In his dissertation from 1993, Meinhard Wohlgemuth constructs the a 4-ended embedded minimal surface of genus 2, which he calls CSSCFF. This was the first complete embedded minimal surface with four ends, generalizing Costa’s surface

Costa-Wohlgemuth Surface

It has two catenoidal and two planar ends and the coordinate planes as symmetry planes. The existence proof is involved, it requires to solve a 2-dimensional period problem. An embeddedness proof is still lacking. 

Dihedralizations also exist, with the same type of ends but higher genus. The original surface for genus 2 is a borderline case of the Hoffman-Meeks conjecture: A complete, embedded minimal surface of finite total curvature with n ends must have genus at least n-2.

It is possible to deform the inner planar ends into catenoidal ends, that will then eventual intersect the outer catenoidal ends, in contrast to the Costa-Hoffman-Meeks family, which is embedded throughout.

Resources

Mathematica Notebook and CDF

PoVRay Sources

M. Wohlgemuth: Vollständige Minimalflächen höheren Geschlechts und endlicher Totalkrümmung, Dissertation, Bonn, 1993.

M. Wohlgemuth: Minimal Surfaces of Higher Genus with Finite Total Curvature, Arch. Rational Mech. Anal. 137 (1997) 1–25

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