Mathysteria

Since last time, we now know the crux of Measure Theory. So, let us proceed to our minimal surface. Its definition is derived quite a bit from its name: a surface with mean curvature as zero and which locally minimizes the area. Minimal surfaces solve the problem of minimizing the surface area in each class of surfaces. For example, in physical problems, a soap film spanning a wire frame naturally forms a minimal surface, as it minimizes the surface area for the given boundary. We went over something called ' countable additivity ' and though its implication remains the same, our application of it changes from a coordinate plane to a sphere (a soap bubble). The advantage of a sphere being Lebesgue measurable is that it can be broken down into smaller regions or sets such that countable additivity holds. One such way to think about it is to decompose the surface into two disjoint hemispheres.  Moreover, this can be applied to multiple connected spheres or in our case, bubb...

Minimal surfaces, otherwise better known in the form of a Mobius strip, sphere, or a helicoid, are inclined to provide the least surface area with respect to some constraint(s). We want to see a connection to Measure Theory, if one exists at all (and it does). To drive our motivation further, we pick the principle example of a soap bubble.  But first, what is Measure Theory? It is built on the concepts researched by Henri Lebesgue. Lebesgue measure theory consists of two key concepts used here- Lebesgue Measure and Lebesgue Integral and stands on the fundamentals of measure of sets and intervals developed by Lebesgue.  We can think of the concept of measure coming from the length between two coordinates or a length of an interval on coordinate plane such that l(I)= b-a So where is the minimal surface here? It would not precisely exist in the coordinate system as definite as above is, but I promise, we will be tempted to draw some common strings once we get acquainted with meas...