TAILIEUCHUNG - Đề tài " The topological classification of minimal surfaces in R3 "

We give a complete topological classification of properly embedded minimal surfaces in Euclidian three-space. 1. Introduction In 1980, Meeks and Yau [15] 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 [6] proved that any two triply periodic minimal surfaces in R3 are properly ambiently isotopic. | Annals of Mathematics The topological classification of minimal surfaces in R3 By Charles Frohman and William H. Meeks III Annals of Mathematics 167 2008 681 700 The topological classification of minimal surfaces in R3 By Charles Frohman and William H. Meeks III Abstract We give a complete topological classification of properly embedded minimal surfaces in Euclidian three-space. 1. Introduction In 1980 Meeks and Yau 15 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 6 proved that any two triply periodic minimal surfaces in R3 are properly ambi-ently isotopic. More recently Frohman and Meeks 9 proved that a properly embedded minimal surface in R3 with one end is a Heegaard surface in R3 and that Heegaard surfaces of R3 with the same genus are topologically equivalent. Hence properly embedded minimal surfaces in R3 with one end are unknotted even when the genus is infinite. These topological uniqueness theorems of Meeks Yau and Frohman are special cases of the following general classification theorem which was conjectured in 9 and which represents the final result for the topological classification problem of properly embedded minimal surfaces in R3 . The space of ends of a properly embedded minimal surface in R3 has a natural linear ordering up to reversal and the middle ends in this ordering have a parity even or odd see Section 2 . Theorem Topological Classification Theorem for Minimal Surfaces . Two properly embedded minimal surfaces in R3 are properly ambiently isotopic if and only if there exists a homeomorphism between the surfaces that preserves the ordering of their ends and preserves the parity of their middle ends. This material is based upon work for the NSF by the first author under Award No. DMS-0405836 and by the second author under Award No. DMS-0703213. Any opinions findings and conclusions or recommendations .

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