Quaderni di Dipartimento

• QDD211 - 04/06/2015
  Pagani,C.D.; Pierotti, D.; Pistoia, A.; Vaira, G.
  Concentration along geodesics for a nonlinear Steklov problem arising in corrosion modelling

  Abstract

  Abstract

  We consider the problem of finding pairs (lambda; u), with lambda > 0 and u a harmonic function in a three dimensional torus-like domain D, satisfying the nonlinear boundary condition partial_n u =  sinh u on partial D. This type of boundary condition arises in corrosion modelling (Butler Volmer condition). We prove existence of solutions which concentrate along some geodesics of the boundary as the parameter  lambda goes to zero.

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• QDD210 - 05/05/2015
  Bertacchi, D; Zucca, F; Ambrosini, R.
  The timing of life history events in presence of soft disturbances

  Abstract

  Abstract

  We study a model for the evolutionarily stable strategy (ESS) used by biological populations for choosing the time of life-history events, such as migration and breeding. In our model we accounted for both intra-species competition (early individuals have a competitive advantage) and a disturbance which strikes at a random time, killing a fraction 1-p of the population. Disturbances include spells of bad weather, such as freezing or heavily raining days It has been shown by Iwasa and Levin (1995), that when p=0, then the ESS is a mixed strategy, where individuals wait for a certain time and afterwards start arriving (or breeding) every day. We remove the constraint $p=0$ and show that if 0 then the ESS still implies a mixed choice of times, but strong competition may lead to a massive arrival at the earliest time possible of a fraction of the population, while the rest will arrive throughout the whole period during which the disturbance may occur. More precisely, given "p", there is a threshold for the competition parameter "a", above which massive arrivals occur and below which there is a behaviour as in Iwasa and Levin (1995). We study the behaviour of the ESS and of the average fitness of the population, depending on the parameters involved. We also discuss how the population may be affected by climate change, in two respects: first, how the ESS should change under the new climate and whether this change implies an increase of the average fitness; second, which is the impact of the new climate on a population that still follows the old strategy. We show that, at least under some conditions, extreme weather events imply a temporary decrease of the average fitness (thus an increasing mortality). If the population adapts to the new climate, the survivors may have a larger fitness.

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• QDD209 - 21/04/2015
  Alpay, D.; Colombo, F.; Sabadini, I.
  Slice Hyperholomorphic Schur Analysis

  Abstract

  Abstract

  The purpose of this work is to define and study the counterpart of Schur functions and Schur analysis in the slice hyperholomorphic setting. There are at least two motivations for such a study, both having in the background the desire to replace the complex numbers by the quaternions. One motivation comes from the theory of linear systems and signal processing. Another motivation is to define new tools and problems in hypercomplex analysis inspired from the complex setting (for instance Nevanlinna-Pick interpolation and the characteristic operator functions). The monograph is divided in three parts: namely Classical Schur analysis, Quaternionic analysis, and Quaternionic Schur analysis. Several parts in the second and third part are original and appear in this work for the first time.

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• QDD208 - 20/04/2015
  Catino, G.; Mastrolia, P.; Monticelli, D.D.
  A variational characterization of flat spaces in dimension three

  Abstract

  	Abstract
  	In this short note we prove that, in dimension three, flat metrics are the only complete metrics with non-negative scalar curvature which are critical for the sigma_2-curvature functional.

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• QDD207 - 20/04/2015
  Catino, G.
  On conformally flat manifolds with constant positive scalar curvature

  Abstract

  	Abstract
  	We classify compact conformally flat n-dimensional manifolds with constant positive scalar curvature and satisfying an optimal integral pinching condition: they are covered isometrically by either S^n with the round metric, S^1×S^{n-1} with the product metric or S^1×S^{n-1} with a rotationally symmetric Derdzinski metric.

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• QDD206 - 13/04/2015
  Catino, G.; Mastrolia, P.; Monticelli, D.D.; Rigoli, M.
  Analytic and geometric properties of generic Ricci solitons

  Abstract

  Abstract

  The aim of this paper is to prove some classification results for generic shrinking Ricci solitons. In particular, we show that every three dimensional generic shrinking Ricci soliton is given by quotients of either ????^3, ?×????^2 or ?^3, under some very weak conditions on the vector field X generating the soliton structure. In doing so we introduce analytical tools that could be useful in other settings; for instance we prove that the Omori-Yau maximum principle holds for the X-Laplacian on every generic Ricci soliton, without any assumption on X.

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• QDD205 - 13/04/2015
  Catino, G.
  Some rigidity results on critical metrics for quadratic functionals

  Abstract

  Abstract

  In this paper we prove rigidity results on critical metrics for quadratic curvature functionals, involving the Ricci and the scalar curvature, on the space of Riemannian metrics with unit volume. It is well-known that Einstein metrics are always critical points. The purpose of this article is to show that, under some curvature conditions, a partial converse is true. In particular, for a class of quadratic curvature functionals, we prove that every critical metric with non-negative sectional curvature must be Einstein.

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• QDD204 - 08/04/2015
  Catino, G.; Mastrolia, P.; Monticelli, D.D.; Rigoli, M.
  Conformal Ricci solitons and related integrability conditions

  Abstract

  Abstract

  In this paper we introduce, in the Riemannian setting, the notion of conformal Ricci soliton, which includes as particular cases Einstein manifolds, conformal Einstein manifolds and (generic and gradient) Ricci solitons. We provide here some necessary integrability conditions for the existence of these structures that also recover, in the corresponding contexts, those already known in the literature for conformally Einstein manifolds and for gradient Ricci solitons. A crucial tool in our analysis is the construction of some appropriate and highly nontrivial (0,3)-tensors related to the geometric structures, that in the special case of gradient Ricci solitons become the celebrated tensor D recently introduced by Cao and Chen. A significant part of our investigation, which has independent interest, is the derivation of a number of commutation rules for covariant derivatives (of functions and tensors) and of transformation laws of some geometric objects under a conformal change of the underlying metric.

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