This book provides an introduction to some aspects of the flourishing field of nonsmooth geometric analysis. In particular, a quite detailed account of the first-order structure of general metric measure spaces is presented, and the reader is introduced to the second-order calculus on spaces – known as RCD spaces – satisfying a synthetic lower Ricci curvature bound. Examples of the main topics covered include notions of Sobolev space on abstract metric measure spaces; normed modules, which constitute a convenient technical tool for the introduction of a robust differential structure in the nonsmooth setting; first-order differential operators and the corresponding functional spaces; the theory of heat flow and its regularizing properties, within the general framework of “infinitesimally Hilbertian” metric measure spaces; the RCD condition and its effects on the behavior of heat flow; and second-order calculus on RCD spaces. The book is mainly intended for young researchers seeking a comprehensive and fairly self-contained introduction to this active research field. The only prerequisites are a basic knowledge of functional analysis, measure theory, and Riemannian geometry.
Lectures on Nonsmooth Differential Geometry / Gigli, Nicola; Pasqualetto, Enrico. - 2(2020), pp. 1-204.
|Titolo:||Lectures on Nonsmooth Differential Geometry|
|Autori:||Gigli, Nicola; Pasqualetto, Enrico|
|Data di pubblicazione:||2020|
|Digital Object Identifier (DOI):||http://dx.doi.org/10.1007/978-3-030-38613-9|
|Appare nelle tipologie:||3.1 Book or scientific treatise|