*Ph.D. Thesis*

**Inserted:** 22 nov 2019

**Last Updated:** 22 nov 2019

**Year:** 2019

**Abstract:**

The thesis is divided in three main parts:

In the first part we introduce the class of optimal Entropy-Transport problems, a recent generalization of optimal transport problems where also creation and destruction of mass is taken into account.

We focus in particular on the metric properties of these problems, computed in terms of an entropy function $F$ and a cost function. Starting from the power-like entropy $F(s)=(s^p-p(s-1)-1)/(p(p-1))$ and a suitable cost depending on a metric $\mathsf{d}$ on a space $X$, our main result ensures that for every $p>1$ the related Entropy-Transport cost induces a distance on the space of finite measures over $X$. Inspired by previous work of Gromov and Sturm, we then use these Entropy-Transport metrics to construct new complete and separable distances on the family of metric measure spaces with finite mass.
We also study in detail the pure entropic setting, that can be recovered as a particular case when the transport is forbidden. In this situation, corresponding to the classical theory of Csiszár $F$-divergences, we analyse some structural properties of these entropic functionals and we highlight the important role played by the class of Matusita's divergences.

The second part is devoted to the study of bounds involving Cheeger's isoperimetric constant $h$ and the first eigenvalue $\lambda_{1}$ of the Laplacian.

A celebrated lower bound of $\lambda_{1}$ in terms of $h$, $\lambda_{1}\geq h^{2}/4$, was proved by Cheeger in 1970 for smooth Riemannian manifolds. An upper bound on $\lambda_{1}$ in terms of $h$ was established by Buser in 1982 (with dimensional constants) and improved (to a dimension-free estimate) by Ledoux in 2004 for smooth Riemannian manifolds with Ricci curvature bounded below.

The goal of this part is two fold. First: by adapting the approach of Ledoux via heat semigroup techniques, we sharpen the inequalities obtaining a dimension-free sharp Buser inequality for spaces with (Bakry-Émery weighted) Ricci curvature bounded below by $K\in \mathbb{R}$ (the inequality is sharp for $K>0$ as equality is obtained on the Gaussian space). Second: all of our results hold in the higher generality of (possibly non-smooth) metric measure spaces with Ricci curvature bounded below in synthetic sense, the so-called $ \mathsf{RCD}(K,\infty)$ spaces.

In the third part, given a complete, connected Riemannian manifold $ \mathbb{M}^n $ with Ricci curvature bounded from below, we discuss the stability of the solutions of a porous medium-type equation with respect to the 2-Wasserstein distance.
We produce (sharp) stability estimates under negative curvature bounds, which to some extent generalize well-known results by Sturm and Otto-Westdickenberg.

The strategy of the proof mainly relies on a quantitative $L^1$--$L^\infty$ smoothing property of the equation considered, combined with the Hamiltonian approach developed by Ambrosio, Mondino and Savaré in a metric-measure setting.

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