| Preface |
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v | |
| Introduction |
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1 | (17) |
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Principal results for area minimizing rectifiable currents |
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1 | (1) |
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Earlier related regularity results for area minimizing surfaces |
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2 | (1) |
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Geometric measure theory and the calculus of variations and the existence of area minimizing surfaces |
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2 | (1) |
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Why the results of I.1 are the best possible and the general necessity of singularities in area minimizing surfaces |
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3 | (1) |
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The spaces Q and Q and the mappings ρ and ξ |
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4 | (1) |
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Relations between currents and Q-valued functions |
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4 | (1) |
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Principal results for Q-valued functions minimizing Dirichlet's integral |
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5 | (1) |
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Relation between the nonparametric area integrand and Dirichlet's integrand |
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6 | (1) |
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The principal parameter of regularity; frequency of oscillation functions and their monotonicity |
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7 | (1) |
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Other applications of Q and Q (P,N)-valued functions |
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7 | (1) |
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Questions not settled and related questions |
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8 | (2) |
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Why this volume is so long |
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10 | (1) |
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10 | (1) |
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Summary of the principal themes by chapters |
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11 | (7) |
| Terminology |
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18 | (819) |
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General terminology (α(m), β(m), A(p, q), γ, γ0, ωT.1, graph, |||. |||, ±) |
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18 | (8) |
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Smoothing by convolution (φε, ψε, *, MF) |
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26 | (2) |
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Some coordinates, mean curvatures, and submanifolds (θ*, M*, NP*, Ms) |
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28 | (6) |
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Basic properties of Q and Q valued functions |
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34 | (57) |
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Definitions (Q, F, G, η, ξ, Q*, ρ, affine approximation) |
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34 | (6) |
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Theorem (the bilipschitzian correspondence ξ: Q → Q*) |
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40 | (4) |
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Theorem (the Lipschitz retraction ρ: RPQ→ Q*) |
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44 | (11) |
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Theorem (affine approximation and differentiation) |
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55 | (3) |
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Extensions of Lipschitz and piecewise affine Q valued functions, and integral currents associated with such functions |
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58 | (11) |
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Mappings of flat chains by Lipschitz Q valued functions |
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69 | (10) |
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Theorem (measurable representation of Lipschitz Q valued functions) |
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79 | (1) |
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Some estimates on tilted planes |
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79 | (4) |
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Theorem (some relations between several generalized ``excesses'') |
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83 | (2) |
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Proposition (continuous Q valued functions with interval domains) |
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85 | (6) |
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Properties of Dir minimizing Q valued functions and tangent cone stratification of mass minimizing rectifiable currents |
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91 | (134) |
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Definitions and terminology (Y2 (A, V), δY2 (A, V), Dir, dir) |
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91 | (1) |
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Theorem (existence of Dir minimizing functions) |
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92 | (1) |
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Elementary deformations of Q valued functions |
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93 | (2) |
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The squash deformations of a Q valued function |
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95 | (2) |
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The squeeze deformations of a Q valued function |
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97 | (3) |
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Theorem (the monotonicity of N and associated estimates on D and H) |
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100 | (7) |
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A comparison surface construction for Q valued functions on 2-dimensional domains |
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107 | (2) |
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A homogeneous comparison surface construction for general Q valued functions |
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109 | (1) |
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A standard modification procedure for members of Q |
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110 | (1) |
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The semi-retraction mappings Φ, Φ*: Q→Q |
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111 | (11) |
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Selection of constants for use in 2.12 |
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122 | (1) |
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Theorem (an estimate of Dir in terms of dir for Dir minimizing functions) |
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123 | (8) |
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Theorem (the Holder continuity of Dir minimizing functions) |
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131 | (4) |
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Theorem (the analyticity of graphs of Dir minimizing Q valued functions except for closed sets of Hausdorff codimension two) |
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135 | (8) |
