6.3 Slicing and projection of currents
6 min read•august 14, 2024
and are key operations in Federer-Fleming Theory. They let us analyze currents by cutting them into smaller pieces or flattening them onto subspaces. These techniques help us understand the geometry and topology of currents in different dimensions.
Slicing gives us cross-sections of currents, while projection collapses them onto lower-dimensional spaces. Both preserve important properties like and boundary structure. Together, they're powerful tools for studying currents and their geometric properties.
Slicing for Currents
Definition and Geometric Interpretation
Top images from around the web for Definition and Geometric Interpretation
Cross product - Wikiversity View original
Is this image relevant?
22.10 Magnetic Force between Two Parallel Conductors – College Physics View original
Is this image relevant?
Frontiers | Closed, Two Dimensional Surface Dynamics | Physics View original
Is this image relevant?
Cross product - Wikiversity View original
Is this image relevant?
22.10 Magnetic Force between Two Parallel Conductors – College Physics View original
Is this image relevant?
1 of 3
Top images from around the web for Definition and Geometric Interpretation
Cross product - Wikiversity View original
Is this image relevant?
22.10 Magnetic Force between Two Parallel Conductors – College Physics View original
Is this image relevant?
Frontiers | Closed, Two Dimensional Surface Dynamics | Physics View original
Is this image relevant?
Cross product - Wikiversity View original
Is this image relevant?
22.10 Magnetic Force between Two Parallel Conductors – College Physics View original
Is this image relevant?
1 of 3
- The slicing operation for a k-dimensional T in R^n is defined as the (k-1)-dimensional current <T, f, z> obtained by intersecting T with the level set {x : f(x) = z}, where f is a Lipschitz function and z is a real number
- Geometrically, the slicing operation can be interpreted as taking a "cross-section" of the current T along the level sets of the function f
- The slice <T, f, z> represents the restriction of T to the preimage f^(-1)(z), which is a codimension-1 subspace of R^n
- The orientation of the slice is determined by the orientation of T and the gradient of f, following the right-hand rule (e.g., if T is a 2-dimensional current in R^3 and f(x, y, z) = z, then the slice <T, f, c> is a 1-dimensional current in the xy-plane with orientation determined by the right-hand rule)
Properties and Linearity
- The slicing operation is linear in T and satisfies the product rule <T, f, z> = <∂T, f, z> + (-1)^k <T, 1, z> ∧ df, where ∂T is the boundary of T
- Linearity means that for currents S and T and scalars a and b, <aS + bT, f, z> = a<S, f, z> + b<T, f, z>
- The product rule relates the slice of a current to the slice of its boundary and the exterior derivative of the slicing function f
- The slicing operation commutes with the pushforward operator, i.e., for a smooth map φ: R^n → R^m, φ_(<T, f, z>) = <φ_T, f∘φ^(-1), z>
- This property allows for the computation of slices in different coordinate systems or under transformations
Slices of Normal and Integral Currents
Existence and Regularity
- For a normal current T, the slice <T, f, z> exists for almost every z in R and is also a normal current
- The set of z for which the slice does not exist has Lebesgue measure zero
- The normality of the slice follows from the fact that the slicing operation preserves the normal current structure
- For an T, the slice <T, f, z> is also an integral current for almost every z
- The slice of an integral current has finite mass and is uniquely determined up to a set of measure zero
- Integral currents are a subclass of normal currents with additional integrality properties
Mass Bounds and Uniqueness
- The mass of the slice <T, f, z> is bounded by the mass of T multiplied by the Lipschitz constant of f, as stated in the inequality M(<T, f, z>) ≤ Lip(f) M(T)
- This inequality provides an upper bound for the mass of slices in terms of the mass of the original current and the regularity of the slicing function
- The Lipschitz constant Lip(f) measures the maximum stretching or contraction of f and controls the distortion of mass under slicing
- For integral currents, the slices are uniquely determined up to a set of measure zero
- If T is an integral current and <T, f, z1> = <T, f, z2> for almost every z1 and z2, then the slices coincide as integral currents
- This uniqueness property is stronger than the uniqueness for normal currents, which allows for differences on sets of measure zero
Mass and Boundary under Slicing
Coarea Formula and Mass Decomposition
- The total mass of the slices is related to the mass of T by the coarea formula: ∫ M(<T, f, z>) dz = ∫ |∇f| dμ_T, where μ_T is the mass measure associated with T
- The coarea formula provides a way to decompose the mass of a current into the masses of its slices
- It relates the integral of the slice masses to the integral of the gradient magnitude of f with respect to the mass measure of T
- The coarea formula can be used to prove the existence of slices with controlled mass and to study the behavior of currents under slicing
- For example, if T is a normal current and f is a Lipschitz function, then the coarea formula implies that the slice <T, f, z> exists and has finite mass for almost every z
