All Chen-Gackstatter surfaces of genus g can be symmetrized, increasing both the genus and the winding number of the ends. In general, this requires solving a (g-1)-dimensional period problem.
Besides the existence, not much is known. Do they converge for fixed symmetry and increasing genus to the symmetric Karcher-Scherk surfaces? Can one add vertical handles as in the toroidal Karcher-Scherk surfaces?
2 thoughts on “Symmetrized Chen-Gackstatter (g=2n)”
Thanks for the interest. I don’t use facebook, so I reply here. There are several layers of “payoff”, depending on your definition of payoff.
The first layer has immediate applications: Minimal surfaces have been used to design tubes to insert into cells in order to transfer certain molecules in and out. This, I am told, has improved the success rate by several order of magnitudes. Material scientists have observed minimal surfaces as interfaces between copolymers. Theoretical insight suggests what other interfaces might be possible, leading to new materials.
The second layer is less immediate. For the research mathematician, one reason why minimal surfaces are interesting is that they are governed by differential equations that are at the borderline of being well understood, and there are tools from many different areas available to research them. Thus theoretical progress for minimal surfaces has the potential to lead to progress for more general types of equations, some of which might have real world significance (structure stability, weather prediction etc).
The third layer is generic for all fundamental research. We try to understand. We are not just trying to answer question, we try to find the right questions that have meaningful answers. Such progress is rare, but immensely significant as it changes our views of the world, of science, what we can understand, and how we teach it.
Let me give you an example. One of the most famous mathematicians of all time, Carl Friedrich Gauss, spent a decade of his life walking through the countryside in order to produce an accurate map for his government (for military purposes, I suppose). During that period, his restless mind produced insights about curved surfaces that had little value for the task at hand. It led, however, his student Bernhard Riemann to develop an even more abstract, streamlined version of this, now called Riemannian geometry. Its concepts are at the foundation of the theory of relativity, which not only reshaped our understanding of physics, but also brought us payoffs like GPS.
Does every piece of pure mathematics lead to revolutions in the future? Certainly not. We try a thousand things, just to have one success.
I’ve posted several of your blog post like the one above on my FB group. The group is eye-candy and visual inspiration for science fiction authors. Your posts are fascinating, but being a layman I’m often entirely lost. So I’ve asked a question in my group and thought I’d ask for your insight. The question and the group below:
“I’m always curious what, if any, is the payoff for this type of math/geometry? Is this something that has real-world applications? Or is this the type of abstraction that we have to wait millennia before it’s applicable?
What say you?”