## Tesi etd-06252018-124752 |

Thesis type

Tesi di laurea magistrale

Author

ANTONELLI, GIOACCHINO

URN

etd-06252018-124752

Title

Limits of Riemannian manifolds with Ricci curvature bounded from below

Struttura

MATEMATICA

Corso di studi

MATEMATICA

Supervisors

**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

13/07/2018;

Consultabilità

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.

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.

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