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Tesi etd-06252018-124752

Thesis type
Tesi di laurea magistrale
Limits of Riemannian manifolds with Ricci curvature bounded from below
Corso di studi
relatore Prof. Ambrosio, Luigi
relatore Dott. Mari, Luciano
controrelatore Prof. Novaga, Matteo
Parole chiave
  • Ricci-limit space
  • Ricci curvature
  • Riemannian manifold
  • Almost splitting theorem
Data inizio appello
Secretata d'ufficio
Riassunto analitico
The aim of this thesis is the study of some structure properties of the so-called Ricci-limit spaces. The main ingredient for the definition of a Ricci-limit space is the the notion of pointed Gromov-Hausdorff convergence, introduced by Gromov in a paper which dates back to 1981: from the compactness criterion proved in this paper and the Bishop-Gromov inequality, it follows that a sequence of complete pointed Riemannian n-dimensional manifolds (M_i,m_i) which satisfy a uniform lower bound on Ricci curvature admits a subsequence which converges, in the pointed Gromov-Hausdorff sense, to some pointed metric space (Y,y). These limit points are the so-called Ricci-limit spaces.

It is predictable that a Ricci-limit space could have good geometric properties: the aim of studying such properties led Cheeger and Colding to publish a series of papers dealing with such spaces. One of the most important topics studied by them is the analysis of tangent spaces to these Ricci-limit spaces.

The main question on which we focus our attention is: what is the Hausdorff dimension of the singular points S, i.e. the points where the Ricci-limit space does not have a unique tangent space which is isometric to some Euclidean space? The answer to the question is given, by Cheeger and Colding, in a particular case: if we assume the non-collapsed condition, then this dimension is no more than n-2.

The most important tools to show this result are the almost splitting theorem, a generalization of a well-known theorem in Riemannian Geometry, i.e. the splitting theorem, and the fact that every tangent space, in the non-collapsed case, is a metric cone. We analyze the proof of the almost splitting theorem with all the details. The fact that every tangent space is a metric cone gives a rigidity to the structure of tangent spaces which makes possible to perform a dimension reduction argument very useful for the proof of the result about the dimension of the singular set.