On the number of minimal surfaces with a given boundary

We generalize the following result of White: Suppose $N$ is a compact, strictly convex domain in \RR^3 with smooth boundary. Let $\Sigma$ be a compact 2-manifold with boundary. Then a generic smooth curve $\Gamma\cong \partial\Sigma$ in $\partial N$ bounds an odd or even number of embedded minimal surfaces diffeomorphic to $\Sigma$ according to whether $\Sigma$ is or is not a union of disks. First, we prove that the parity theorem holds for any compact riemannian 3-manifold $N$ such that $N$ is strictly mean convex, $N$ is homeomorphic to a ball, $\partial N$ is smooth, and $N$ contains no closed minimal surfaces. We then further relax the hypotheses by allowing $N$ to be weakly mean convex and to have piecewise smooth boundary. We extend the parity theorem yet further by showing that, under an additional hypothesis, it remains true for minimal surfaces with prescribed symmetries. The parity theorems are used in an essential way to prove the existence of embedded genus-$g$ helicoids in \SS^2\times \RR. We give a very brief outline of this application. (The full argument will appear elsewhere.)Comment: 13 pages Dedicated to Jean Pierre Bourguignon on the occasion of his 60th birthday. One tex 'newcommand' revised because arxiv version had an error. Two illustrations and one proof have been added. May 2009: Abstract, key words, MSC codes added. One typo fixed. Paper has been published in Asterisqu

Genus-One Helicoids from a Variational Point of View

We prove by variational means the existence of a complete, properly embedded, genus-one minimal surface in R^3 that is asymptotic to a helicoid at infinity. We also prove existence of surfaces that are asymptotic to a helicoid away from the helicoid's axis, but that have infinitely many handles arranged periodically along the axis. Finally, we prove some new properties of such helicoid-like surfaces.Comment: 36 pages, 5 figures. Revised version: typos corrected, references added, proof of Thm 6.1 made more self-contained, several paragraphs added to the proof of Theorem 6.

The Geometry of Genus-One Helicoids

We prove: a properly embedded, genus-one minimal surface that is asymptotic to a helicoid and that contains two straight lines must intersect that helicoid precisely in those two lines. In particular, the two lines divide the surface into two connected components that lie on either side of the helicoid. We prove an analogous result for periodic helicoid-like surfaces. We also give a simple condition guaranteeing that an immersed minimal surface with finite genus and bounded curvature is asymptotic to a helicoid at infinity.Comment: 22 pages. This updated version (Apr 17, 2009) contains a much simplified statement and proof of Lemma 3.2. This version will appear in Comm. Math. Hel

Helicoidal minimal surfaces of prescribed genus, I

For every genus g, we prove that S^2 x R contains complete, properly embedded, genus-g minimal surfaces whose two ends are asymptotic to helicoids of any prescribed pitch. We also show that as the radius of the S^2 tends to infinity, these examples converge smoothly to complete, properly embedded minimal surfaces in Euclidean 3-space R^3 that are helicoidal at infinity. In a companion paper, we prove that helicoidal surfaces in R^3 of every prescribed genus occur as such limits of examples in S^2 x R.Comment: 53 pages, 5 figure

MediaCommons: Social Networking Tools for Digital Scholarly Communication

New York University, working with the Institute for the Future of the Book, seeks Level II funding in order build a working prototype of a set of networking tools that will serve as the membership system for MediaCommons, an all-electronic scholarly publishing network in the digital humanities. This set of tools, which one might imagine as bringing together the functionalities of e-portfolio software, social networking systems, and electronic publishing platforms, will enable the users of MediaCommons to find one another, collaborate, and disseminate their work in new ways. Within this social network, scholars would be able to make available a wide range of their work, including published texts ranging from the monograph to the article, works-in-progress, blogs and other more informal online writing, and other activities that often go unnoticed as forms of scholarly production, such as reviews of other scholars' work, as well as syllabi and other teaching resources