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Theorem (various a priori estimates on Dir minimizing and nearly Dir minimizing Q valued functions obtained from compactness type arguments) |
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143 | (32) |
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Theorem (observations about and examples of Dir minimizing functions) |
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175 | (6) |
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Definitions (rectifiable currents associated with Dir minimizing functions whose graphs have finite area) |
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181 | (1) |
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Theorem (reference statement of approximation of a nearly flat mass minimizing rectifiable current by the graph of a Lipschitz Q valued function) |
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182 | (1) |
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Theorem (limits of blowups of Dir decreasing sequences of Q valued approximations to increasingly flat mass minimizing rectifiable currents are Dirminimizing) |
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183 | (17) |
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Theorem (a sliced holomorphic branched covering is a Dir minimizing Q valued function) |
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200 | (5) |
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Examples and remarks (phenomena illustrated by holomorphic varieties) |
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205 | (3) |
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Theorem (an estimate of extra area in a cylinder compared with area in a ball for nearly flat rectifiable currents with controlled area density ratios) |
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208 | (5) |
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Corollary (E-1/2 blowups of Q valued functions approximating nearly flat mass minimizing rectifiable currents having density Q at 0 are a priori bounded) |
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213 | (4) |
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Corollary (relation between points where a nearly flat mass minimizing rectifiable current has density Q and points where the E-1/2 blowup of the Q valued approximating function has multiplicity Q) |
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217 | (1) |
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Definitions and terminology for Theorem 2.26 (generalized stratification of a mass minimizing rectifiable current according to the maximum dimension of the linear subspaces of maximal density among the, possibly several, tangent cones) |
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217 | (2) |
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Theorem (Hausdorff dimension estimates associated with the generalized stratification of mass minimizing rectifiable currents) |
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219 | (4) |
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Corollary (Hausdorff dimension estimates associated with the generalized stratification of mass minimizing rectifiable currents; except for a set of Hausdorff codimension two, each interior point of a mass minimizing rectifiable current has at least one oriented tangent cone which is an integral multiple of an oriented linear subspace) |
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223 | (1) |
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Remark (Hausdorff dimension estimates associated with a generalized stratification of integral varifolds having bounded generalized mean curvatures) |
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224 | (1) |
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Approximation in mass of nearly flat rectifiable currents which are mass minimizing in manifolds by graphs of Lipschitz Q valued functions which can be weakly nearly Dir minimizing |
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225 | (106) |
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Assumptions for sections 3.1 through 3.9 (special points Y in the support of an integral varifold having summable mean curvatures) |
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225 | (1) |
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Construction of the sets Yi, Xj, Xjk (special subsets of Y) |
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226 | (1) |
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Construction of the index set J |
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227 | (1) |
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Some estimates (distance and diameter estimates) |
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227 | (1) |
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Theorem (mass estimates associated with density estimates at Y) |
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228 | (3) |
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Theorem (separation of stationary integral varifolds with zero excess) |
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231 | (1) |
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Some choices of constants and properties of surfaces having small first variation distributions |
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232 | (1) |
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Theorem (initial approximation of nearly flat nearly stationary integral varifolds by graphs of locally Lipschitz Q valued functions) |
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233 | (7) |
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Corollary (global mass approximation of nearly flat nearly stationary integral varifolds by graphs of Lipschitz Q valued functions) |
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240 | (1) |
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Corollary (relations between the accuracy of global mass approximations of nearly flat nearly stationary integral varifolds and the Lipschitz constants of the approximating Q valued functions) |
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240 | (1) |
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Corollary (further accuracy---Lipschitz constant approximation relations) |
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241 | (1) |
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Corollary (further accuracy---Lipschitz constant approximation relations in terms of the E excess) |
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241 | (1) |
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242 | (1) |
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Corollary (further accuracy---Lipschitz constant approximation relations in terms of excess E and radius S for V stationary on M associated with Ms) |
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242 | (1) |
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Corollary (final accuracy---Lipschitz approximation relations in terms of E and S for V stationary on M associated with Ms) |
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243 | (1) |
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Theorem (mass relations between Q(l) valued functions and the associated Q valued functions induced by Φ §) |
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244 | (3) |
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Corollary (a comparison surface estimate for a piece of a mass minimizing rectifiable current) |
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247 | (2) |
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Corollary (a further comparison surface estimate) |
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249 | (1) |
