**Contemporary Mathematics**

Volume: 599;
2013;
254 pp;
Softcover

MSC: Primary 53; 58; 32; 44; 52; 35; 49; 76;
Secondary 47

Print ISBN: 978-0-8218-9149-0

Product Code: CONM/599

List Price: $89.00

AMS Member Price: $71.20

MAA Member Price: $80.10

**Electronic ISBN: 978-1-4704-1081-0
Product Code: CONM/599.E**

List Price: $89.00

AMS Member Price: $71.20

MAA Member Price: $80.10

# Geometric Analysis, Mathematical Relativity, and Nonlinear Partial Differential Equations

Share this page *Edited by *
*Mohammad Ghomi; Junfang Li; John McCuan; Vladimir Oliker; Fernando Schwartz; Gilbert Weinstein*

This volume presents the proceedings of the Southeast Geometry Seminar
for the meetings that took place bi-annually between the fall of 2009
and the fall of 2011, at Emory University, Georgia Institute of
Technology, University of Alabama Birmingham, and the University of
Tennessee. Talks at the seminar are devoted to various aspects of
geometric analysis and related fields, in particular, nonlinear
partial differential equations, general relativity, and geometric
topology.

Articles in this volume cover the following topics: a new set of
axioms for General Relativity, CR manifolds, the Mañé
Conjecture, minimal surfaces, maximal measures, pendant drops, the
Funk-Radon-Helgason method, ADM-mass and capacity, and extrinsic
curvature in metric spaces.

#### Readership

Graduate students and researchers interested in geometric analysis, nonlinear partial differential equations, and general relativity.

# Table of Contents

## Geometric Analysis, Mathematical Relativity, and Nonlinear Partial Differential Equations

- Preface vii8 free
- On Dark Matter, Spiral Galaxies, and the Axioms of General Relativity 110 free
- 1. Overview 110
- 2. Geometric Motivation 413
- 3. Cosmological Predictions 918
- 4. The Galactic Scale 1120
- 5. Spiral Galaxies 1827
- 6. Elliptical Galaxies 4150
- 7. Open Problems 4857
- 8. Acknowledgments 5059
- Appendix A. A Brief Introduction to General Relativity 5160
- Appendix B. Derivation of the Einstein-Klein-Gordon Equations with a Cosmological Constant from Axiom 1 . 5463
- Appendix C. A Bigger Geometric Theory 5968
- Appendix D. The Coldness of Scalar Field Dark Matter in the Homogeneous and Isotropic Case 6069
- References 6372

- Embedded three-dimensional CR manifolds and the non-negativity of Paneitz operators 6574
- Aubry sets, Hamilton-Jacobi equations, and the Mañé Conjecture 8392
- 1. Mañé critical value and critical subsolutions 8392
- 2. Lax-Oleinik semigroup and Weak KAM Theorem 8594
- 3. Peierls barrier and Aubry set 8796
- 4. The uniqueness issue 8998
- 5. Regularity of critical subsolutions and weak KAM solutions 9099
- 6. The Mañé conjecture 91100
- 7. Some results on the Mañé conjecture 92101
- References 103112
- References 103112

- Minimal surfaces and eigenvalue problems 105114
- On the maximal measure of sections of the 𝑛-cube 123132
- 1. Product measures and sections of the cube 123132
- 2. Formulas for the measure of cube sections 125134
- 3. Distribution functions of the Fourier transform 128137
- 4. Properties of the Fourier transform of ℎ 129138
- 5. Proof of the main integral inequality 136145
- 6. The complex case 140149
- 7. Appendix: The case 𝑎₁>1/√2 143152
- References 155164

- Extremities of stability for pendant drops 157166
- On the Funk-Radon-Helgason Inversion Method in Integral Geometry 175184
- Inequalities for the ADM-mass and capacity of asymptotically flat manifolds with minimal boundary 199208
- Bounded extrinsic curvature of subsets of metric spaces 213222