- The formula also provides a way to estimate the total mass of slices in terms of the mass of T and the regularity of f
Boundary Commutation and Structure Preservation
- The boundary of the slice is the slice of the boundary, i.e., ∂<T, f, z> = <∂T, f, z>, which allows for the computation of the boundary of slices using the boundary of the original current
- This commutation property relates the slicing and boundary operations and preserves the structure of the current under slicing
- It implies that the boundary of a slice can be obtained by slicing the boundary of the current at the same level set
- The slicing operation commutes with the , preserving the structure of the current
- If T is a normal (or integral) current, then ∂T is also a normal (or integral) current, and the slices of ∂T are the boundaries of the slices of T
- This property is crucial for the study of the homological properties of currents and their slices, as it allows for the computation of homology groups and the analysis of topological invariants
Projection vs Slicing
Definition and Relation to Slicing
- The projection of a k-dimensional current T onto a subspace V of R^n is defined as the current π_V(T) obtained by pushing forward T by the orthogonal projection map onto V
- The projection operation can be seen as a way to "flatten" or "collapse" the current onto a lower-dimensional subspace
- It is a linear operation that preserves the normal (or integral) current structure
- The projection operation can be expressed in terms of slicing using the formula π_V(T) = ∫ <T, dist_V, z> dz, where dist_V is the distance function to the subspace V
- This formula expresses the projection as an integral of slices taken along the level sets of the distance function to V
- It provides a connection between the projection and slicing operations and allows for the study of projections using the properties of slices
Mass Inequality and Boundary Commutation
- The mass of the projected current satisfies the inequality M(π_V(T)) ≤ M(T), showing that projection does not increase mass
- This inequality is a consequence of the fact that the projection map is a contraction and does not stretch or increase distances
- It implies that the mass of a current can only decrease or remain the same under projection onto a subspace
- The projection operation commutes with the boundary operator, i.e., ∂(π_V(T)) = π_V(∂T)
- This commutation property relates the projection and boundary operations and preserves the structure of the current under projection
- It allows for the computation of the boundary of a projected current using the projection of the boundary of the original current
Applications and Geometric Analysis
- The projection operation can be used to study the behavior of currents under linear transformations and to analyze their geometric properties in lower-dimensional spaces
- For example, the projection of a current onto a hyperplane can be used to study its intersection with the hyperplane and to compute geometric invariants such as the intersection number or the linking number
- The projection onto a line can be used to study the distribution of mass along the line and to analyze the one-dimensional structure of the current
- Projections can also be used to construct new currents from existing ones and to study the relationship between currents in different dimensions
- For example, the cone construction, which extends a current T by one dimension using the formula C(T) = π_*([0, 1] × T), where π is the projection onto the first factor, can be analyzed using the properties of projections and slices
- The projection and slicing operations provide a powerful set of tools for the geometric analysis of currents and their applications in various areas of mathematics and physics
Key Terms to Review (17)
Allard's Regularity Theorem: Allard's Regularity Theorem is a result in geometric measure theory that provides conditions under which a varifold can be approximated by smooth submanifolds. It establishes the regularity of varifolds under certain geometric assumptions, ensuring that singular points are controlled and finite in measure. This theorem is pivotal when dealing with currents and their projections, as it allows for a deeper understanding of their structure and behavior in higher-dimensional spaces.
Approximation by smooth forms: Approximation by smooth forms refers to the process of representing a given current or geometric object using smooth differential forms, which are infinitely differentiable functions. This concept is crucial in geometric measure theory as it allows for the analysis of more complex geometric objects through simpler, well-understood smooth forms. The ability to approximate currents using smooth forms facilitates operations like slicing and projection, and aids in the characterization of rectifiable currents, ultimately contributing to results like the closure theorem.
Boundary Operator: The boundary operator is a mathematical tool used in geometric measure theory to define the boundary of chains and currents. It transforms a chain or current into its boundary, effectively capturing how the geometry of a space is structured and changing as one considers its boundaries. This operator plays a crucial role in various concepts, including the relationship between flat chains and cochains, the slicing and projection of currents, and the formulation of approximation theorems.
