## Duke Mathematical Journal

### Symmetry of embedded genus 1 helicoids

#### Abstract

In this article, we use the Lopez-Ros deformation to show that any embedded genus $1$ helicoid (or “genus-one helicoid”) must be symmetric with respect to rotation by $180^\circ$ around a normal line. This partially answers a conjecture of Bobenko. We also show that this symmetry holds for an embedded genus $k$ helicoid $\Sigma$, provided that the underlying conformal structure of $\Sigma$ is hyperelliptic.

#### Article information

Source
Duke Math. J. Volume 159, Number 1 (2011), 83-97.

Dates
First available in Project Euclid: 11 July 2011

https://projecteuclid.org/euclid.dmj/1310416363

Digital Object Identifier
doi:10.1215/00127094-1384791

Mathematical Reviews number (MathSciNet)
MR2817649

Zentralblatt MATH identifier
1228.53008

#### Citation

Bernstein, Jacob; Breiner, Christine. Symmetry of embedded genus 1 helicoids. Duke Math. J. 159 (2011), no. 1, 83--97. doi:10.1215/00127094-1384791. https://projecteuclid.org/euclid.dmj/1310416363.

#### References

• A. Alarcón, L. Ferrer, and F. Martí N, A uniqueness theorem for the singly periodic genus-one helicoid, Trans. Amer. Math. Soc. 359, no. 6 (2007), 2819–2829.
• J. Bernstein and C. Breiner, Conformal structure of minimal surfaces with finite topology, Comment. Math. Helv. 86 (2011), 353–381.
• A. I. Bobenko, Helicoids with handles and Baker-Akhiezer spinors, Math. Z. 229 (1998), 9–29.
• H. M. Farkas and I. Kra, Riemann Surfaces, 2nd ed., Grad. Texts in Math. 71, Springer, New York, 1992.
• L. Hauswirth, J. Pérez, and P. Romon, Embedded minimal ends of finite type, Trans. Amer. Math. Soc. 353, no. 4 (2001), 1335–1370.
• D. Hoffman, H. Karcher, and F. Wei, The singly periodic genus-one helicoid, Comment. Math. Hel. 74 (1999), 248–279.
• D. Hoffman and B. White, Genus-one helicoids from a variational point of view, Comment. Math. Helv. 83 (2008), 767–813.
• F. J. López and A. Ros, On embedded complete minimal surfaces of genus zero, J. Differential Geom. 33 (1991), 293–300.
• W. H. Meeks Iii, and H. Rosenberg, The geometry of periodic minimal surfaces, Comment. Math. Helv. 68 (1993), 538–578.
• J. Pérez and A. Ros, Some uniqueness and nonexistence theorems for embedded minimal surfaces, Math. Ann. 295 (1993), 513–525.
• B. Solomon and B. White, A strong maximum principle for varifolds that are stationary with respect to even parametric elliptic functionals, Indiana Univ. Math. J. 38 (1989), 683–691.
• M. Weber, D. Hoffman, and M. Wolf, An embedded genus-one helicoid, Ann. of Math. (2) 169 (2009), 347–448.
• B. White, Existence of smooth embedded surfaces of prescribed genus that minimize parametric even elliptic functionals on $3$-manifolds, J. Differential Geom. 33 (1991), 413–443.