# The computer-aided discovery of new embedded minimal surfaces

@article{Hoffman1987TheCD, title={The computer-aided discovery of new embedded minimal surfaces}, author={David Hoffman and Henri Matisse}, journal={The Mathematical Intelligencer}, year={1987}, volume={9}, pages={8-21} }

In 1984, Bill Meeks and I established the existence of an infinite family of complete embedded minimal surfaces in R 3. For each k > 0, there exists an example which is homeomorphic to a surface of genus k from which three points have been removed. Figure 30-1 is a picture of the genus-one example. The equations for this remarkable surface were established by Celsoe Costa in his thesis, but they were so complex that the underlying geometry was obscured. We used the computer to numerically…

## 64 Citations

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A summary of the computer-aided discoveries in minimal surface theory is given, including a new example of a complete minimal surface of finite total curvature found in 1982 and later proved to be properly embedded.

### Complete embedded minimal surfaces of finite total curvature

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### Properties of properly embedded minimal surfaces of finite topology

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Until the recent discovery of a sequence of properly embedded minimal surfaces with finite topology (Hoffman [4, 5]; Hoffman and Meeks [6, 7]), the only known examples were the plane, the catenoid…

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### From Sketches to Equations to Pictures: Minimal Surfaces and Computer Graphics

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In 1984, mathematician David Hoffman and computer scientist Jim Hoffman sat before a computer screen and saw a crude rendition of a new minimal surface unfold itself and it was proved that this surface, which had been discovered by C. Costa, was embedded.

### Loop Group Methods for Constant Mean Curvature Surfaces

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This is an elementary introduction to a method for studying harmonic maps into symmetric spaces, and in particular for studying constant mean curvature (CMC) surfaces, that was developed by J.…

### Computing minimal surfaces with differential forms

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A new algorithm is described that solves a classical geometric problem: Find a surface of minimal area bordered by an arbitrarily prescribed boundary curve using the Alternating Direction Method of Multiplier (ADMM) to find global minimal surfaces.

### Computing minimal surfaces with differential forms

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A new algorithm is described that solves a classical geometric problem: Find a surface of minimal area bordered by an arbitrarily prescribed boundary curve using the Alternating Direction Method of Multiplier to find global minimal surfaces.

### About the cover: Early images of minimal surfaces

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The survey by Meeks and Pérez in this issue opens with a score of images of complete minimal surfaces in space; pictures like these powerfully and succinctly communicate the subtle richness of the…

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We will survey what is known about minimal surfaces S ⊂ R 3, which are complete, embedded, and have finite total curvature: \(\int_s {|K|} dA < \infty \). The only classically known examples of such…

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