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 (18th century)

Examples (19th century)

Examples (20th century)



Singly periodic

Doubly periodic

Triply periodic

Examples (21st century)


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, Fusashi Nakamura, Wayne Rossman, and Seong-Deog Yang for valuable comments about drawing minimal surfaces.