8,135 research outputs found
On the number of minimal surfaces with a given boundary
We generalize the following result of White: Suppose is a compact,
strictly convex domain in \RR^3 with smooth boundary. Let be a
compact 2-manifold with boundary. Then a generic smooth curve in bounds an odd or even number of embedded
minimal surfaces diffeomorphic to according to whether is or
is not a union of disks. First, we prove that the parity theorem holds for any
compact riemannian 3-manifold such that is strictly mean convex, is
homeomorphic to a ball, is smooth, and contains no closed
minimal surfaces. We then further relax the hypotheses by allowing 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- 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
Embedded minimal ends asymptotic to the helicoid
The ends of a complete embedded minimal surface of {\em finite total
curvature} are well understood (every such end is asymptotic to a catenoid or
to a plane). We give a similar characterization for a large class of ends of
{\em infinite total curvature}, showing that each such end is asymptotic to a
helicoid. The result applies, in particular, to the genus one helicoid and
implies that it is embedded outside of a compact set in
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
Sparse Stochastic Inference for Latent Dirichlet allocation
We present a hybrid algorithm for Bayesian topic models that combines the
efficiency of sparse Gibbs sampling with the scalability of online stochastic
inference. We used our algorithm to analyze a corpus of 1.2 million books (33
billion words) with thousands of topics. Our approach reduces the bias of
variational inference and generalizes to many Bayesian hidden-variable models.Comment: Appears in Proceedings of the 29th International Conference on
Machine Learning (ICML 2012
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