4.1 Introduction to currents and their properties

Currents are powerful tools in geometric measure theory, extending integration and differentiation to non-smooth settings. They generalize oriented submanifolds, allowing us to study objects with singularities or non-smooth boundaries like fractals and soap films.

Currents have key properties: linearity, continuity, locality, and a boundary operator. They also have a notion of mass, measuring their total variation. These properties make currents ideal for tackling complex geometric problems and variational principles.

Currents in Geometric Measure Theory

Definition and Role of Currents

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• Currents are continuous linear functionals on the space of smooth differential forms with compact support
• Generalize the concept of oriented submanifolds
  • Provide a framework for studying geometric objects with singularities or non-smooth boundaries (fractals, soap films)
• The space of currents is a dual space to the space of smooth differential forms
  • Allows for the application of functional analysis techniques
• Play a crucial role in geometric measure theory by extending the notion of integration and differentiation to non-smooth settings (Lebesgue integration, distributional derivatives)
• Enable the development of a calculus on singular spaces and the analysis of geometric variational problems (minimal surfaces, isoperimetric problem)

Properties of Currents

• Linearity
  • For any two currents $T_1$ and $T_2$ and scalars $a$ and $b$, $(aT_1 + bT_2)(\omega) = aT_1(\omega) + bT_2(\omega)$ for any differential form $\omega$
• Continuity
  • Continuous with respect to the weak topology on the space of differential forms
  • If a sequence of differential forms $\omega_n$ converges to $\omega$, then $T(\omega_n)$ converges to $T(\omega)$ for any current $T$
• Locality
  • The value of a current $T$ on a differential form $\omega$ depends only on the values of $\omega$ in the support of $T$
  • Allows for the study of local properties of currents (density, tangent spaces)
• Boundary operator
  • The boundary of a current $T$, denoted by $\partial T$, is defined by $(\partial T)(\omega) = T(d\omega)$, where $d$ is the exterior derivative
  • Allows for the study of the topology of currents (homology, cohomology)
• Mass
  • The mass of a current $T$, denoted by $M(T)$, is a non-negative real number that measures the total variation of $T$
  • Defined as the supremum of $T(\omega)$ over all differential forms $\omega$ with sup-norm less than or equal to 1
  • Provides a notion of size or magnitude for currents (area, volume)

Currents and Differential Forms

Relationship between Currents and Differential Forms

• Currents are defined as continuous linear functionals on the space of smooth differential forms with compact support
• The duality between currents and differential forms allows for the extension of classical operations to non-smooth settings
  • Integration (action of a current on a differential form)
  • Differentiation (exterior derivative of a differential form corresponds to the boundary of a current)
• The action of a current $T$ on a differential form $\omega$ is denoted by $T(\omega)$ and can be interpreted as a generalized notion of integration
  • Extends the concept of integration of differential forms over smooth submanifolds to non-smooth objects (rectifiable sets, varifolds)
• The space of currents is a larger space than the space of smooth submanifolds
  • Includes objects with singularities and non-smooth boundaries that can still be represented by currents (fractals, soap films)

Solving Problems with Currents

Applications of Currents in Geometric Measure Theory

• Study the existence and regularity of minimal surfaces
  • Formulate the problem in terms of finding stationary points of the mass functional on the space of currents
  • Plateau problem: find a surface of minimal area spanning a given boundary curve by minimizing the mass of currents with the prescribed boundary
• Model the geometry of soap films and bubbles
  • Objects can be modeled as currents that minimize the mass functional subject to certain constraints (area, volume)
• Investigate the existence and structure of singular minimizers in various geometric variational problems
  • Isoperimetric problem: find a set of given volume with minimal surface area
  • Willmore problem: find a surface that minimizes the total squared mean curvature
• Define and study the concept of rectifiable sets
  • Sets that can be approximated by Lipschitz images of subsets of Euclidean space
  • Allows for the extension of geometric measure theory to more general spaces (metric spaces, Banach spaces)