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| Aubin T. — Nonlinear Analysis on Manifolds: Monge-Ampere Equations |
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| Предметный указатель |
 spaces 78
A priori estimate 146 150—153 157 160—168 188
Absolutely continuous 77
Adjoint operator 27 75
Admissible function 141
Almost everywhere 76
Arc length 5
Ascoli’s Theorem 74
Atlas, classe  1
Atlas, definition 1
Atlas, equivalent 1
Banach space 70
Banach — Steinhaus theorem 73
Banach’s theorem 74
Berger’s problem 119
Best constants 43 58 60 65 68
Best constants in Sobolev spaces 41 100 129
Best constants, definition 99
Betti numbers 29 30 142
Bianchi’s identities 6
Boundary 25
Bracket 2
Calabi’s conjecture 142 150
Calderon — Zygmund inequality 90
Cauchy’s Theorem 72
Chart 1
Chern class 140
Christoffel symbols 3
Clarkson’s inequalities 89
Closed see Differentiable
Co-differential 27
Cohomology class, definition 140
Cohomology class, positive definite 142
Compact mapping 74
Complex manifold 139
Conjugate point 17 18
Connection 3
Connection, Riemannian 6
Continuity method 115 145 147 158 159
Convex ball 11 19
Convex function 159 174
Convex function, inequality for 178
Convex set 157
Convolution product 89
coordinates 1
Coordinates, geodesic 9
Coordinates, normal 7
Covariant derivative 4
Covering manifold 24
Critical point 40
Curvature, definition 3
Curvature, Ricci 7 140
Curvature, scalar 7
Curvature, sectional 7
Curvature, tensor 4 6 140
Cut locus 14 18
Degenerate 40
Density problem 33—35
Diffeomorphism 72
Differential operator 83
Differential operator, elliptic 83 84
Differential operator, leading part of 83
Differential operator, linear 83
Differential operator, uniformly elliptic 83
Differentiate 71
Differentiate form 2
Differentiate form, closed 28
Differentiate form, co-exact 28
Differentiate form, coclosed 28
Differentiate form, exact 28
Differentiate form, harmonic 28
Differentiate form, homologous 28
Differentiate, manifold 1
Differentiate, mapping 1
Dirichlet problem 157 158 182
Distance 5
Divergence form 85
Dual space 70
Eigenfunction 101 102
Eigenvalue, definition 75 102
Eigenvalue, of the Laplacian 31 101 102 134
Eigenvector 75
Einstein metric 7
Einstein — Kahler metric 143 146 150
Elliptic see Differential operator and linear
Equicontinuous 74
Euler equation 42 102 103 120 122 123
Euler — Poincare characteristic X 29 120
Exact 28 see
Exceptional case 63 68
Exponential mapping 9
Exterior, differential p-form 2
Exterior, differentiation 3
Fatou’s theorem 77
First Chern class 140 143
First fundamental form 139
Fixed point method 115
FORM see Differentiable
Formal adjoint 84
Fredholm nonlinear theorem 123
Fredholm theorem 75
Fubini’s Theorem 78
Functional 105 120 123 174 183
Gauss Bonnet theorem 120
Geodesic 8 11
Geodesic coordinates 9
Global scalar product 28
Green’s formula 107
Green’s function, definition 107
Green’s function, existence properties 108 112
Hahn — Banach theorem 73
Harmonic see Differentiable Spherical
Hermitian metric 139
Hilbert space 70
Hodge decomposition theorem 29
Hoelder’s inequality 88
Homeomorphism 72
Homologous 28
Hopf — Rinow theorem 13
Imbedding 1
Immersion 1
Implicite function theorem 72
Improvement of the best constants 57 68
Index form 18
Index inequality 19
Inequality, Clarkson’s 89
Inequality, Hoelder’s 88
Inequality, index 19
Inequality, interpolation 93
| Inequality, isoperimetric 40
Inequality, Minkowski 177
Inequality, optimal 50
Inequality, plurisubharmonic function 184 185
Injectivity radius 15
Inner product 70
Integrable 76
Integration over Riemannian manifolds 23 24 29
Interpolation inequalities 93
Inverse function theorem 72
Isoperimetric inequality 40
Jacobi field 17
Jacobian matrix 71
Kaehler, Einstein metric 143 146 150 see
Kaehler, manifold 139 140
Kaehler, metric 139 141
Kondrakov Theorem 53 55
Korn — Lichtenstein theorem 90
Kronecker’s symbol 4
Laplacian 27 106
Leading part 83
Lebesgue integral 75 77
Lebesgue measure 77
Lebesgue theorems 76 77
Length, arc 5
Length, minimizing 11
Length, second variation 15
Leray — Schauder theorem 74
Linear elliptic equation 101 113
Linear elliptic equation, existence of solution for 104 105 113
Linear mapping 70
Linear tangent mapping 2
Lower solution 97 see
Lower solution, method of 115 138 139 154 169
Manifold with boundary 25
Manifold, complete 13 14 16 45
Manifold, complex 139
Manifold, definition 1
Manifold, differentiable 1
Manifold, Riemannian 4
Mapping, differentiable 1
Mapping, rank 1
Maximum principle 96 97
Maximum principle, generalized 98
Mean value theorem 71
Measurable 76
Measure 76
METHODS 115
Metric, Einstein 7
Metric, Hermitian 139
Metric, Kaehler 139
Metric, Riemannian 4
Metric, space 5
Metric, tensor, estimates on 20
Minimizing curve 11
Monge — Ampere equation 144 157
Moser’s theorems 65 122
Myers’ theorem 16
Neumann problem 88
Nirenberg’s problem 122
Norm 70
Normal coordinates 7
Normed space 70
Open mapping theorem 73
Optimal inequalities 50
Orientable 23
Orientation 23
Parallel displacement 8
Parallel vector field 8
Parametrix for the Laplacian 106
Plurisubharmonic function 182
Plurisubharmonic function, inequality 184 185
Precompact 74
Projective space, complex 147 150
Projective space, real 23 122
Rademacker’s theorem 77
Radon measure 75 169 183
Radon’s theorem 81
Rank 71
Reflexive 70 81
Regularity up to the boundary 87
Regularity, interior 85 86
Regularization 80
Ricci curvature 7
Ricci form 140
Ricci tensor 7
Riemannian connection 6
Riemannian manifold 4
Riemannian metric 4
Scalar curvature 7
Schauder fixed point theorem 74
Schauder interior estimates 88
Second variation of the length integral 15
Sectional curvature 7
Semi-norm 73
Sense of distribution 84
Sobolev imbedding theorem 35 44 45 50
Sobolev imbedding theorem, proof 37—39 46 47 49 51
Sobolev lemma 37
Sobolev space 32
Spectrum 31
Spherical harmonic 133
Steepest descent 115
Subsolution 97 see
Supersolution 97 see Lower
Tangent space 2
Tangent vector 2
Taylor’s Formula 72
Tensor field 2
Topological methods 115
Torsion 3
Trace 69
Tubular neighborhood 15
Uniformly, convex 81
Uniformly, locally finite cover 48
Upper solution 97 see
Variational problem 42 101 105 115 116 119 120 126 179 185
Vector field 2
Volume element, oriented 26
Volume element, Riemannian 30
Weak, convergence 74
Weak, derivative 81 84
Weak, solution 84 85
Weakly sequentially compact 74
Weyl tensor 132
Whitney’s theorem 4
Yamabe’s equation 115
Yamabe’s problem 125
Yamabe’s theorem 126
Yang — Mills field vii
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