**Proceedings of Symposia in Pure Mathematics**

Volume: 44;
1986;
464 pp;
Hardcover

MSC: Primary 00;
Secondary 28; 49; 53

**Print ISBN: 978-0-8218-1470-3
Product Code: PSPUM/44**

List Price: $102.00

AMS Member Price: $81.60

MAA Member Price: $91.80

**Electronic ISBN: 978-0-8218-9336-4
Product Code: PSPUM/44.E**

List Price: $96.00

AMS Member Price: $76.80

MAA Member Price: $86.40

# Geometric Measure Theory and the Calculus of Variations

Share this page *Edited by *
*William K. Allard; Frederick J. Almgren, Jr.*

These twenty-six papers survey a cross section of current work in
modern geometric measure theory and its applications in the calculus
of variations. Presently the field consists of a jumble of new ideas,
techniques and intuitive hunches; an exchange of information has been
hindered, however, by the characteristic length and complexity of
formal research papers in higher-dimensional geometric analysis. This
volume provides an easier access to the material, including
introductions and summaries of many of the authors' much longer works
and a section containing 80 open problems in the field. The papers
are aimed at analysts and geometers who may use geometric measure-theoretic
techniques, and they require a mathematical sophistication
at the level of a second year graduate student.

The papers included were presented at the 1984 AMS Summer Research
Institute held at Humboldt State University. A major theme of this
institute was the introduction and application of multiple-valued
function techniques as a basic new tool in geometric analysis,
highlighted by Almgren's fundamental paper Deformations and
multiple-valued functions. Major new results discussed at the
conference included the following: Allard's integrality and regularity
theorems for surfaces stationary with respect to general elliptic
integrands; Scheffer's first example of a singular solution to the
Navier-Stokes equations for a fluid flow with opposing force; and
Hutchinson's new definition of the second fundamental form of a
general varifold.

#### Readership

# Table of Contents

## Geometric Measure Theory and the Calculus of Variations

- Contents v6 free
- Preface vii8 free
- List of Speakers and Titles of Their Talks ix10 free
- List of Participants xiii14 free
- An integrality theorem and a regularity theorem for surfaces whose first variation with respect to a parametric elliptic integrand is controlled 118 free
- Deformations and multiple-valued functions 2946
- Local estimates for minimal submanifolds in dimensions greater than two 131148
- Second variation estimates for minimal orbits 139156
- Index theory for operator ranges and geometric measure theory 149166
- MACSYMA and minimal surfaces 163180
- Sur les chaines maximalement complexes de bord donné 171188
- Index and total curvature of complete minimal surfaces 207224
- The structure of stable minimal hypersurfaces near a singularity 213230
- Some regularity results in plasticity 239256
- Tangential regularity near the C[sup(1)]-boundary 245262
- Solving Plateau's problem for hypersurfaces without the compactness theorem for integral currents 255272
- Complex analytic geometry and measure theory 261278
- Mean curvature contraction of convex hypersurfaces 275292
- C[sup(1)],[sup(a)] multiple function regularity and tangent cone behaviour for varifolds with second fundamental form in LP 281298
- Pointwise pinched manifolds are space forms 307324
- The multiplicity of generic projections of n-dimensional surfaces in R[sup(n)]+[sup(k)] (n + k & #8804; 4) 329346
- Deformation of Riemannian metrics and manifolds with bounded curvature ratios 343360
- A regularity condition at the boundary for weak solutions of some nonlinear elliptic systems 353370
- Solutions to the Navier-Stokes inequality with singularities on a Cantor set 359376
- Asymptotic behaviour of minimal submanifolds and harmonic maps 369386
- Complete catalog of minimizing embedded crystalline cones 379396
- Liouville theorems for stable harmonic maps into either strongly unstable, or δ-pinched, manifolds 405422
- A regularity theorem for minimizing hypersurfaces modulo p 413430
- Regularity of quasi-minima and obstacle problems 429446
- Some open problems in geometric measure theory and its applications suggested by participants of the 1984 AMS Summer Institute 441458