A reading guide
Geometric Measure Theory
Books, lecture notes and papers, from the basic language of surfaces to anisotropic variational problems and the geometry of singularities.
Where to start
For a first overview, begin with Pigati’s summer school notes. Then choose a direction: Maggi for sets of finite perimeter, the currents references for oriented surfaces, or Simon and De Lellis for varifolds and regularity.
The anisotropic and tangent-cone sections are intended as a next step after this foundation. The comments on the introductory references reflect my own experience and taste.
Useful background: measure theory, multivariable analysis and basic functional analysis; differential forms for currents, and elliptic PDE for regularity.
Introductory lecture notes
A compact introduction to the main objects of GMT, with a final lecture connecting the subject to diffuse interfaces.
Alessandro Pigati’s SLMath summer school notes
I was the TA for the second half of the SLMath summer school, June 17–26, 2024. The second week was an introduction to geometric measure theory taught by Alessandro Pigati, using the following notes written by him.
Sets of finite perimeter
Sets of finite perimeter, also called Caccioppoli sets, provide a way to treat hypersurfaces as boundaries of measurable sets. This is a natural starting point for isoperimetric problems and codimension-one regularity.
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Sets of Finite Perimeter and Geometric Variational Problems
This is my favorite book to start learning from: it develops the theory with many examples and detailed proofs. Begin with the measure-theoretic foundations, then move to finite perimeter, compactness and lower semicontinuity, and the regularity of perimeter minimizers. Measure theory and some functional analysis are useful background.
Currents & regularity
Currents encode orientation and multiplicity and give a flexible formulation of Plateau’s problem. After the definitions, the key milestones are compactness, existence of minimizers, and ε-regularity.
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Cartesian Currents in the Calculus of Variations I
Although the book is devoted to Cartesian currents, it also gives an excellent introduction to currents and a useful crash course in measure theory and real analysis. The foundational material can be read before the specialized applications.
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Geometric Integration Theory
An accessible development from measure theory to currents, with detailed regularity arguments. This is a useful reference when working through the ε-regularity theory for area-minimizing integral currents.
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Center Manifold: A Case Study
Appendix A is a useful place to study De Giorgi’s ε-regularity theorem. Read it alongside a first course on area-minimizing currents, before moving to the more involved center-manifold constructions in higher codimension.
Varifolds & Allard regularity
Varifolds retain tangent-plane information without requiring an orientation. First variation, monotonicity, rectifiability and Allard’s theorem form the core of this part of the subject.
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Introduction to Geometric Measure Theory
A concise, systematic reference for readers with some graduate analysis. It develops the theory of varifolds and gives a detailed treatment of Allard regularity in modern notation; it also provides the background for the tangent-cone references below.
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Allard’s Interior Regularity Theorem: An Invitation to Stationary Varifolds
A gentle introduction to integral varifolds and Allard’s regularity theorem, with a particularly useful presentation of the tilt-excess decay argument. A good companion to Simon’s text.
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On the First Variation of a Varifold
The foundational treatment of first variation, rectifiability, compactness and regularity for varifolds. It is helpful to read an exposition first, then return to the original paper for the full framework and hypotheses.
Anisotropic geometric measure theory
An anisotropic surface energy assigns a cost that can depend on position and tangent plane. The Wulff shape is the basic isoperimetric model; rectifiability, regularity and min–max theory require understanding what replaces the isotropic tools.
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Sets of Finite Perimeter and Geometric Variational Problems
Return to Chapter 20, “Anisotropic surface energies,” after learning finite-perimeter compactness. Section 20.2 treats the Wulff problem and gives a concrete first encounter with direction-dependent surface energies.
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Rectifiability of Varifolds with Locally Bounded First Variation with Respect to Anisotropic Surface Energies
An extension of Allard’s rectifiability theory to anisotropic energies. The central structural hypothesis is the atomic condition; in codimension one, the paper relates it to strict convexity of the integrand.
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The Anisotropic Min-Max Theory: Existence of Anisotropic Minimal and CMC Surfaces
A route from anisotropic first variation to a min–max construction in closed three-manifolds. The paper constructs surfaces of prescribed constant anisotropic mean curvature, smooth away from at most one point.
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Existence and Regularity of Min-Max Anisotropic Minimal Hypersurfaces
Develops the min–max theory in higher dimensions, with control of the singular set. A key ingredient is a uniform upper bound for density ratios; in dimension three the construction gives a smooth anisotropic minimal surface.
Tangent cones & singularities
Blowing up near a point reveals the local geometry of a surface. For area-minimizing currents, monotonicity and compactness lead to tangent cones; proving that the cone is unique is a separate, much subtler question.
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Introduction to Geometric Measure Theory
Start with monotonicity and blow-ups, then Chapter 7, §3, “Tangent Cones and Densities.” Keep existence of a subsequential limit separate from uniqueness of the limit.
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Asymptotics for a Class of Non-Linear Evolution Equations, with Applications to Geometric Problems
A foundational source for the analytic approach to asymptotic uniqueness. Its geometric applications include uniqueness of tangent cones in settings with a smooth multiplicity-one link; this is a natural next step after the basic blow-up theory.
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Uniqueness of Tangent Cones for 2-Dimensional Almost Minimizing Currents
A treatment of uniqueness for two-dimensional almost area-minimizing currents, extending the ideas of White’s theorem for area-minimizing currents. Useful for seeing how quantitative comparison arguments produce uniqueness.
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Uniqueness of Certain Cylindrical Tangent Cones
Studies uniqueness when an area-minimizing hypersurface has tangent cone C × ℝ, with C the Simons cone over S³ × S³. The singular axis and the failure of integrability make this a different problem from cones with an isolated singularity.