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

Homotopical Adjoint Lifting Theorem

This paper provides a homotopical version of the adjoint lifting theorem in category theory, allowing for Quillen equivalences to be lifted from monoidal model categories to categories of algebras over colored operads. The generality of our approach allows us to simultaneously answer questions of rectification and of changing the base model category to a Quillen equivalent one. We work in the setting of colored operads, and we do not require them to be $\Sigma$-cofibrant. Special cases of our main theorem recover many known results regarding rectification and change of model category, as well as numerous new results. In particular, we recover a recent result of Richter-Shipley about a zig-zag of Quillen equivalences between commutative $H\mathbb{Q}$-algebra spectra and commutative differential graded $\mathbb{Q}$-algebras, but our version involves only three Quillen equivalences instead of six. We also work out the theory of how to lift Quillen equivalences to categories of colored operad algebras after a left Bousfield localization.Comment: This is the final, journal versio

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.