Mathematical Physics and Astrophysics

# Research on Surface with Constant Mean Curvature -Why are bubbles in a spherical shape?

Shinpei Kobayashi , Associate Professor

Faculty of Science, Graduate School of Science

High school : Osaka Prefectural Ikeda High School

Academic background : Graduate School of Science and Technology, Kobe University

- Research areas
- Mathematics
- Research keywords
- Differential Geometry, Integrable Systems
- Website
- https://sites.google.com/site/kobayashishimpeisite/home

## Why are Bubbles in a Spherical Shape?

Bubbles in the Natural World

You probably made bubbles from soapy water and played with them when you were kid. So, why are bubbles in a spherical shape? Why aren’t they in a cubical or rugby-ball shape? There is a mathematical reason hidden in the answer. Actually, bubbles have a surface with a minimum surface area under the condition of a constant enclosed volume. We can say that bubbles have the most efficient shape. This type of surface with a minimum surface area is called a surface with constant mean curvature. There are so many similar things in the world with this surface other than a bubble shape.

The concept of extreme values is required to mathematically understand the surface with constant mean curvature (around which the minimum or the maximum is achieved). An extreme value of a given function can be determined by finding the point where the derivative of the function is equal to zero (0) and investigating the behavior of the function around the point (or finding if twice-differentiation gives positive or negative value at the point). The surfaces with constant mean curvature in the real world (bubbles) can be expressed as a surface which gives an extreme value of a functional as the generalization of an extreme value of a function. This type of problem is called a variational problem.

## Is Research on Surfaces with Constant Mean Curvature Helpful in the Real World?

Yes, it is. One might think finding the theory behind the spherical shape of bubbles is useless, so let’s look at an example that is useful in another area. In the field of chemistry, there are polymer compounds made through the covalent binding of different types of polymers called block copolymers. The surface of the boundary of these block copolymers is known to become a surface with constant mean curvature under certain conditions (such as temperature or blend ratio). For example, there is a triply periodic surface called a gyroid. In addition, the application of a surface with constant mean curvature to architectural structures is popular (roof of the Olympic Park in Munich). In this way, surfaces with constant mean curvature have been applied to various areas and have been the focus of research.

## Are There Unknowns Even after Hundreds of Years of Research?

Unfortunately yes, there are many. If anything, I would say that we hardly know anything. For example, let’s think about the following question which was submitted by a Swiss mathematician Heinz Hopf in 1956.

“Is a finite and closed surface with constant mean curvature limited to spherical surfaces (bubbles)?”

In the era when this question was submitted, mathematicians thought the answer was absolutely yes. This is because the following two facts are known: “A closed surface without holes with constant mean curvature is limited to a spherical surface (H. Hopf).” “A closed curve surface with constant mean curvature which doesn’t cross itself is limited to a spherical surface (A. D. Aleksandrov).” However, in 1984, Henry C. Wente found a new surface with constant mean curvature and solved this problem with a negative answer. That surface is a surface with constant mean curvature with one hole which crosses itself (called Wente torus).

## How Do You Do Research?

It’s not that I go out and collect bubbles around me (I might amuse myself by making bubbles though.). Mathematicians mainly conduct calculations (or logical reasoning) using sheets of paper and pencils.

As mentioned above, the research on the surface with constant mean curvature has a long history, studied intensely by notable mathematicians, with only difficult problems left.

However, recently a new method has been developed to investigate the surface with constant mean curvature. This is called the method of integrable system. Let me explain it a little. I mentioned that a variational problem is the generalization of an extreme value of a function. The extreme value of a function requires finding the point where the derivative of a function is equal to zero (0), which can be generalized as finding the point where the derivative of a functional is equal to zero (0).

In fact, the point where the derivative of a functional is equal to zero (0) is related to a differential equation called Euler-Lagrange equation (A differential equation means the equality formulated between functions and the differentials. It was discovered by noted Swiss mathematician L. Euler). Consequently, investigating a surface with constant mean curvature requires solving the differential equation. Generally, differential equations are very difficult to solve, or in some cases impossible. A method of integrable system can provide sophisticated solutions for certain kinds of differential equations. This method can be applied to the differential equation for a surface with constant mean curvature, and it facilitates better understanding on various aspects of this subject.

## References

(1) Hildebrandt and Tromba, “Rule of Shape – Shape and Pattern in the Natural World - (Katachi no Hosoku – Shizenkai no Katachi to Pattern –),” Tokyo Kagaku Dojin.

(Original: Stefan Hildebrandt and Anthony J. Tromba, Mathematics and Optimal Form: Coll.Pa (Scientific American Library), December 1, 1984.)

Katsuei Kenmotsu “Lecture on Theory of Surfaces: Introduction to Surface with Constant Mean Curvature (Kyokumenron Kogi Heikin Kyokuritsu Itteikyokumen Nyumon),” Bifukan.