Geometric Measure Theory

  title={Geometric Measure Theory},
  author={Toby C. O’Neil},

Geometric Measure Theory

The regularity of minimal surfaces in higher codimension

The Plateau’s problem investigates those surfaces of least area spanning a given contour. It is one of the most classical problems in the calculus of variations, it lies at the crossroad of several

Lectures on chainlet geometry - new topological methods in geometric measure theory

These draft notes are from a graduate course given by the author in Berkeley during the spring semester of 2005. They cover the basic ideas of a new, geometric approach to geometric measure theory.

Discrete methods in geometric measure theory

The thesis addresses problems from the field of geometric measure theory. It turns out that discrete methods can be used efficiently to solve these problems. Let us summarize the main results of the

Differential geometry of generalized submanifolds

In the recent past a lot of authors have devoted systematic efforts to indagate problems related to compactness and degeneration phenomenons in classes of surfaces with integral bounds on curvature,

Effective Fractal Dimension in Algorithmic Information Theory

Hausdorff dimension assigns a dimension value to each subset of an arbitrary metric space. In Euclidean space, this concept coincides with our intuition that smooth curves have dimension 1 and smooth

Ravello lecture notes on geometric calculus -- Part I

In these notes of lectures at the 2004 Summer School of Mathematical Physics in Ravello, Italy, the author develops an approach to calculus in which more efficient choices of limits are taken at key

Geometry and nonlinear analysis

Nonlinear analysis has played a prominent role in the recent developments in geometry and topology. The study of the Yang-Mills equation and its cousins gave rise to the Donaldson invariants and more


1.1. In studying the differential geometry of a hypersurface M in euclidean space E + 1 it has often proved fruitful to view the integral of the Gauss-Kronecker curvature (or "Gauss-Bonnet

Lectures on the Geometric Group Theory

Preface The main goal of this book is to describe several tools of the quasi-isometric rigidity and to illustrate them by presenting (essentially self-contained) proofs of several fundamental

Geometric Measure Theory: A Beginner's Guide

Geometric Measure Theory: A Beginner's Guide, Fifth Edition provides the framework readers need to understand the structure of a crystal, a soap bubble cluster, or a universe. The book is essential

Geometry of Sets and Measures in Euclidean Spaces: Fractals and Rectifiability

Acknowledgements Basic notation Introduction 1. General measure theory 2. Covering and differentiation 3. Invariant measures 4. Hausdorff measures and dimension 5. Other measures and dimensions 6.

Measure theory and fine properties of functions

GENERAL MEASURE THEORY Measures and Measurable Functions Lusin's and Egoroff's Theorems Integrals and Limit Theorems Product Measures, Fubini's Theorem, Lebesgue Measure Covering Theorems

Fractured fractals and broken dreams : self-similar geometry through metric and measure

1. Basic definitions 2. Examples 3. Comparison 4. The Heisenberg group 5. Background information 6. Stronger self-similarity for BPI spaces 7. BPI equivalence 8. Convergence of metric spaces 9. Weak

On the first variation of a varifold

Suppose M is a smooth m dimensional Riemannian manifold and k is a positive integer not exceeding m. Our purpose is to study the first variation of the k dimensional area integrand in M. Our main

The structure of singularities in soap-bubble-like and soap-film-like minimal surfaces

In this paper we provide a complete classification of the local structure of singularities in a wide class of two-dimensional surfaces in R3 collected under the adjective (M, i, a) minimal by Almgren


If X is a subset of R , let Hk(X) denote the k-dimensional hausdorff measure of X . We write Ik(X) = 0 if H(πKX) = 0 for almost every k-plane K in R , where πK : R → K is orthogonal projection.

Plateau's problem : an invitation to varifold geometry

The phenomena of least area problems Integration of differential forms over rectifiable sets Varifolds Variational problems involving varifolds References Additional references Index.

Proof of the Double Bubble Conjecture

We prove that the standard double bubble provides the least-area way to enclose and separate two regions of prescribed volume in R 3 .