Embedded minimal surfaces of finite total curvature

Here are computer generated images of complete, embedded minimal surfaces of finite total curvature. These surfaces are computed by opening the nodes on a Riemann surface with nodes. This implements an abstract construction developped in the paper An embedded minimal surface with no symmetries, Journal of Diff. Geom. 60, p 103--153. More details about how these surfaces are computed may be found in Exploring the space of embedded minimal surfaces of finite total curvature, Experimental Mathematics 17:2, p 205--221. The data to construct the noded Riemann surface is a configuration of points in the plane. The configurations are represented on the right. Roughly speaking, each point gives the position of a catenoidal neck.

A Costa Hoffman Meeks surface (genus one, three ends). I was able to push the parameter quite far. Note that our surfaces are computed without taking advantage of the symmetries.

Use the mouse to rotate, and the mouse with SHIFT pressed to scale. Type "s" to get a stereographic pair.

This is the configuration used to construct the noded Riemann surface. The symbols , and (on the following examples) represent the level of the neck, from top to bottom.

A genus 4 example with four ends and three vertical planes of symmetry. Configuration of type 1,3,3 with order 3 dihedral symmetry. This is probably in the same family as some examples constructed by M. Weber and M. Wolf.

This example has the same number of necks at each level as the previous one, but is less symetric. It has only one vertical plane of symmetry. Note : the necks moved quite a bit while opening the nodes.

Another example with only one vertical plane of symmetry. This example has four ends and genus 5. Configuration of type 1,4,3.

An example with four ends, genus 7 and with no symmetries at all. I was not able to push the parameter as far as in the previous examples. Configuration of type 1,5,4. All examples on this page are embedded : the logarithmic growths of the ends have the right ordering.