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 $n$, 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.Comment: 40 page

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 $\phi: M \to \mathbb{R}^n$ 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 $\phi$ 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 $\phi$ is an isometric immersion, we also show that, if $M$ is sufficiently well-behaved and has non-positive sectional curvature, $\phi(M)$ cannot be contained into a non-degenerate cone of $\mathbb{R}^{2m-1}$.Comment: 19 pages, to appea

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.Comment: 20 pages, submitte

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 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

Some geometric properties of hypersurfaces with constant rr-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}.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 $L u\geq b(x) f(u) \ell(|\nabla u|)$ 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

On the 1/H1/H-flow by pp-Laplace approximation: new estimates via fake distances under Ricci lower bounds

In this paper we show the existence of weak solutions $w : M \rightarrow \mathbb{R}$ of the inverse mean curvature flow starting from a relatively compact set (possibly, a point) on a large class of manifolds satisfying Ricci lower bounds. Under natural assumptions, we obtain sharp estimates for the growth of $w$ and for the mean curvature of its level sets, that are well behaved with respect to Gromov-Hausdorff convergence. The construction follows R. Moser's approximation procedure via the $p$-Laplace equation, and relies on new gradient and decay estimates for $p$-harmonic capacity potentials, notably for the kernel $\mathcal{G}_p$ of $\Delta_p$. These bounds, stable as $p \rightarrow 1$, are achieved by studying fake distances associated to capacity potentials and Green kernels. We conclude by investigating some basic isoperimetric properties of the level sets of $w$.Comment: 61 pages. Revised version. Section 3.2 (properness under volume doubling and weak Poincar\'e inequalities, p.41-45) was rewritten, and the main Theorems 1.4 and 4.6 changed accordingl