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Corollary (an estimate for the amount of mass excess above a small set in a nearly flat rectifiable current which is mass minimizing in M associated with Ms for small S) |
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249 | (1) |
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First comparison surface constructions |
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250 | (12) |
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Portions of comparison currents obtained from the deformation theorem construction with isoperimetric inequality mass estimates |
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262 | (2) |
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Theorem (an estimate on the amount of Dir associated with steep slopes in a Q valued function mass approximating a mass minimizing rectifiable current) |
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264 | (4) |
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Terminology for sections 3.24, 3.25, and 3.26 |
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268 | (1) |
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Theorem (an estimate on the amount of Dir associated with either steep slopes or a small domain in a Q valued function mass approximating a rectifiable current which is mass minimizing in M associated with Ms for small S) |
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269 | (9) |
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Second comparison surface construction |
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278 | (21) |
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Theorem (a mass excess estimate over a small set for a mass minimizing rectifiable current) |
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299 | (3) |
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Summary of constant selection and some new terminology |
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302 | (2) |
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First approximation theorem (a refined estimate for the mass approximation of nearly flat rectifiable currents which are mass minimizing in M by graphs of Lipschitz Q valued functions) |
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304 | (11) |
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Corollary (approximation estimates when M = Rm+l) |
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315 | (1) |
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Corollary (second approximation theorem with fewer hypotheses than 3.28) |
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316 | (2) |
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Lemma (an integral height estimate on approximating Q valued functions when the current approximated has a point with density Q) |
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318 | (2) |
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Theorem (mass minimizing rectifiable currents with small excess and relatively thick supports contain no interior points of density Q) |
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320 | (7) |
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Theorem (Q valued approximations to nearly flat rectifiable currents which are mass minimizing in M associated with Ms with S2 << are weakly nearly Dir minimizing) |
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327 | (4) |
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Approximation in mass of a nearly flat rectifiable current which is mass minimizing in a manifold by the image of a Lipschitz Q(Rm+n) valued function defined on a center manifold |
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331 | (352) |
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Proposition (a basic nonparametric area variational formula) |
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331 | (5) |
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Assumptions and terminology for sections 4.2 through 4.7 (a mass minimizing rectifiable current T is approximated by the graph of Lipschitz Q valued F with f = {r1 x {0}}# ˆ F and g = η ˆ f) |
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336 | (2) |
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Proposition (formulas for special first variations of T) |
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338 | (7) |
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Corollary (estimates on the Laplacian of ψε * g) |
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345 | (2) |
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Corollary (estimates on the Laplacian of ψε * g in weak form) |
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347 | (3) |
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Corollary (approximation of ψε * g) |
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350 | (3) |
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First center manifold approximation theorem (approximation of g by the sum of a harmonic function and a quadratic polynomial) |
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353 | (1) |
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Theorem on iterated spherical harmonic estimates |
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354 | (10) |
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First main approximation theorem (η ˆ F is approximatable by a cubic polynomial with errors depending on the number of homothetic expansions occuring earlier in an iteration process) |
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364 | (9) |
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Bases for Rm+n determined by linear mappings, orthonormalization, and orthogonal mappings determined by orthonormal bases |
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373 | (3) |
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Proposition (relations between linear subspaces and various associated mappings) |
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376 | (8) |
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Theorem on centerings of Q fold coverings in different coordinate systems |
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384 | (10) |
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Definitions and terminology (DIR, LIP) |
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394 | (1) |
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Theorem on Lipschitz constants and Dirichlet integrals of Q valued functions in different coordinate systems |
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395 | (5) |
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Proposition (first power integral inequalities on linear functions with rotated graphs) |
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400 | (2) |
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Assumptions, definitions and terminology for the remainder of Chapters 4 and 5 and some estimates |
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402 | (5) |
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Theorem (compatibility of the approximations of 4.9 with rotation and translations of coordinates and changes of scale) |
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407 | (21) |
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Some cubes in Rm, cubical subdivisions of [-1, 1]m, interpolation functions, admissible partitionings, and useful collections of cubes |
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428 | (4) |
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Center manifold differentiability criterion theorem |
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432 | (10) |
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Geodesic coordinates and associated estimates for surfaces which are nearly flat |
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442 | (9) |
