Support

Definition

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.

5 Must Know Facts For Your Next Test

1. The support of a current is crucial for understanding its topological and geometric characteristics, as it identifies where the current contributes to integrals and geometric quantities.
2. For normal currents, the support is a closed set that can be locally approximated by smooth manifolds, giving rise to important relationships between geometry and analysis.
3. The slicing operation involves looking at cross-sections of currents, and the support helps to determine where these slices exist meaningfully within the ambient space.
4. When projecting currents onto lower-dimensional spaces, the support plays a vital role in understanding how mass is distributed and retained after projection.
5. In varifolds, the concept of support extends to studying configurations of sets with density functions, aiding in the exploration of variational problems in geometric measure theory.

Review Questions

• How does understanding the support of a current enhance our comprehension of its geometric properties?
  • Understanding the support of a current provides insight into where significant geometric features occur. It indicates regions contributing to integrals and helps in approximating currents by rectifiable sets. This understanding is vital in applications like analyzing singularities or evaluating intersections in geometric contexts.
• Discuss how the slicing operation utilizes the concept of support when analyzing normal currents.
  • The slicing operation relies heavily on knowing where a normal current's support lies. By focusing on these regions, one can determine meaningful cross-sections that retain mass from the original current. This allows for evaluating properties like curvature and other geometric features within those slices.
• Evaluate the implications of varying supports when projecting currents onto lower-dimensional spaces.
  • When projecting currents onto lower-dimensional spaces, differing supports can lead to various outcomes regarding mass retention and distribution. If the support overlaps significantly with the lower-dimensional space, much of the original mass can be preserved. Conversely, if there's little overlap, it may indicate loss of mass or essential geometric information, which affects applications in variational calculus and optimization problems in geometric measure theory.

Related terms

Rectifiable Current: A rectifiable current is a current that can be represented as a finite sum of smooth differential forms over rectifiable sets, allowing for a well-defined integration process.

Varifold: A varifold is a generalized notion of a manifold that allows for the study of geometrical properties of sets without the need for smoothness, often used in the context of minimal surfaces and geometric measure theory.

Mass: The mass of a current or varifold is a measure of its 'size' or 'extent', defined through integration over the support, which directly relates to the concept of support.