📏Geometric Measure Theory Unit 4 – Currents and Varifolds

Key Concepts and Definitions

• Currents generalize the notion of a submanifold to allow for singularities and boundary
• Varifolds extend the concept of a submanifold to include multiplicity and orientation
  • Multiplicity refers to the number of times a point is counted in the varifold
  • Orientation distinguishes between the two sides of a submanifold
• Rectifiable currents are currents that can be represented by integration over rectifiable sets
  • Rectifiable sets are sets that can be covered by a countable union of Lipschitz images of subsets of Euclidean space
• Normal currents are currents that have finite mass and whose boundary is also a current
• Integral currents are rectifiable currents with integer multiplicity and integer-multiplicity boundary
• Flat norm measures the distance between currents by considering both the mass and the boundary mass
• Compactness theorems for currents and varifolds guarantee the existence of convergent subsequences under certain conditions

Historical Context and Development

• Currents and varifolds emerged as tools to study geometric variational problems in the 1960s
• Federer and Fleming introduced normal and integral currents in their seminal paper "Normal and integral currents" (1960)
• Almgren developed the theory of varifolds in his PhD thesis "Plateau's Problem: An Invitation to Varifold Geometry" (1966)
• The development of currents and varifolds was motivated by the need to extend classical calculus of variations to more general settings
  • Classical calculus of variations dealt primarily with smooth submanifolds
  • Currents and varifolds allow for singularities, boundaries, and multiplicities
• The theory of currents and varifolds has been instrumental in solving long-standing problems in geometric measure theory
  • Examples include the proof of the regularity of area-minimizing hypersurfaces (De Giorgi, 1961) and the solution of the Plateau problem (Reifenberg, 1960)

Mathematical Foundations

• Currents are linear functionals on the space of differential forms satisfying certain continuity and locality properties
  • The space of $k$-dimensional currents on a manifold $M$ is denoted by $\mathcal{D}_k(M)$
  • The boundary of a $k$-current $T$ is the $(k-1)$-current $\partial T$ defined by $\partial T(\omega) = T(d\omega)$, where $d$ is the exterior derivative
• Varifolds are Radon measures on the Grassmann bundle $Gr(M)$ of a manifold $M$
  • The Grassmann bundle $Gr(M)$ consists of pairs $(x,P)$, where $x \in M$ and $P$ is a tangent plane to $M$ at $x$
  • The space of $k$-dimensional varifolds on $M$ is denoted by $\mathbf{V}_k(M)$
• The mass of a current $T$ is defined as $\mathbf{M}(T) = \sup\{T(\omega) : |\omega| \leq 1\}$
  • The mass of a varifold $V$ is defined similarly using the total variation of the Radon measure
• The flat norm of a current $T$ is defined as $\mathbf{F}(T) = \inf\{\mathbf{M}(T-\partial S) + \mathbf{M}(S)\}$, where the infimum is taken over all currents $S$
• Compactness theorems for currents and varifolds rely on the Federer-Fleming compactness theorem and the Allard compactness theorem, respectively

Types of Currents and Varifolds

• Normal currents are currents with finite mass and boundary mass
  • The space of normal currents is denoted by $\mathbf{N}_k(M)$
  • Normal currents form a Banach space under the mass norm
• Integral currents are normal currents with integer multiplicity and integer-multiplicity boundary
  • The space of integral currents is denoted by $\mathbf{I}_k(M)$
  • Integral currents are closed under boundary and compactly supported pushforward
• Rectifiable currents are currents that can be represented by integration over rectifiable sets
  • The space of rectifiable currents is denoted by $\mathbf{R}_k(M)$
  • Rectifiable currents have a notion of approximate tangent plane at almost every point
• Integral varifolds are varifolds that can be represented by integration over rectifiable sets with integer multiplicity
  • The space of integral varifolds is denoted by $\mathbf{IV}_k(M)$
• General varifolds include non-rectifiable and non-integer multiplicity varifolds
  • General varifolds can be used to model more complex geometric objects, such as fractals and soap films with higher multiplicity

