De Lellis C. Rectifiable sets, densities, and tangent measures. - Zürich: EMS, 2008. - vi, 124 p. - (Zürich lectures in advanced mathematics). - ISBN 978-3-03719-044-9  Место хранения:   013 | Институт математики СО РАН | Новосибирск | Библиотека

Оглавление / Contents

1. Introduction ................................................. 1 2. Notation and preliminaries ................................... 4 2.1. General notation and measures ........................... 4 2.2. Weak* convergence of measures ........................... 5 2.3. Covering theorems and differentiation of measures ....... 9 2.4. Hausdorff measures ..................................... 10 2.5. Lipschitz functions .................................... 11 2.6. The Stone-Weierstrass Theorem .......................... 12 3. Marstrand's Theorem and tangent measures .................... 13 3.1. Tangent measures and Proposition 3.4 ................... 16 3.2. Lemma 3.7 and some easy remarks ........................ 21 3.3. Proof of Lemma 3.8 ..................................... 22 3.4. Proof of Corollary 3.9 ................................. 25 4. Rectifiability .............................................. 27 4.1. The Area Formula I: Preliminary lemmas ................. 29 4.2. The Area Formula II .................................... 32 4.3. The Geometric Lemma and the Rectifiabihty Criterion .... 35 4.4. Proof of Theorem 4.8 ................................... 37 5. The Marstrand-Mattila Rectifiabihty Criterion ............... 40 5.1. Preliminaries: Purely unrectifiable sets and projections ............................................ 42 5.2. The proof of the Marstrand-Mattila rectifiabihty criterion .............................................. 47 5.3. Proof of Theorem 5.1 ................................... 53 6. An overview of Preiss' proof ................................ 56 6.1. The cone {x42 = x12 + x22 + x32} ......................... 59 6.2. Part A of Preiss' strategy ............................. 63 6.3. Part В of Preiss' strategy: Three main steps ........... 65 6.4. From the three main steps to the proof of Theorem 6.10 ........................................... 66 7. Moments and uniqueness of the tangent measure at infinity ... 70 7.1. From Proposition 7.7 to the uniqueness of the tangent measure at infinity ............................ 74 7.2. Elementary bounds on bk,s and the expansion (7.5) ...... 76 7.3. Proof of Proposition 7.7 ............................... 79 8. Flat versus curved at infinity .............................. 85 8.1. The tangent measure at infinity is a cone .............. 88 8.2. Conical uniform measures ............................... 88 8.3. Proof of Proposition 8.5 ............................... 91 9. Flatness at infinity implies flatness ....................... 95 9.1. Proofs of (ii) and (iv) ................................ 99 9.2. An integral formula for tr(b2(2)LV) .................... 100 9.3. An intermediate inequality ............................ 103 9.4. Proof of (9.7) and conclusion ......................... 106 10.Open problems .............................................. 110 Appendix A. Proof of Theorem 3.11 ............................. 117 Appendix B. Gaussian integrals ................................ 122 Bibliography .................................................. 125 Index ......................................................... 127