Current: A current is a mathematical object used in geometric measure theory to generalize the concept of oriented submanifolds, allowing for a flexible and powerful framework to analyze geometric properties. Currents can be seen as distributions that represent generalized surfaces and allow for the integration of differential forms over them, facilitating the study of geometric structures and properties in higher dimensions.
Differential Forms: Differential forms are mathematical objects that generalize the concept of functions and vectors, providing a framework to perform integration on manifolds. They play a crucial role in various areas of analysis, geometry, and physics, allowing us to express integrals over curves, surfaces, and higher-dimensional spaces in a concise and elegant way.
Flat convergence: Flat convergence is a notion in geometric measure theory that describes the convergence of a sequence of currents in a specific, controlled manner, particularly when slicing and projecting these currents. This type of convergence is essential for analyzing how geometric properties of currents behave under various transformations, including how they change when viewed from different perspectives or dimensions.
Homology Class: A homology class is an equivalence class of chains in a topological space, where chains represent formal sums of simplices or other geometric shapes. Each homology class captures a unique topological feature of the space, reflecting how different shapes can be combined or decomposed. Understanding homology classes is crucial for analyzing the properties of currents, as they provide insight into the way these currents can be manipulated and projected within various geometric contexts.
Integral Current: An integral current is a generalized notion of a geometric object that allows for the representation of singular and flat chains in a way that measures oriented area or volume with integer coefficients. Integral currents can be seen as the mathematical objects that arise when working with various geometric structures, helping to connect concepts like slicing and projection, as well as rectifiability in more complex geometric settings such as sub-Riemannian manifolds.
Mass: In the context of geometric measure theory, mass refers to a generalized notion of size that extends the concept of measure to higher-dimensional spaces, particularly in relation to currents. Mass plays a critical role in understanding properties of normal and rectifiable currents, where it quantifies how much 'weight' a current carries and allows for comparisons between different currents and their behaviors under various operations.
Normal Vector: A normal vector is a vector that is perpendicular to a given surface or manifold at a specific point. This concept is crucial for understanding geometric properties, as it helps in defining how surfaces interact with their surrounding space. Normal vectors play a key role in various mathematical contexts, such as when considering slicing and projections of currents, and in calculating the first variation of varifolds to determine mean curvature.
Projection: In geometric measure theory, projection refers to the operation of mapping a geometric object or current onto a lower-dimensional space. This operation preserves certain properties of the original object while simplifying its structure, making it easier to analyze. Projections play a crucial role in understanding the relationships between different currents and their dimensions, especially when it comes to slicing and manipulating them within various contexts.
Rectifiable Current: A rectifiable current is a mathematical object in geometric measure theory that generalizes the notion of oriented surfaces. It is defined as a current that can be represented as a finite sum of integral currents, which have well-defined area and orientation properties. This concept connects to various properties of currents, including how they can be sliced and projected, as well as their role in the closure theorem that establishes conditions for their convergence and compactness.
Slicing: Slicing refers to the process of taking a 'slice' or cross-section of a geometric object, particularly in the context of currents, where it helps analyze the behavior and properties of these objects in lower dimensions. By looking at these lower-dimensional aspects, one can gain insights into how higher-dimensional structures behave, enabling better understanding and manipulation of currents in geometric measure theory.
Smooth Current: A smooth current is a generalized notion of integration over smooth manifolds, which allows for the integration of differential forms on these manifolds. It extends the idea of currents to include more regular behaviors and properties, enabling a deeper understanding of geometric measure theory and related concepts. Smooth currents facilitate operations like slicing and projection, highlighting their importance in the study of geometric structures and analysis.
Support: In geometric measure theory, the support of a current or a varifold refers to the closure of the set where the current or varifold has non-zero mass. This concept is essential as it helps to determine the regions where the current or varifold has significant contribution and provides insight into their geometric properties, including how they can be approximated by rectifiable currents or how they behave under various operations like slicing and projection.
Transport of currents: Transport of currents refers to the process of moving or pushing currents, which are generalized surfaces that can represent various geometric and physical phenomena, through different spaces. This concept is crucial when considering how currents interact with the underlying geometry and can change when mapped or transformed, especially in relation to slicing and projecting these currents onto lower-dimensional spaces.
Varifolds: Varifolds are generalizations of smooth surfaces used in geometric measure theory, allowing for a broader framework to study geometric objects with singularities or varying dimensions. They provide a way to analyze and represent sets that may not be rectifiable, making them essential for understanding more complex geometric structures.