This textbook originated from the teaching experience of the first author at the Scuola
Normale Superiore, where a course on optimal transport and its applications has been given …

The goal of the paper is to prove an exact representation formula for the Laplacian of the
distance (and more generally for an arbitrary 1-Lipschitz function) in the framework of metric …

We generalize most of the known Ricci flow invariant non-negative curvature conditions to
less restrictive negative bounds that remain sufficiently controlled for a short time. As an …

Consider a compact Riemannian manifold of dimension $\geq 3$ with strictly convex
boundary, such that the manifold admits a strictly convex function. We show that the …

The goal of the paper is four-fold. In the setting of non-smooth spaces with Ricci curvature
lower bounds (more precisely RCD (K, N) metric measure spaces):-we develop an intrinsic …

Abstract We prove matrix Li–Yau–Hamilton estimates for positive solutions to the heat
equation and the backward conjugate heat equation, both coupled with the Ricci flow. We …

We find a local solution to the Ricci flow equation under a negative lower bound for many
known curvature conditions. Under a non-collapsing assumption, the flow exists for a …

In recent years, the study of the interplay between (fully) non-linear potential theory and
geometry received important new impulse. The purpose of this work is to move a step further …

This book demonstrates the influence of geometry on the qualitative behaviour of solutions
of quasilinear PDEs on Riemannian manifolds. Motivated by examples arising, among …

Nonlinear sigma models are quantum field theories describing, in the large deviation sense,
random fluctuations of harmonic maps between a Riemann surface and a Riemannian …