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Definitions, terminology, and various estimates on several families of functions and associated sets |
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451 | (34) |
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Theorem on separation of mass minimizing integral currents |
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485 | (3) |
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Q(Rm+n) valued functions on a manifold N ((N, r) admissible functions f, interpolation procedures and estimates, evaluation of integrals) |
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488 | (19) |
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Further estimates on mass minimizing integral currents under conditions which permit approximation by Dir minimizing Q valued functions |
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507 | (22) |
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A refined procedure for approximating mass minimizing integral currents by Q(Rm+n) valued functions, especially when E is small |
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529 | (45) |
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A global procedure for approximating mass minimizing integral currents by Q(Rm+n) valued functions on a given manifold under various local approximation assumptions |
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574 | (35) |
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Some definitions and terminology for the center manifold construction (HT, EX) |
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609 | (12) |
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Some basic center manifold estimates and further estimates associated with HT and EX stopped cubes |
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621 | (27) |
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Construction of a cascade of functions for T4.27 |
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648 | (5) |
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Construction of the center manifolds |
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653 | (3) |
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Additional center manifold terminology and estimates |
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656 | (9) |
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Some further definitions and terminology and associated observations (J4.32, S4.32, s(j), R4.32(j), Z4.32(j)) |
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665 | (4) |
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Main approximation theorem for the approximation of a nearly flat mass minimizing integral current by a Q(Rm+n) valued function defined on a center manifold (f, N1, N2, Nr, δNr) |
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669 | (9) |
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Theorem (additional parameters associated with the main approximation theorem) |
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678 | (5) |
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Bounds on the frequency functions and the main interior regularity theorem |
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683 | (154) |
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Some definitions, terminology, and assumptions for sections 5.1 through 5.20 |
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683 | (2) |
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Definitions, terminology, observations, and estimates on the frequency functions (D, H, Dj, Hj, Nj, Bj, Cj) |
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685 | (3) |
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Theorem estimating Nj2 in terms of Nj1 and the Bj and Cj |
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688 | (2) |
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Some derivative computations and estimates on H'(r) and Cj(s) |
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690 | (12) |
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A general first variation formula for T |
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702 | (1) |
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Estimates from the squash deformations |
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703 | (7) |
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Estimates from the squeeze deformations |
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710 | (19) |
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Some definitions and terminology (index of terms appearing in the formulas of 5.6 and 5.7) |
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729 | (1) |
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Proposition (a formula for B*(s) - D'j(s)/Dj(s)) |
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730 | (2) |
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Some further terminology (L(r), HT(r), EX(r))L |
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732 | (1) |
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Some fundamental inequalities used to estimate B*j(s) - D'j(s)/Dj(s) |
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733 | (6) |
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Some conditional estimates on B*j(s) - D'j(s)/Dj(s) |
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739 | (16) |
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The basic inequality for the frequency functions Nj(s) |
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755 | (6) |
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Theorem (construction of a comparison surface of less mass in case the θm∞-2+ε measure of the set of points of density Q is relatively substantial) |
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761 | (28) |
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Theorem (regularity is implied by coincidence with the center manifold at any radius) |
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789 | (3) |
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Theorem (an a priori Nj(s) bound associated with radii near s4.32) |
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792 | (8) |
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Definitions and terminology (R*m, F*, mass minimizing tangent cones) |
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800 | (1) |
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Theorem (an a priori Nj(s) bound associated with nonflat tangent cones) |
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801 | (16) |
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Theorem (an a priori Nj(s) bound associated with radii near s4.32, Dir estimates associated with Nj(s), and a centering inequality) |
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817 | (2) |
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Selection and specification of various constants |
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819 | (3) |
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Theorem (at a density Q point admitting some flat tangent cone a mass minimizing rectifiable current is either regular there or the upper m-2+&epse; density there of &phis;m∞-2+&eps; restricted to its points of density Q is small) |
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822 | (10) |
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Main interior regularity theorem |
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832 | (3) |
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Boundary regularity and connectedness of mass minimizing integral currents (W.K. Allard) |
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835 | (1) |
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Almost sure uniqueness of mass minimizing integral currents in Rm+n (F. Morgan) |
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836 | (1) |
