This book deals with the new class of one-dimensional variational problems the problems with branching solutions. Instead of extreme curves (mappings of a segment to a manifold) we investigate extreme networks, which are mappings of graphs (one-dimensional cell complexes) to a manifold. Various applications of the approach are presented, such as several generalizations of the famous Steiner problem of finding the shortest network spanning given points of the plane.
Preface vii Preliminary Results 1(20) Graphs 1(6) Topological and framed graphs, their equivalence 2(1) Operations on graphs 3(2) Boundary of graph, local graph 5(1) Smooth structure on topological graph 6(1) Parametric networks 7(2) Main definitions 7(1) Classes of networks smoothness 8(1) Network-traces 9(4) Networks-traces and their canonical representatives 10(3) Stating of variational problem 13(8) Construction of edge functionals 13(1) Classical variational functional 14(2) Classical functional of Bolza 16(1) Construction of edge functionals for networks with fixed topology 17(4) Networks Extremality Criteria 21(18) Local structure of extreme parametric networks 22(5) Local structure of extreme networks-traces 27(12) Smooth Lagrangians 27(2) Quasiregular Lagrangians 29(10) Linear Networks in RN 39(38) Mutually parallel linear networks with a given boundary 40(5) Geometry of planar linear trees 45(2) Twisting number of planar linear tree 46(1) Main theorem 47(1) On the proof of Theorem 3.2 47(30) Planar polygonal lines I: the case of general position 48(1) Twisting and turning 49(2) A pair of polygonal lines in general position 51(4) Caps 55(1) Planar polygonal lines II: the general case 56(8) Twisting number of a planar linear tree 64(1) Proper linear trees 64(1) Quasi-geodesics 65(1) Caps 66(2) Proof of Theorem 3.2 68(1) The case p = q 69(3) The case p < q 72(5) Extremals of Length Type Functionals: The Case of Parametric Networks 77(22) Parametric networks extreme with respect to Riemannian length functional 77(6) Local structure of weighted extreme parametric networks 83(2) Polyhedron of extreme weighted networks in space, having some given type and boundary 85(9) Structure of the set of extreme weighted networks 87(5) Immersed extreme weighted Steiner networks in the plane 92(2) Global structure of planar extreme weighted trees 94(1) Geometry of planar embedded extreme weighted binary trees 95(4) Twisting number of embedded planar weighted binary trees 95(4) Extremals of the Length Functional: The Case of Networks-Traces 99(154) Minimal networks on Euclidean plane 100(60) Correspondence between planar binary trees and diagonal triangulations 101(2) Structural elements of diagonal triangulations 103(1) Tiling realization of binary trees whose twisting number is at most five 104(2) Tilings and their properties 106(1) Decompositions of a tiling into a skeleton and growths 106(2) Decomposition of a tree skeleton into branching points and linear parts 108(1) Axis 108(2) Structural elements of skeletons from WP5 110(1) Branching points of a tree skeletons 110(1) Linear parts 111(1) Operations of reduction and antireduction 112(1) Cutting and pasting 112(1) Reduction of planar binary tree 113(1) Reduction of tilings from WP5 114(1) Profiles and their properties 115(1) Definition of profiles 115(1) Relations between the twisting numbers of profiles and the twisting number of the tiling 116(1) Terminological remark 116(1) Classification Theorem for skeletons from WP5 116(2) Directions of ending linear parts of skeletons from WP5 118(1) Codes of non-degenerate 6-skeletons 118(1) Location of the growths of tilings from WP5 on their skeletons 119(1) Theorem of realization 120(1) Minimal binary trees with regular boundary 121(1) Complete classification of local minimal skeletons with regular boundaries 121(2) Some properties of tilings from WP5 having RM-realization 123(1) Growths and linear parts of minimal networks with convex boundaries 124(1) Ending growths 125(1) Ending vertices 126(1) Geometry of ending linear parts 127(1) The length of the tongue: the ending snake has growths 128(3) The length of the tongue: the ending snake is not attached with growths, but the ending linear part has a break 131(1) Characteristic half-planes 132(1) Estimations on the length of the tongue 133(2) The length of the tail: the ending linear part has a break, but the ending snake is attached with growths 135(4) Mutual location of ending