Geometric flows have many applications in physics and geometry. The mean curvature flow occurs in the description of the interface evolution in certain physical models. This is related to the property that such a flow is the gradient flow of the area functional and therefore appears naturally in problems where a surface energy is minimized. The mean curvature flow also has many geometric applications, in analogy with the Ricci flow of metrics on abstract riemannian manifolds. One can use this flow as a tool to obtain classification results for surfaces satisfying certain curvature conditions, as well as to construct minimal surfaces. Geometric flows, obtained from solutions of geometric parabolic equations, can be considered as an alternative tool to prove isoperimetric inequalities. On the other hand, isoperimetric inequalities can help in treating several aspects of convergence of these flows. Isoperimetric inequalities have many applications in other fields of geometry, like hyperbolic manifolds.
Author(s): Manuel Ritoré, Carlo Sinestrari, Vicente Miquel, Joan Porti
Series: Advanced Courses in Mathematics - CRM Barcelona
Edition: 1
Publisher: Birkhäuser Basel
Year: 2009
Language: English
Pages: 124
Cover......Page 1
Advanced Courses in Mathematics CRM Barcelona......Page 3
Mean Curvature Flow and Isoperimetric Inequalities......Page 4
9783034602129......Page 5
Foreword......Page 6
Contents......Page 8
I: Formation of Singularities in the Mean Curvature Flow - Carlo Sinestrari......Page 10
1 Introduction ......Page 12
2 Geometry of hypersurfaces ......Page 14
3 Examples ......Page 18
4 Local existence and formation of singularities ......Page 19
5 Invariance properties ......Page 25
6 Singular behaviour of convex surfaces ......Page 29
7 Convexity estimates ......Page 32
8 Rescaling near a singularity ......Page 34
9 Huisken’s monotonicity formula ......Page 37
10 Cylindrical and gradient estimates ......Page 41
11 Mean curvature flow with surgeries ......Page 45
Bibliography ......Page 48
II: Geometric Flows, Isoperimetric Inequalities and Hyperbolic Geometry - Manuel Ritoré......Page 54
Preface ......Page 58
1.2.1 Area and volume ......Page 62
1.2.2 Variational formulas ......Page 64
1.2.3 The isoperimetric profile ......Page 65
1.2.4 Isoperimetric regions ......Page 66
1.3 The isoperimetric inequality in Euclidean space ......Page 67
1.3.1 A calibration argument ......Page 68
1.3.2 Schwarz symmetrization ......Page 70
1.3.3 Another proof of the isoperimetric inequality ......Page 75
2.1 Introduction ......Page 78
2.1.1 The Gauss–Bonnet Theorem ......Page 79
2.2.1 Basic results ......Page 81
2.2.2 The avoidance principle ......Page 83
2.3.1 Planes ......Page 84
2.3.2 Spheres ......Page 91
3.1 Introduction ......Page 94
3.2 H^k-flows and isoperimetric inequalities......Page 95
3.3.1 Euclidean spaces ......Page 98
3.3.2 3-dimensional Hadamard manifolds......Page 100
3.4 Singularities in the volume-preserving mean curvature flow ......Page 105
4.1 Introduction ......Page 108
4.2 Bounds on the Heegaard genus of a hyperbolic manifold ......Page 109
4.3 The isoperimetric profile for small volumes ......Page 111
Bibliography ......Page 114
