"This fourth edition of Geometric Measure Theory: A Beginner's Guide presents the latest results on soap bubble clusters and double bubbles in spheres, tori, and Gauss space. Gauss space, defined as Euclidean space with Gaussian density, of long import to probabilists, appears in Perelman's original paper on the Poincare Conjecture. This edition also describes general manifolds with density and their relationship to Perelman's paper.Throughout there are updates, new illustrations, new exercises and solutions, and new references. Morgan emphasizes geometry over proofs and technicalities to provide the most accessible introduction to the subject."--BOOK JACKET.