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Marco Rigoli

Publications and source records attributed to Marco Rigoli.

48 records · Page 3Linked to original sources
arXiv2011

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 $g(t)$ of $g"+Kg=0$, subjected to the initial condition $g(0)=0$, $g'(0)=1$. 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 $K\ge 0$, in borderline cases they improve on previous works of Calabi, Hille-Nehari and Moore.

math.DG↗
arXiv2011

Spacelike hypersurfaces of constant higher order mean curvature in generalized Robertson-Walker spacetimes

In this paper we analyze the problem of uniqueness for spacelike hypersurfaces with constant higher order mean curvature in generalized Robertson-Walker spacetimes. We consider first the case of compact spacelike hypersurfaces, completing some previous results given in [2]. We next extend these results to the complete noncompact case. In that case, our approach is based on the use of a generalized version of the Omori-Yau maximum principle for trace type differential operators, recently given in [3].

math.DG↗
arXiv2010

On the Geometry of complete Ricci Solitons

In this paper we establish three basic equations for a general soliton structure on the Riemannian manifold $(M, <, >)$. We then draw some geometric conclusions with the aid of the maximum principle.

math.DG↗
arXiv2010

Maps from Riemannian manifolds into non-degenerate Euclidean cones

Let $M$ be a connected, non-compact $m$-dimensional Riemannian manifold. In this paper we consider smooth maps $ϕ: M \to \mathbb{R}^n$ with images inside a non-degenerate cone. Under quite general assumptions on $M$, we provide a lower bound for the width of the cone in terms of the energy and the tension of $ϕ$ and a metric parameter. As a side product, we recover some well known results concerning harmonic maps, minimal immersions and Kähler submanifolds. In case $ϕ$ is an isometric immersion, we also show that, if $M$ is sufficiently well-behaved and has non-positive sectional curvature, $ϕ(M)$ cannot be contained into a non-degenerate cone of $\mathbb{R}^{2m-1}$.

math.DG↗
arXiv2010

Some geometric properties of hypersurfaces with constant $r$-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 $L_r$ associated with the $r$-th Newton tensor of $f$. This appears in the Jacobi operator for the variational problem of minimizing the $r$-mean curvature $H_r$. Two natural applications are found. The first one ensures that, under the mild condition that the integral of $H_r$ 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 $(r+1)$-mean curvature. Under similar growth assumptions, we prove that the affine tangent spaces $f_*T_pM$, $p\in M$, fill the whole $\erre^{m+1}$.

math.DG↗
arXiv2010

Keller-Osserman type conditions for differential inequalities with gradient terms on the Heisenberg group

The aim of this paper is to study the qualitative behaviour of non-negative entire solutions of certain differential inequalities involving gradient terms on the Heisenberg group. We focus our investigation on the two classes of inequalities of the form $Δ^ϕu \ge f(u)l(|\nabla u|)$ and $Δ^ϕu \ge f(u) - h(u) g(|\nabla u|)$, where $f,l,h,g$ are non-negative continuous functions satisfying certain monotonicity properties. The operator $Δ^ϕ$, called the $ϕ$-Laplacian, can be viewed as a natural generalization of the $p$-Laplace operator recently considered by various authors in this setting. We prove some Liouville theorems introducing two new Keller-Osserman type conditions, both extending the classical one which appeared long ago in the study of the prototype differential inequality $Δu \ge f(u)$ in $\erre^m$. Furthermore, we show sharpness of our conditions when we specialize to the case of the $p$-Laplacian. Needless to say, our results continue to hold, with the obvious minor modifications, also in the Euclidean space.

math.DG↗
arXiv2010

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 $(vz')'+Avz = 0$, where $A,v$ 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 Schrödinger 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.

math.DG↗
arXiv2010

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 Möbius 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 $n$, 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.

math.DG↗
arXiv2010

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.

math.DG↗
arXiv2009

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.

math.AP↗
arXiv2007

Existence and non-existence results for a logistic-type equation on manifolds

We study the steady state solutions of a generalized logistic type equation on a complete Riemannian manifold. We provide sufficient conditions for existence, respectively non-existence of positive solutions, which depend on the relative size of the coefficients and their mutual interaction with the geometry of the manifold, which is mostly taken into account by means of conditions on the volume growth of geodesic balls.

math.DG↗