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
- 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)