En jämförelse mellan två bärstolsmodeller : En kartläggning av åsikter

Syftet med studien var att kartlägga prehospitala personalens åsikter vid Mellersta Ny-lands räddningsverk angående två olika bärstolsmodeller. En av modellerna kräver att användaren fysiskt lyfter stolen vid patienttransport ner för trappor. Den andra modellen var utrustad med en nedgångscylinder som möjliggör förflyttning av en patient ner för trappor utan att fysiskt behöva lyfta stolen. Specifikt strävade arbetet till att besvara följande frågeställningar: 1. Vilken stolmodell upplevs vara bättre? 2. Vilka för- och nackdelar hade stolmodellerna? Bärstolsmodellerna användes under ett simuleringstillfälle i rollen som ledare och följare. Ledaren var den som gick framför stolen medan följaren gick bakom stolen. Informanterna fyllde efter simuleringstillfället i en medföljande enkät. Bärstolsmodellen utrustad med en nedgångscylinder överträffade modellen utan nedgångscylinder när det kom till upplevd: fysisk påfrestning, minskande av skaderisker samt kontrollerbarhet (styra, ändra hastighet, balansera). Majoriteten av informanterna som besvarade enkäten föredrog att i framtiden använda bärstolsmodellen utrustad med en nedgångcylinder i arbetslivet. En tabell med bärstolsmodellernas för- och nackdelar togs också fram. Denna studie samt tidigare forskning gällande bärstolsmodeller rekommenderar att en modell som fysiskt inte behöver lyftas, utan kan ”glida” ner för trappor, används för att minska på den fysiska påfrestningen och eventuellt skaderisken vid patientförflyttningar ner för trappor med bärstol. Den tidigare forskningen har jämfört stolar som behöver lyftas med stolar som kan ”glida” ner för trappor. Framtida forskning kunde jämföra stolar som kan ”glida” ner med varandra och utreda vilka av dessa modeller som orsakar minst fysisk påfrestning, har den minsta relativa skaderisken och upplevs vara bäst av användaren.The aim of the study was to map out the opinions of pre-hospital personnel at Keski-Uudenmaan pelastuslaitos regarding two different models of stair chairs. One of the models required the operators to physically carry the chair. The other model was equipped with a stair cylinder which allowed the operators to transfer the patient down the stairs without physically lifting the chair. Specifically, the study aimed to answer the following research questions: 1. Which stair chair model is perceived better? 2. What are the pros and cons of each stair chair model? The stair chairs were used during a simulation, the informants tried each chair in the role as leader and follower. The leader was positioned in front of the chair while the follower was located behind the chair. A survey was answered after the simulation. The stair chair equipped with the stair cylinder was found superior compared to the stair chair without one. The experience of physical stress and risk of injury was decreased while the experience of controllability (control, change of speed, balance) was increased. The majority of the informants preferred the stair chair equipped with the stair cylinder as a stair chair to be used in future work. A pro-con chart for each stair chair model was also developed. This study, as well as other studies regarding stair chairs, recommend the usage of a model that can “slide” down the stairs to decrease physical stress and the possible risk of injury in patient transfer situations. Previous studies have compared stair chairs that require the operator to physically lift the chair with stair chairs that the operators can “slide” down the stairs. Future studies could compare models that “slide” down the stairs with each other to examine the physical stress, relative injury risk and preferability between these models.Opinnäytetyön tarkoituksena oli kartoittaa Keski-Uudenmaan pelastuslaitoksen prehospitaalisen henkilöstön mielipiteitä kahdesta kantotuolimallista. Yksi kantotuoleista vaati käyttäjiltä tuolin fyysisen nostamisen potilassiirrossa portaita alas. Toinen kantotuolimalli oli varusteltu rappulaskusylinterillä, joka mahdollisti potilaan siirtämisen portaita alas nostamatta itse tuolia. Tarkennettuna oli opinnäytetyön tarkoituksena vastata seuraaviin kysymyksiin: 1. Kumpi kantotuoleista koetaan parempana? 2. Mitä hyviä ja huonoja puolia kullakin tuolilla oli? Kantotuolimalleja käytettiin simulaatiotilaisuudessa johtajan ja seuraajan rooleissa. Johtaja oli sijoittunut tuolin eteen ja seuraaja tuolin taakse. Vastaajat täyttivät tilaisuuden jälkeen kyselyn aiheesta. Kantotuolimalli rappulaskusylinterillä varusteltuna koettiin parempana kun huomioitiin seuraavat asiat: fyysinen kuormittavuus, loukkaantumisriskien väheneminen sekä kontrolloitavuus (ohjaus, nopeuden muuttaminen, tasapaino). Suurin osa vastaajista koki että rappulaskusylinterillä varusteltu kantotuoli olisi mieluisempi vaihtoehto työelämässä. Molempien kantotuolimallien hyödyt ja haitat nostettiin esille taulukoissa. Kyseinen tutkimus, sekä aiemmat tutkimukset kantotuolimalleihin liittyen, suosittelee käytettävän kantotuolimallia jota ei fyysisesti tarvitse nostaa, vaan jota voi liu`uttaa rappusia alas potilassiirroissa. Kyseisellä mallilla fyysinen rasittuvuus sekä mahdollinen loukkaantumisriski vähenisi. Aikaisemmat tutkimukset ovat vertailleet kantotuolimalleja, jotka vaativat käyttäjien fyysisesti nostavan tuolia, malleihin joita voi liu´uttaa rappusia alas. Tulevaisuuden tutkimukset voisivat vertailla kantotuolimalleja joita voi liu´uttaa toisiinsa, tutkiakseen tuolien aiheuttamaa fyysistä kuormittavuutta, loukkaantumisriskiä sekä paremmuutta

Charger-mediated energy transfer for quantum batteries: an open system approach

The energy charging of a quantum battery is analyzed in an open quantum setting, where the interaction between the battery element and the external power source is mediated by an ancilla system (the quantum charger) that acts as a controllable switch. Different implementations are analyzed putting emphasis on the interplay between coherent energy pumping mechanisms and thermalization