## Minimal Surfaces in $\mathbb{R}^3$

### Weierstrass Representation

Let $M$ be a Riemann surface and $(g,\eta)$ a pair of meromorphic function and holomorphic 1-form on $M$ such that \[ \big(1+|g|^2\big)^2\eta\overline{\eta} \] gives a Riemannian metric on $M$. Then \[ f(p)=\mathrm{Re}\int_{p_0}^p\big(1-g^2,\,i(1+g^2),\,2g\big)\eta \] gives a conformal minimal immersion (possibly multi-valued on $M$) into $\mathbb{R}^3$, where $p_0$ is a fixed point on $M$.

### Examples

- Enneper surface.
- Catenoid.
- Helicoid (singly periodic).
- Scherk's first surface (doubly periodic).
- Scherk's second surface (associate surface of catenoid).
- Scherk's third surface (branched).
- Scherk's fourth surface (branched, singly periodic).
- Scherk's fifth surface (singly periodic).
- Jorge-Meeks $n$-noid.
- Moebius strip by Meeks and Oliveira.
- Another Moebius strip (cf. Meeks-Weber).
- Double Enneper of genus $k$.
- Costa-Hoffman-Meeks surface of genus $k$.
- Fujimori-Shoda of genus $k$.
- This website is no longer updated. Find more examples at my current website.

### Acknowledgements

This site is inspired by Minimal Surface Museum and Minimal Surface Repository maintained by Matthias Weber. I thank him, and I also thank Peter Connor, Shimpei Kobayashi, Wayne Rossman, and Seong-Deog Yang for valuable comments about drawing minimal surfaces.

### Links

- Minimal Surface Repository by Matthias Weber
- Gallery of Minimal Surfaces by Sisto Baldo
- My old minimal surface gallery