Topics in modern regularity theory

Cover......Page 1
Title Page \r......Page 4
Copyright Page \r......Page 5
Table of Contents \r......Page 6
Introduction......Page 9
The Willmore functional......Page 12
1.1 Introduction......Page 13
1.2 Conformal invariance......Page 16
1.3 The Euler Lagrange equation......Page 20
References......Page 26
2.1 Monotonicity formula......Page 27
References......Page 34
3.1 Introduction......Page 36
3.2 Estimation of the maximal existence time......Page 38
3.3 Blow up......Page 46
3.4 Convergence to a sphere......Page 48
3.A Blow up radius......Page 56
References......Page 57
4.1 Conformal parametrization......Page 59
4.B Auxiliary results......Page 69
4.C Huber’s theorem for simply connected surfaces in codimension one......Page 74
References......Page 77
5.1 C1,α-regularity for point singularities......Page 79
5.2 Power-decay......Page 84
5.3 Higher regularity for point singularities......Page 85
5.4 Applications......Page 91
References......Page 94
6.1 Non-compact invariance group......Page 95
6.2 Selection of the Möbius transformation......Page 98
6.3 Proof of the compactness theorem......Page 100
References......Page 106
7.1 ConstrainedWillmore surfaces......Page 107
7.2 Existence of constrained minimizers......Page 109
7.A The space STT......Page 115
References......Page 116
8.1 The large genus limit......Page 118
References......Page 121
1 Introduction......Page 123
2 Conformally invariant coercive Lagrangians with quadratic growth, in dimension 2......Page 127
3.1 Constant mean curvature equation (CMC)......Page 137
3.2 Harmonic maps with values in the sphere Sn......Page 144
3.3 H´elein’s moving frames method and the regularity of harmonic maps mapping into a manifold......Page 145
4 The regularity of critical points to quadratic, coercive and conformally invariant Lagrangians in two dimension: the proof of Hildebrandt’s conjecture......Page 151
5 A PDE version of the constant variation method for Schrödinger Systems with anti-symmetric potentials......Page 165
5.1 Concluding remarks......Page 170
References......Page 171
1 Introduction......Page 174
2 Statement of the existence and regularity results......Page 176
3 The interior ball condition......Page 180
4 A decay estimate for the gradient of u......Page 187
5 Proof of Theorem 2.5......Page 205
References......Page 208
Back matter......Page 210