generalizations of Calabi compactness theorem
In this paper we obtain generalized Calabi-type compactness criteria for
complete Riemannian manifolds that allow the presence of negative amounts of
Ricci curvature. These, in turn, can be rephrased as new conditions for the
positivity, for the existence of a first zero and for the
nonoscillatory-oscillatory behaviour of a solution of ,
subjected to the initial condition , . A unified approach for
this ODE, based on the notion of critical curve, is presented. With the aid of
suitable examples, we show that our new criteria are sharp and, even for , in borderline cases they improve on previous works of Calabi, Hille-Nehari
and Moore.Comment: 20 pages, submitte
Splitting theorems, symmetry results and overdetermined problems for Riemannian manifolds
Our work proposes a unified approach to three different topics in a general
Riemannian setting: splitting theorems, symmetry results and overdetermined
elliptic problems. By the existence of a stable solution to the semilinear
equation on a Riemannian manifold with non-negative Ricci
curvature, we are able to classify both the solution and the manifold. We also
discuss the classification of monotone (with respect to the direction of some
Killing vector field) solutions, in the spirit of a conjecture of De Giorgi,
and the rigidity features for overdetermined elliptic problems on submanifolds
with boundary
Spectral radius, index estimates for Schrodinger operators and geometric applications
In this paper we study the existence of a first zero and the oscillatory
behavior of solutions of the ordinary differential equation ,
where are functions arising from geometry. In particular, we introduce a
new technique to estimate the distance between two consecutive zeros. These
results are applied in the setting of complete Riemannian manifolds: in
particular, we prove index bounds for certain Schr\"odinger operators, and an
estimate of the growth of the spectral radius of the Laplacian outside compact
sets when the volume growth is faster than exponential. Applications to the
geometry of complete minimal hypersurfaces of Euclidean space, to minimal
surfaces and to the Yamabe problem are discussed.Comment: 48 page
On the geometry of curves and conformal geodesics in the Mobius space
This paper deals with the study of some properties of immersed curves in the
conformal sphere \mathds{Q}_n, viewed as a homogeneous space under the action
of the M\"obius group. After an overview on general well-known facts, we
briefly focus on the links between Euclidean and conformal curvatures, in the
spirit of F. Klein's Erlangen program. The core of the paper is the study of
conformal geodesics, defined as the critical points of the conformal arclength
functional. After writing down their Euler-Lagrange equations for any , we
prove an interesting codimension reduction, namely that every conformal
geodesic in \mathds{Q}_n lies, in fact, in a totally umbilical 4-sphere
\mathds{Q}_4. We then extend and complete the work in (Musso, "The conformal
arclength functional", Math Nachr.) by solving the Euler-Lagrange equations for
the curvatures and by providing an explicit expression even for those conformal
geodesics not included in any conformal 3-sphere.Comment: 40 page
Some geometric properties of hypersurfaces with constant -mean curvature in Euclidean space
Let f:M\ra \erre^{m+1} be an isometrically immersed hypersurface. In this
paper, we exploit recent results due to the authors in \cite{bimari} to analyze
the stability of the differential operator associated with the -th
Newton tensor of . This appears in the Jacobi operator for the variational
problem of minimizing the -mean curvature . Two natural applications
are found. The first one ensures that, under the mild condition that the
integral of over geodesic spheres grows sufficiently fast, the Gauss map
meets each equator of \esse^m infinitely many times. The second one deals
with hypersurfaces with zero -mean curvature. Under similar growth
assumptions, we prove that the affine tangent spaces , , fill
the whole \erre^{m+1}.Comment: 10 pages, corrected typo
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 and , where is a non-linear diffusion-type operator.
Prototypical examples of these operators are the -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 and
. 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
- …