Geometric analysis, Ricci flow and applications

Abstract

Geometric Analysis is a frontier area of Mathematics that, as the name implies, relates Analysis to Geometry. In a nutshell, Geometric Analysis uses the theory of Partial Differential Equations (especially Elliptic theory) to solve problems of Geometry and/or Differential Topology.The use of analytical techniques in Geometry is not new, but it gained much prominence from the 80's (and remains in prominence today) with the works of S.T. Yau, R. Schoen and R.Hamilton. Among other important contributions, we highlight the Ricci flow introduced by Hamilton in the article [8] of 82, which led to the demonstration of important generalizations of the Sphere Theorem. Twenty years later, this flow created by Hamilton would become the fundamental technique of the demonstration, given by Grygori Perelman, of the Thurston Geometrization Conjecture and, consequently, of the Poincaré Conjecture.The objective of this master's project is to provide the student a solid training to become a researcher in the field of Geometric Analysis. The first part of this project has the general culture of the student as its main objective. In it we will study the necessary prerequisites of partial differential equations, variational techniques, maximum principles, comparison theorems, important formulas, operators and problems of eigenvalues, classic results on minimal and constant mean curvature surfaces, etc.The second part of the project will be dedicated to the study of the Ricci flow. Existence, important properties, study of the evolution of geometrical quantities, Ricci solitons and other special solutions, isoperimetric estimates and singularity analysis.The last part of the project will be devoted to some applications of Ricci flow. Our main attention will be directed to several versions of the Sphere Theorem existing in the literature. If the project develops faster than we expect, the remaining time will be used for an introductory study of Yamabe's work. (AU)