Minimal Surfaces

Minimal surfaces are stunning objects, and the mathematics used to understand, classify, and create new ones are equally rich and beautiful.  For my thesis, I showed how to construct a singly periodic genus one helicoid as a limit of doubly periodic genus one Scherk Surfaces, much of which was published in the Journal of Differential Geometry in 2014.

This result was obtained using the method of flat structures, a technique devised by my advisor, Dr. Mike Wolf, and one of his collaborators, Dr. Mattias Weber.  In theory, one can build minimal surfaces by specifying Weierstrass data, i.e. a triple \displaystyle \left(M, g, dh\right).  Here M is a Riemann surface, g is a meromorphic function, and dh is a holomorphic 1-form whose zeroes balance the poles of g and 1/g.  This triple will provide a minimal immersion \Phi:M \to \mathbb{R}^3 provided certain period conditions are satisfied:

\displaystyle \int_{\gamma} \frac{1}{g}\,dh = \overline{\int_{\gamma} g\,dh}

\text{Re} \int_{\gamma} \! dh = 0

for all non-trivial \gamma \in H_1(M, \mathbb{Z}).  These conditions can be difficult to satisfy, especially when the genus of M is high.

Instead, one can reverse this process, beginning with a desired minimal surface (typically one that is highly symmetric) and using its shape to determine the 1-forms gdh = \omega_1 and (1/g)dh = \omega_2.  In fact, one can associate pictures to these forms by using the developing maps

F_i(z) =  \displaystyle \int_{z_0}^z \! \omega_i\,.

When a Riemann surface, M, can be conformally immersed in \mathbb{R}^3 as a highly symmetric minimal surface each of these maps develop M into a polygonal region in \mathbb{C}; if the minimal surface is not very symmetric, some of these lines will be identified.  Because the surface is minimal, though, these \omega_1 and \omega_2 flat structures will enjoy a special relationship: images of \omega_1 geodesics under F_1 will be straight lines that are conjugate to the images of \omega_2 geodesics under F_2.

Rather than solve the aforementioned period problems, one can instead start with polygonal flat structures that enjoy this conjugate relationship.  This set-up has the first period condition assumed, but leaves one to check that the flat structures “came from the same Riemann surface.”  One then has to prove that the flat structures are conformally equivalent.  (Lastly, one also has to verify the last period condition.)