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: , , , and a . They also have a notion of , 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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Top images from around the web for Definition and Role of Currents
Visualization of currents in neural models with similar behavior and different conductance ... View original
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Deformations of Conically Singular Cayley Submanifolds | The Journal of Geometric Analysis View original
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Some Classes of Invariant Submanifolds of LP-Sasakian Manifolds View original
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Deformations of Conically Singular Cayley Submanifolds | The Journal of Geometric Analysis View original
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Currents are continuous linear functionals on the space of smooth with compact
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 T1 and T2 and scalars a and b, (aT1+bT2)(ω)=aT1(ω)+bT2(ω) for any differential form ω
Continuity
Continuous with respect to the weak topology on the space of differential forms
If a sequence of differential forms ωn converges to ω, then T(ωn) converges to T(ω) for any T
Locality
The value of a current T on a differential form ω depends only on the values of ω in the support of T
Allows for the study of local properties of currents (density, tangent spaces)
Boundary operator
The T, denoted by ∂T, is defined by (∂T)(ω)=T(dω), 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(ω) over all differential forms ω 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 ω is denoted by T(ω) 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, )
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)