The marriage of analytic power to geometric intuition drives many of today's mathematical advances, yet books that build the connection from an elementary level remain scarce. This engaging introduction to geometric measure theory bridges analysis and geometry, taking readers from basic theory to some of the most celebrated results in modern analysis. The theory of sets of finite perimeter provides a simple and effective framework. Topics covered include existence, regularity, analysis of singularities, characterization and symmetry results for minimizers in geometric variational problems, starting from the basics about Hausdorff measures in Euclidean spaces and ending with complete proofs of the regularity of area-minimizing hypersurfaces up to singular sets of codimension 8. Explanatory pictures, detailed proofs, exercises and remarks providing heuristic motivation and summarizing difficult arguments make this graduate-level textbook suitable for self-study and also a useful reference for researchers. Readers require only undergraduate analysis and basic measure theory.

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Sets of Finite Perimeter and Geometric Variational Problems: An Introduction to Geometric Measure Theory

The rigorous microscopic theory of equilibrium crystal shapes has made enormous progress during the last decade. We review here the main results that have been obtained, both in two and higher dimensions. In particular, we describe how the phenomenological Wulff and Winterbottom constructions can be derived from the microscopic description provided by the equilibrium statistical mechanics of lattice gases. We focus on the main conceptual issues and describe the central ideas of the existing approaches.

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Rigorous probabilistic analysis of equilibrium crystal shapes

In this paper we introduce a new method for proving area-minimization which we call "paired calibrations." We begin with the simplest application, the cone over the tetrahedron, which appears in soap films. We then discuss immiscible fluid interfaces, crystal surfaces, and one-dimensional networks minimizing other norms. 1. Introduction In her classification of soap-film singularities [Tl], Jean Taylor proved only by the process of elimination that the cone over the edges of the regular tetrahedron minimizes area among surfaces separating the four faces. We give a direct proof which applies to regular simplices in all dimensions. See Figure 1.0.1. Configurations of several immiscible fluids try to minimize an energy proportional to interfacial surface area, but the constant of proportionality varies for each pair of fluids. Chapter 2 proves that certain cones minimize such weighted areas. The surface energy of a crystal depends on direction, as given by a norm Φ on unit normals. Chapter 3 proves certain cones Φminimizing, such as a cone over a triangular prism. The hypotheses involve basic geometric questions, such as the number of possible cardinalities of equilateral sets (i.e., sets of pairwise equidistant points) for a norm on Rn. We also consider 1-dimensional Φ-minimizing networks for differentiable norms Φ. It is well-known that length-minimizing networks meet in threes at 120° angles. Chapter 4 classifies the singularities in Φ-minimizing networks in Rn and establishes n + 1 as the sharp bound on the number of segments that can meet at a point.

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Paired calibrations applied to soap films, immiscible fluids, and surfaces or networks minimizing other norms

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

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Proof of the Double Bubble Conjecture

In this article, we present a diffusion-generated approach for evolving volume-preserving motion by mean curvature. Our algorithm alternately diffuses and sharpens characteristic functions to produce a normal velocity which equals the mean curvature minus the average mean curvature. This simple algorithm naturally treats topological mergings and breakings and can be made very fast even when the volume constraint is enforced to double precision (or more). Two dimensional numerical studies are provided to demonstrate the convergence of the method for smooth and nonsmooth problems.

A Simple Scheme for Volume-Preserving Motion by Mean Curvature

Of course the circle is the least-perimeter way to enclose a region of prescribed area in the plane. This paper proves that a certain standard «double bubble» is the least-perimeter way to enclose and separate two regions of prescribed areas. The solution for three regions remains conjectural

https://msp.org/pjm/1993/159-1/pjm-v159-n1-p03-p.pdf

The standard double soap bubble in R2 uniquely minimizes perimeter

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The structure of singularities in $\Phi$-minimizing networks in ${\bf R}^2$.

/pdf/existence-of-shortest-directed-networks-in-bf-r-2-dipqfn9ctl.pdf

Manuel Alfaro

Papers

The standard double soap bubble in R2 uniquely minimizes perimeter

The structure of singularities in $\Phi$-minimizing networks in ${\bf R}^2$.

Existence of shortest directed networks in ${\bf R}^2$.