linear parts 139(6) Quasiregular polygons which cannot be spanned by minimal binary trees 145(2) Non-degenerate minimal networks with convex boundary. Cyclical case 147(1) Planar minimal realization of non-degenerate graphs and Steiner networks 148(1) Fundamental cycles of non-degenerate minimal networks with convex boundaries. Trivial networks 149(1) Twisting number of trivial networks 149(1) Tiling realization of a trivial network whose twisting number does not exceed five 150(1) Description of tilings of general form 151(1) Skeletons and growths 151(1) Tiling hulls and kernels 151(1) Branching points 152(1) Linear parts 153(1) Structural elements 154(1) Macroelements and ends 155(1) Skeletons from P5 156(1) Structural elements 156(1) Directions of the ending macroelements of a skeleton from P5 156(2) Codes of skeletons from P5 158(1) Polygrowths 158(1) Location of growths on skeletons of tilings from P5 159(1) Final remark 160(1) Closed minimal networks on closed surfaces of constant curvature 160(32) Minimal networks on surfaces of constant positive curvature 162(1) Closed minimal networks on S2 162(1) Closed minimal networks on P2 163(1) Classification of closed minimal networks on flat tori 164(1) Description of flat metrics on a two-dimensional torus 165(1) Flat tori translations groups, lattices, and universal coverings 166(1) Net geodesics 167(3) The type of a network 170(4) Characteristic triangle 174(1) Classification theorems 175(5) Classification of closed minimal networks on flat Klein bottles 180(1) Description of flat metrics on a Klein bottle 180(1) The universal covering of a flat Klein bottle 181(1) The covering of a flat Klein bottle by a flat torus 182(2) Regular networks 184(1) Classification theorems 185(2) Closed networks on two-dimensional surfaces of negative curvature 187(1) Metric restrictions on the structure of closed networks 187(1) Examples of closed minimal networks on surfaces of negative curvature 188(2) Enumeration of closed local minimal networks on surfaces of constant negative curvature up to topological equivalence 190(2) Closed local minimal networks on surfaces of polyhedra 192(26) General properties of local minimal networks on polyhedra 192(2) Developments 194(2) Local geodesics 196(2) Local structure of minimal networks on polyhedra 198(1) The Gauss-Bonnet theorem for polyhedra 199(1) Metric and topological restrictions on the structure of closed minimal networks 199(1) Existence of closed local minimal networks on convex polyhedra 200(1) T. V. Pavlyukevich Cells of networks on convex polyhedra 201(2) The case of regular polyhedra 203(2) Classification of closed minimal networks on regular tetrahedron 205(1) The branching covering of tetrahedra by the plane and by flat tori 205(3) Regular networks 208(2) Classification theorems 210(1) ``Reproduction algorithm for closed local minimal networks on polyhedra 211(5) Closed geodesics on the cube 216(2) Morse indices of local minimal networks 218(14) M. V. Pronin Introduction 218(1) Index form 218(6) Minimal networks on non-positive curvature manifolds 224(2) Minimal networks on the sphere 226(2) Index Theorem 228(4) Morse theory for planar linear networks 232(21) G. A. Karpunin Introduction 232(2) Morse theory for simplicial complexes 234(1) Morse theory for special metric spaces 235(5) Morse theory for minimal networks 240(7) Some applications 247(6) Extremals of Functionals Generated by Norms 253(60) Norms of general form 256(24) Local minimal and extreme networks 256(1) The first variation of straight segment length in a normalized space 257(6) Structure of extreme curves 263(1) Local structure of extreme linear parametric networks 264(7) Networks-traces extremality criterion 271(9) Stability of extreme binary trees under deformations of the boundary 280(3) Planar norms with strictly convex smooth circles 283(14) Extremality criterion for networks-traces 283(5) Geometry of extreme networks-traces 288(1) Relations with Euclidean norm case 288(1) Geometry of boundary sets and twisting number 289(2) Set of extreme networks having a fixed type 291(1) Surfaces 292(4) Norms with ellipsoidal circles 296(1) Manhattan local minimal and extreme networks 297(16) General properties 298(1) Extreme networks and linear networks 299(1) Extreme networks on the Manhattan plane 300(13) Appendix: Some Unsolved Problems 313(10) Bibliography 323(8) Index 331
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