Geometric Measure Theory Explained

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Geometric measure theory is the study of the geometric properties of sets that are typically in Euclidean space. When calculating a coordinate, it is necessary to have three specific points available in the two-dimensional Euclidean plane to determine a specific location. This is a process that is similar to triangulation. Where the three lines connect become the point in Euclidean space that is being examined.

This is different from a standard special coordinate, which would require six specific points to determine a specific location within the space. That is because standard special coordinates work in three dimensions instead of two dimensions.

By studying the geometric properties of sets, it becomes possible to use various geometric tools to surfaces that may not be smooth and would normally be difficult to interpret otherwise.

What Was Geometric Measure Theory Developed

Geometric measure theory was developed out of the need to solve the Plateau problem. This problem asks for every smooth closed curve to have a surface which exists that is the least area among all surfaces when the boundary equals a given curve. It was a problem that was first proposed in 1760 and not solved until the 1930s.

The problem is named after a 19th century physicist named Joseph Plateau. He studied soap films and found that many patterns in nature followed these four specific laws.

  1. They are made of an unbroken smooth surface.
  2. The mean curvature of a film portion is everywhere constant at any point.
  3. Films always meet in threes along an edge and do so at a specific angle.
  4. The borders meet in fours at a vertex and do so at a specific angle as well.

These laws hold for minimal surfaces. Geometric measure theory has been able to prove these laws are factual mathematically.

What Is Central to the Geometric Measure Theory?

There are 4 objects that are considered to be central when looking at the geometric measure theory.

  • Radon measures, or rectifiable sets, have the least possible regularity required to admit an approximate tangent space.
  • Integral manifold currents are present, possibly with a boundary.
  • Flat chains are an alternative to the manifold currents and may also have a possible boundary.
  • Sets of locally finite perimeter, sometimes called Caccioppoli sets, generalize the concept of the manifold currents and the Divergence theorem applies.

There are also four theorems or concepts that are considered to be central to the implementation of the geometric measure theory.

  • Area Formula. This generalizes the concept of variable change during integration.
  • Coarea Formula. This generalizes and then adapts Fubini’s theorem to the geometric measure theory.
  • Isoperimetric Inequality. This states that the smallest circumference that is possible for any given area is that of a circle which is round.
  • Flat Convergence. This generalizes the concept of manifold convergence.

What About Unorientated Surfaces?

The geometric measure theory does an excellent job of investigated surfaces that are orientated. For unorientated surfaces, however, certain problems arise with the equations. This led to the developed of the Varifolds theory to work in conjunction with the geometric measure theory. By incorporating varifolds, it becomes possible to gain information about the rectifiability and the degree of smoothness within the calculation.

It should also be noted that there are several variants that can operate within and without the geometric measure theory that may produce similar results. The success of this theory to calculate Euclidean spaces, however, is beyond dispute. It’s accuracy suggests that the ideas presented within the theory could be applied to other spaces that are more generalized, making it potentially possible to calculate 6 points instead of just 3 with predictability.

The geometric measure theory has made the language of mathematics become more accessible to humanity. Through its study and application, it becomes possible to study dimensional spaces with more accuracy, allowing for the possibility of proving past ideas and theories. By finding the minimal area spanning a surface on a boundary curve, we unlock more ways to explore the universe.