Chapter1 Preliminary 1
1.1 Notations 1
1.2 Some useful properties 4
1.3 Graphical submanifolds 7
1.4 Interior Holder estimates 10
Chapter2 Hβ-flow for h-convex Hypersurfaces in H n k＋1 13
2.1 Introduction and main results 13
2.2 Short-time existence and evolution equations 16
2.3 Preserving h-convex 22
2.4 Long-time existence 27
2.5 Contraction to a point 32
Chapter3 Hβ-flow for Pinched Hypersurfaces in H n k＋1 34
3.1 Introduction 34
3.2 Preserving pinching of curvature 38
3.3 The pinching estimate 47
3.4 The normalized equations 51
3.5 Convergence to a unit geodesic sphere 55
3.6 Exponential convergence 63
Chapter4 Volume-preserving Hβ m-flow in H n k＋1 66
4.1 Introduction 66
4.2 Short-time existence and evolution equations 71
4.3 Preserving pinching 78
4.4 Upper bound on F 84
4.5 Long-time existence 93
4.6 Exponential convergence to a geodesic sphere 103
Chapter5 φ（H）-flow in Rn＋1 107
5.1 Introduction and main results 107
5.2 Short-time existence and evolution equations 111
5.3 Long-time existence 115
5.4 Preserving convexity 118
Chapter6 φ（H）-flow in H n k＋1 125
6.1 Introduction and main results 125
6.2 Short-time existence and evolution equations 128
6.3 Preserving h-convex 133
6.4 Long-time existence 139
6.5 Contraction to a point 145
Chapter7 Mixed Volume Preserving Fβ-flow in Rn＋1 146
7.1 Introduction and main results 146
7.2 Short-time existence and evolution equations 151
7.3 Preserving pinching 155
7.4 Upper bound on φ（F） 164
7.5 Long-time existence 169
7.6 Exponential convergence to the sphere 175
Chapter8 Forced MCF of Submanifolds in Rn＋p 184
8.1 Introduction 184
8.2 Evolution equations 189
8.3 Relationship with the mean curvature flow 191
8.4 Asymptotic behavior of submanifolds 194
Bibliography 201
编后记 209