Isoperimetric inequalities : differential geometric and analytic perspectives

Part I. Introduction: 1. The isoperimetric problem; 2. The isoperimetric inequality in the plane; 3. Preliminaries; 4. Bibliographic notes; Part II. Differential Geometric Methods: 1. The C2 uniqueness theory; 2. The C1 isoperimetric inequality; 3. Bibliographic notes; Part III. Minkowski Area and Perimeter: 1. The Hausdorff metric on compacta; 2. Minkowski area and Steiner symmetrization; 3. Application: the Faber-Krahn inequality; 4. Perimeter; 5. Bibliographic notes; Part IV. Hausdorff Measure and Perimeter: 1. Hausdorff measure; 2. The area formula for Lipschitz maps; 3. Bibliographic notes; Part V. Isoperimetric Constants: 1. Riemannian geometric preliminaries; 2. Isoperimetric constants; 3. Discretizations and isoperimetric inequalities; 4. Bibliographic notes; Part VI. Analytic Isoperimetric Inequalities: 1. L2-Sobolev inequalities; 2. The compact case; 3. Faber-Kahn inequalities; 4. The Federer-Fleming theorem: the discrete case; 5. Sobolev inequalities and discretizations; 6. Bibliographic notes; Part VII. Laplace and Heat Operators: 1. Self-adjoint operators and their semigroups; 2. The Laplacian; 3. The heat equation and its kernels; 4. The action of the heat semigroup; 5. Simplest examples; 6. Bibliographic notes; Part VIII. Large-Time Heat Diffusion: 1. The main problem; 2. The Nash approach; 3. The Varopoulos approach; 4. Coulhon's modified Sobolev inequality; 5. The denoument: geometric applications; 6. Epilogue: the Faber–Kahn method; 7. Bibliographic notes; Bibliography.