Properties and Characteristics

• Currents and varifolds have a notion of support, which is the smallest closed set containing the object
  • The support of a current $T$ is denoted by $\text{spt}(T)$
  • The support of a varifold $V$ is denoted by $\text{spt}(V)$
• Currents and varifolds can be pushed forward under smooth maps
  • If $f: M \to N$ is a smooth map and $T$ is a current on $M$, then the pushforward $f_\sharp T$ is a current on $N$
  • If $V$ is a varifold on $M$, then the pushforward $f_\sharp V$ is a varifold on $N$
• Currents and varifolds have a notion of boundary
  • The boundary of a current $T$ is the current $\partial T$
  • The boundary of a varifold $V$ is not well-defined in general, but can be defined for integral varifolds
• Currents and varifolds can be restricted to open sets
  • If $T$ is a current and $U$ is an open set, then the restriction $T \llcorner U$ is a current
  • If $V$ is a varifold and $U$ is an open set, then the restriction $V \llcorner U$ is a varifold
• Currents and varifolds have a notion of convergence
  • A sequence of currents $T_i$ converges to a current $T$ if $T_i(\omega) \to T(\omega)$ for all differential forms $\omega$
  • A sequence of varifolds $V_i$ converges to a varifold $V$ if $\int \phi dV_i \to \int \phi dV$ for all continuous functions $\phi$ on $Gr(M)$

Applications in Geometric Measure Theory

• Currents and varifolds are used to study the existence and regularity of minimal surfaces
  • The Plateau problem seeks a surface of minimal area spanning a given boundary curve
  • The solution to the Plateau problem can be obtained by minimizing the mass of integral currents or varifolds with the given boundary
• Currents and varifolds are used to study the regularity of area-minimizing hypersurfaces
  • The regularity theory shows that area-minimizing hypersurfaces are smooth outside a small singular set
  • The dimension of the singular set is controlled by the dimension of the ambient space
• Currents and varifolds are used to study the structure of soap films and bubbles
  • Soap films and bubbles can be modeled as stationary varifolds with respect to the area functional
  • The structure of soap films and bubbles can be analyzed using the first variation formula for varifolds
• Currents and varifolds are used to study the isoperimetric problem and its generalizations
  • The isoperimetric problem seeks a region of maximal volume among all regions with a given boundary area
  • Generalizations of the isoperimetric problem involve other geometric functionals, such as the capacity and the Cheeger constant

Computational Methods and Techniques

• Discrete exterior calculus provides a framework for discretizing currents and varifolds
  • Discrete differential forms are defined on simplicial complexes and obey discrete versions of the Stokes theorem
  • Discrete currents and varifolds can be represented as chains and cochains in the simplicial complex
• Finite element methods can be used to approximate solutions to variational problems involving currents and varifolds
  • The finite element spaces consist of piecewise polynomial differential forms and functions on the Grassmann bundle
  • The variational problems are reduced to finite-dimensional optimization problems
• Multi-scale methods can be used to efficiently represent and manipulate currents and varifolds
  • Wavelet transforms can be used to decompose currents and varifolds into components at different scales
  • Multi-scale representations can be used to compress data and speed up computations
• Machine learning techniques can be used to analyze and classify currents and varifolds
  • Convolutional neural networks can be used to learn features and patterns in the data
  • Unsupervised learning methods, such as clustering and dimensionality reduction, can be used to discover structure in the data

Advanced Topics and Current Research

• The theory of metric currents and metric integral currents extends the classical theory to metric spaces
  • Metric currents are defined using duality with Lipschitz functions instead of differential forms
  • The Ambrosio-Kirchheim theory provides a framework for studying metric currents and their properties
• The theory of $\mathbf{G}$-currents and $\mathbf{G}$-varifolds incorporates group symmetries and equivariance
  • $\mathbf{G}$-currents and $\mathbf{G}$-varifolds are currents and varifolds that are invariant under the action of a group $\mathbf{G}$
  • The theory of $\mathbf{G}$-currents and $\mathbf{G}$-varifolds can be used to study geometric objects with symmetries, such as crystals and quasicrystals
• The theory of currents and varifolds in sub-Riemannian geometry extends the classical theory to non-Riemannian settings
  • Sub-Riemannian geometry studies manifolds with a non-integrable distribution of tangent planes
  • Currents and varifolds in sub-Riemannian geometry can be used to study minimal surfaces and isoperimetric problems in this setting
• The theory of currents and varifolds in infinite-dimensional spaces extends the classical theory to function spaces and shape spaces
  • Currents and varifolds in infinite-dimensional spaces can be used to study the geometry of shapes and patterns
  • The theory of currents and varifolds in infinite-dimensional spaces has applications in computer vision, pattern recognition, and machine learning