In this paper we will prove the Calabi-Yau conjectures for embedded surfaces (i.e., surfaces without self-intersection). In fact, we will prove considerably more. The heart of our argument is very general and should apply to a variety of situations, as will be more apparent once we describe the main steps of the proof later in the introduction.
Annals of Mathematics
This paper is the third in a series where we describe the space of all embedded minimal surfaces of ﬁxed genus in a ﬁxed (but arbitrary) closed 3-manifold. In [CM3]–[CM5] we describe the case where the surfaces are topologically disks on any ﬁxed small scale. Although the focus of this paper, general planar domains, is more in line with [CM6], we will prove a result here (namely, Corollary III.
This paper is the ﬁrst in a series where we describe the space of all embedded minimal surfaces of ﬁxed genus in a ﬁxed (but arbitrary) closed Riemannian 3-manifold. The key for understanding such surfaces is to understand the local structure in a ball and in particular the structure of an embedded minimal disk in a ball in R3 (with the ﬂat metric). This study is undertaken here and completed in [CM6]. These local results are then applied in [CM7] where we describe the general structure of ﬁxed genus surfaces in 3-manifolds. There are two local models for...
The space of embedded minimal surfaces of ﬁxed genus in a 3-manifold II; Multi-valued graphs in disks
By Tobias H. Colding and William P. Minicozzi II*
0. Introduction This paper is the second in a series where we give a description of the space of all embedded minimal surfaces of ﬁxed genus in a ﬁxed (but arbitrary) closed 3-manifold. The key for understanding such surfaces is to understand the local structure in a ball and in particular the structure of an embedded minimal disk in a ball in R3 . We show here that if the curvature of such a disk...
PVC tubing, creating the effect of diaphanous
surfaces of flowing plastic hair that create shade and
accommodate program. The sensuous lines are a
constructive solution that cumulatively define the
larger surfaces and representationally echo the digital
method that made them. That is, the lines define
the physical surface in the same way that embedded
surface curves, or isoparms, make up a digitally ruled
or lofted, one.
Massie's method coordinates well with
conventional building materials.
This paper is the fourth in a series where we describe the space of all embedded minimal surfaces of ﬁxed genus in a ﬁxed (but arbitrary) closed 3manifold. The key is to understand the structure of an embedded minimal disk in a ball in R3 . This was undertaken in [CM3], [CM4] and the global version of it will be completed here; see the discussion around Figure 12 for the local case and [CM15] for some more details. Our main results are Theorem 0.1 (the lamination theorem) and Theorem 0.2 (the one-sided curvature estimate). ...
In this paper we study some properties of reducible surfaces, in particular of unions of planes. When the surface is the central ﬁbre of an embedded ﬂat degeneration of surfaces in a projective space, we deduce some properties of the smooth surface which is the general ﬁbre of the degeneration from some combinatorial properties of the central ﬁbre. In particular, we show that there are strong constraints on the invariants of a smooth surface which degenerates to conﬁgurations of planes with global normal crossings or other mild singularities. ...
We give a complete topological classiﬁcation of properly embedded minimal surfaces in Euclidian three-space. 1. Introduction In 1980, Meeks and Yau  proved that properly embedded minimal surfaces of ﬁnite topology in R3 are unknotted in the sense that any two such homeomorphic surfaces are properly ambiently isotopic. Later Frohman  proved that any two triply periodic minimal surfaces in R3 are properly ambiently isotopic.
The first block is responsible for receiving and amplifying the brain signal, allocating
electrodes into specific places on the scalp in the case of the use of electrodes on the surface, or inside
brain in the intracortical use cases, in the second block signal is sampled, the quantity
and periodic system of time to digitize it, to simplify the following
digitized phase signal can be filtered, for example to reduce the noise level is
Better SNR or signal identification and processing artifacts....
Porcine reproductive and respiratory syndrome (PRRS) is one of the most significant swine diseases
worldwide. Despite its relevance, serum biomarkers associated with early-onset viral infection, when clinical signs
are not detectable and the disease is characterized by a weak anti-viral response and persistent infection, have not
yet been identified. Surface-enhanced laser desorption ionization time of flight mass spectrometry (SELDI-TOF MS)
is a reproducible, accurate, and simple method for the identification of biomarker proteins related to disease in
In this chapter, the analytical embedded atom method and calculating Gibbs free energy
method are introduced briefly. Combining these methods with molecular dynamic and
Monte Carlo techniques, thermodynamics of nano-silver and alloy particles have been
The correctness of many systems and devices in our modern society depends not only on the effects or results they produce but also on the time at which these results are produced. These real-time systems range from the anti-lock braking controller in automobiles to the vital-sign monitor in hospital intensive-care units.
We study random surfaces which arise as height functions of random perfect matchings (a.k.a. dimer conﬁgurations) on a weighted, bipartite, doubly periodic graph G embedded in the plane. We derive explicit formulas for the surface tension and local Gibbs measure probabilities of these models. The answers involve a certain plane algebraic curve, which is the spectral curve of the Kasteleyn operator of the graph. For example, the surface tension is the Legendre dual of the Ronkin function of the spectral curve.
In this paper we will discuss the geometry of ﬁnite topology properly embedded minimal surfaces M in R3 . M of ﬁnite topology means M is homeomorphic to a compact surface M (of genus k and empty boundary) minus a ﬁnite number of points p1 , ..., pj ∈ M , called the punctures. A closed neighborhood E of a puncture in M is called an end of M . We will choose the ends suﬃciently small so they are topologically S 1 × [0, 1) and hence, annular. We remark that M is orientable since M is properly...
(BQ) Machining speed and surface integrity continue to be issues of focus in current wire EDM research. In this light, the proof-of-concept of a hybrid wire EDM process that utilizes a wire embedded with electrically non-conducting abrasives is presented. Material removal in this novel process is realized through electrical erosion that is augmented by two-body abrasion. This is shown to bring about a significant improvement in the removal rate and generate surfaces with minimal recast material, in comparison to an equivalent wire EDM process.