1 Integral Formulae ............................................ 1
1.1 Introduction ............................................ 1
1.2 Preliminaries ........................................... 4
1.2.1 Hypersurfaces of Riemannian Manifolds ............ 4
1.2.2 Invariants of the Weingarten Operator ............ 5
1.2.3 Leaf-Wise Divergence of Operators Ak and Tr(A) ... 8
1.2.4 Leaf-Wise Divergence of Vector Fields h(A)Z and
Tr(A)Z .......................................... 11
1.3 Integral Formulae for Codimension-One Foliations ....... 14
1.3.1 New Integral Formulae ........................... 14
1.3.2 Some Consequences of Integral Formulae .......... 15
1.3.3 Foliations Whose Leaves Have Constant σ2 ........ 17
2 Variational Formulae ........................................ 19
2.1 Introduction ........................................... 19
2.2 Auxiliary Results ...................................... 20
2.2.1 Biregular Foliated Coordinates .................. 20
2.2.2 Foliations with a Time-Dependent Metric ......... 24
2.2.3 A Differential Operator ......................... 26
2.3 Variational Formulae for Codimension-One Foliations .... 31
2.3.1 Variations of Extrinsic Geometric Quantities .... 31
2.3.2 Variations of General Functionals ............... 35
2.3.3 Variations of Particular Functionals ............ 43
2.4 Applications and Examples .............................. 47
2.4.1 Variational Formulae for Umbilical Foliations ... 47
2.4.2 The Energy and Bending of the Unit Normal
Vector Field .................................... 49
3 Extrinsic Geometric Flows ................................... 53
3.1 Introduction ........................................... 53
3.2 The Systems of PDEs Related to EGFs .................... 55
3.3 Auxiliary Results ...................................... 58
3.3.1 Diffeomorphism Invariance of EGFs ............... 58
3.3.2 Quasi-Linear PDEs ............................... 59
3.3.3 Generalized Companion Matrices .................. 60
3.4 Existence and Uniqueness Results (Main Theorems) ....... 67
3.5 The General Case ....................................... 69
3.5.1 Searching for Power Sums ........................ 70
3.5.2 Local Existence of Metrics (Proofs of the Main
Theorems) ....................................... 72
3.5.3 Proofs of the Corollaries ....................... 75
3.6 Global Existence of EGFs (Time Estimation) ............. 77
3.7 Variational Formulae for EGFs .......................... 80
3.7.1 The Normalized EGFs ............................. 81
3.7.2 First Derivatives of Functionals ................ 83
3.8 Extrinsic Geometric Solitons ........................... 85
3.8.1 Introducing EGS ................................. 86
3.8.2 Canonical Form of EGS ........................... 90
3.8.3 Umbilical EGS ................................... 91
3.9 Applications and Examples .............................. 96
3.9.1 Extrinsic Ricci Flow ............................ 96
3.9.2 Extrinsic Ricci Solitons ........................ 99
3.9.3 EGS on Foliated Surfaces ....................... 100
3.9.4 EGS on Hypersurfaces of Revolution ............. 106
References ............................................ 109
Index ......................................................... 113
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