Extrinsic Geometry of Convex SurfacesAmerican Mathematical Soc., 1973 - 669 pages |
Table des matières
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The auxiliary surface and its plane sections | 153 |
Monotypy of closed convex surfaces | 167 |
Monotypy of convex surfaces with boundary Maximum principle | 175 |
Monotypy of unbounded convex surfaces with total curva ture 2π | 182 |
Monotypy of unbounded convex surfaces with total curvature less than 2 | 189 |
INFINITESIMAL BENDINGS OF CONVEX SURFACES | 201 |
Bending fields of general convex surfaces | 202 |
Fundamental Lemma on bending fields of convex surfaces | 211 |
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Gluing theorem Other existence theorems | 33 |
Convex surfaces in spaces of constant curvature | 35 |
Manifolds of bounded curvature | 38 |
REGULARITY OF CONVEX SURFACES WITH REGULAR METRIC | 44 |
Extrinsic properties of geodesic lines on a convex surface | 45 |
Special resolution of the radiusvector in the neighborhood of an arbitrary initial point | 53 |
Convex surfaces of bounded specific curvature | 57 |
Construction of a convex surface with infinite upper curvature on a given set of points | 64 |
Auxiliary surface and some of its properties | 69 |
Monotypy of convex caps with regular metric | 78 |
Intrinsic estimates for some geometric quantities along the boundary of an analytic cap | 83 |
Estimate for normal curvatures at interior points of a regular convex cap | 89 |
Existence of an analytic convex cap realizing a given analytic metric | 95 |
Regularity of convex surfaces with regular metric | 101 |
Estimates for the derivatives of the solution of a secondorder elliptic partial differential equation | 110 |
MONOTYPY OF CONVEX SURFACES | 119 |
Curves with i g c of bounded variation | 120 |
On convergence of isometric convex surfaces | 131 |
Mixture of isometric surfaces | 137 |
On isometric convex surfaces in canonical position | 144 |
Bending field of a convex surface with prescribed vertical component along the boundary | 219 |
Special approximation of a bending field of a general convex surface | 233 |
Estimation of certain integrals | 239 |
Proof of the Fundamental Lemma | 252 |
Rigidity of convex surfaces | 260 |
Applications of the rigidity theorems | 266 |
Convex Surfaces IN SPACES OF CONSTANT CURVATURE | 270 |
Elliptic space | 271 |
Convex bodies and convex surfaces in elliptic space | 282 |
Mappings of congruent figures | 291 |
Isometric surfaces | 300 |
Infinitesimal bendings of surfaces in elliptic space | 308 |
Monotypy of general convex surfaces in elliptic space | 314 |
Regularity of convex surfaces with regular metric | 322 |
Regularity of convex surfaces with regular metric in a hyper | 331 |
bolic space | 344 |
CONVEX SURFACES IN RIEMANNIAN SPACE | 345 |
CONVEX SURFACES WITH GIVEN SPHERICAL IMAGE | 435 |
GEOMETRIC THEORY OF MONGEAMPERE EQUATIONS | 503 |
SURFACES OF BOUNDED EXTRINSIC CURVATURE | 571 |
APPENDIX UNSOLVED PROBLEMS | 652 |
BIBLIOGRAPHY | 660 |
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Expressions et termes fréquents
a₁ analytic angle arbitrarily small arbitrary point assume bending field Borel set boundary bounded variation closed convex surface coefficients component condition congruent contains converge convex polyhedron coordinates corresponding points curve defined denote differentiable dihedral angle direction disk distance domain G dx dy elliptic space endpoints equation estimate euclidean space exists exterior normals extrinsic curvature F and F F₁ and F₂ follows function G₁ G₂ Gauss curvature geodesic curvature Hence Hölder condition homeomorphic infinitesimal bending integral intrinsic metric isometric Lemma Let F Let G line element manifold mapping maximum mean curvature neighborhood normal curvature P₁ parallel point on F point Xo polygon polyhedra polyhedron prove R₁ R₂ satisfy second derivatives semitangent sequence solution specific curvature spherical image straight lines straight-line segment subarcs sufficiently close sufficiently small supporting plane surface F tangent plane theorem triangle vertices X₁ xy-plane y₁ z-axis zero ди