| Appendices |
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837 | (116) |
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A.1 Functions whose distribution first derivatives are square summable, harmonic functions, spherical harmonics, and inequalities relating Dirichlet integrals and zeros of functions |
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837 | (39) |
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A.1.1 Definitions and terminology (functions with square summable first derivatives) |
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837 | (2) |
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A.1.2 Theorem (general facts about functions with square summable first derivatives) |
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839 | (5) |
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A.1.3 Definitions and terminology (harmonic functions and spherical harmonics) |
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844 | (1) |
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A.1.4 Theorem (general facts about harmonic functions and spherical harmonics) |
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845 | (3) |
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A.1.5 Some terminology and elementary facts (including bilipschitzian parametrizations of hemispheres) |
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848 | (3) |
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A.1.6 Theorem (inequalities relating Dirichlet integrals and zeros of functions) |
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851 | (10) |
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861 | (1) |
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A.1.8 Theorem (relations between Dirichlet integrals and Holder continuity) |
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861 | (5) |
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A.1.9 Proposition (supremum and integral relations for polynomial functions) |
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866 | (1) |
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A.1.10 Theorem (inequalities relating Dirichlet integrals and &phis;m∞-2+&eps; estimates on the zeros of functions) |
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867 | (9) |
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A.2 Convolution inequalities and interpolation estimates |
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876 | (9) |
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A.2.1 Theorem (convolution and square integrals) |
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876 | (5) |
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A.2.2 Proposition (interpolation and square integrals) |
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881 | (1) |
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A.2.3 Proposition (convolution, square integrals, and small sets) |
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882 | (2) |
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A.2.4 Proposition (a singular integral estimate) |
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884 | (1) |
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A.3 Estimates relating derivatives of various functions including functions with rotated graphs |
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885 | (31) |
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A.3.1 Definitions, terminology, and some observations (relations involving various derivatives of functions) |
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885 | (2) |
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A.3.2 Corollary (some inequalities for derivatives of functions) |
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887 | (1) |
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A.3.3 Implicit function theorem type estimates for functions with rotated graphs |
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888 | (6) |
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A.3.4 Lemma (G. Hunt) (orthogonal interchange of planes) |
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894 | (1) |
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A.3.5 Corollary to Theorem A.3.3 |
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895 | (1) |
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A.3.6 First estimates on nearest point retractions |
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896 | (10) |
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A.3.7 Second estimates on nearest point retractions |
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906 | (5) |
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A.3.8 Proposition (estimates on first derivatives in curvilinear coordinate systems) |
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911 | (3) |
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A.3.9 Proposition (first estimates on linear functions with rotated graphs) |
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914 | (1) |
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A.3.10 Estimates on linear functions in terms of rotations of their graphs |
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915 | (1) |
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A.4 The nonparametric area integrand, mean curvatures, and Laplacians and further nearest point retraction estimates |
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916 | (9) |
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A.4.1 Proposition (the nonparametric area integrand) |
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916 | (1) |
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A.4.2 Proposition on mean curvature and Laplacians |
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917 | (2) |
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A.4.3 Third estimates on nearest point retractions |
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919 | (3) |
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A.4.4 Retractions onto a manifold preserving submanifold normals |
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922 | (3) |
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A.5 Some differential geometry of submanifolds of Euclidean space |
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925 | (15) |
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A.5.1 Some differential geometry of manifolds which are graphs of functions |
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925 | (11) |
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A.5.2 The differentiability of the exponential mapping of an embedded submanifold of Euclidean space (W.K. Allard) |
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936 | (4) |
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A.6 Varifolds and level sets and gradients of functions |
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940 | (7) |
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A.6.1 Varifolds determined by level sets of functions |
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940 | (6) |
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A.6.2 Estimates on gradients of functions |
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946 | (1) |
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A.7 Some measure and derivative estimates |
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|
947 | (6) |
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A.7.1 An inequality for size ∞ approximating measures |
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947 | (3) |
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A.7.2 Proposition (a derivative estimate for functions with deformed images |
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950 | (3) |
| References |
|
953 | |
Other versions by this Author
