Geometric Analysis

This volume includes expanded versions of the lectures delivered in the Graduate Minicourse portion of the 2013 Park City Mathematics Institute session on Geometric Analysis. The papers give excellent high-level introductions, suitable for graduate students wishing to enter the field and experienced researchers alike, to a range of the most important areas of geometric analysis. These include: the general issue of geometric evolution, with more detailed lectures on Ricci flow and Kahler-Ricci flow, new progress on the analytic aspects of the Willmore equation as well as an introduction to the recent proof of the Willmore conjecture and new directions in min-max theory for geometric variational problems, the current state of the art regarding minimal surfaces in $R^3$, the role of critical metrics in Riemannian geometry, and the modern perspective on the study of eigenfunctions and eigenvalues for Laplace-Beltrami operators. This volume includes expanded versions of the lectures delivered in the Graduate Minicourse portion of the 2013 Park City Mathematics Institute session on Geometric Analysis. The papers give excellent high-level introductions, suitable for graduate students wishing to enter the field and experienced researchers alike, to a range of the most important areas of geometric analysis. These include: the general issue of geometric evolution, with more detailed lectures on Ricci flow and Kähler-Ricci flow, new progress on the analytic aspects of the Willmore equation as well as an introduction to the recent proof of the Willmore conjecture and new directions in min-max theory for geometric variational problems, the current state of the art regarding minimal surfaces in $R^3$, the role of critical metrics in Riemannian geometry, and the modern perspective on the study of eigenfunctions and eigenvalues for Laplace–Beltrami operators. Contents 6 Preface 14 Heat Diffusion in Geometry • Gerhard Huisken 18 Applications of Hamilton’s Compactness Theorem for Ricci Flow • Peter Topping 32 The Kähler-Ricci flow on compact Kähler manifolds • Ben Weinkove 68 Park City lectures on Eigenfunctions • Steve Zelditch 126 Critical Metrics for Riemannian Curvature Functionals • Jeff A. Viaclovsky 212 Min-max theory and a proof of the Willmore conjecture • Fernando C. Marques and André Neves 292 Weak immersions of surfaces with L2-bounded second fundamental form • Tristan Riviére 318 Introduction to Minimal Surface Theory • Brian White 402 Cover -- Title page -- Contents -- Preface -- Introduction -- Heat diffusion in geometry -- Applications of Hamilton's compactness theorem for Ricci flow -- The Kähler-Ricci flow on compact Kähler manifolds -- Park City lectures on eigenfunctions -- Critical metrics for Riemannian curvature functionals -- Min-max theory and a proof of the Willmore conjecture -- Weak immersions of surfaces with 2-bounded second fundamental form -- Introduction to minimal surface theory -- Back Cover Provides expanded versions of the lectures delivered in the Graduate Minicourse portion of the 2013 Park City Mathematics Institute session on Geometric Analysis. The papers give excellent high-level introductions, suitable for graduate students wishing to enter the field and experienced researchers alike, to a range of the most important areas of geometric analysis.