Spacelike Mean Curvature 1 Surfaces of Genus 1 with Two Ends in De Sitter 3-space
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چکیده
We give a mathematical foundation for, and numerical demonstration of, the existence of mean curvature 1 surfaces of genus 1 with either two elliptic ends or two hyperbolic ends in de Sitter 3-space. An end of a mean curvature 1 surface is an “elliptic end” (resp. a “hyperbolic end”) if the monodromy matrix at the end is diagonalizable with eigenvalues in the unit circle (resp. in the reals). Although the existence of the surfaces is numerical, the types of ends are mathematically determined. Introduction The global theories of minimal surfaces in Euclidean 3-space R and constant mean curvature (CMC) 1 surfaces in hyperbolic 3-space H are well understood, as they possess representation formulas using meromorphic functions and so benefit from the theory of complex analysis. In contrast to this, the global theory of spacelike maximal surfaces in Minkowski 3-space R1 and spacelike CMC 1 surfaces in de Sitter 3-space S 3 1 are not well explored yet, even though they possess similar representation formulas. This is perhaps because the only complete spacelike maximal immersions in R1 and spacelike CMC 1 immersions in S1 are flat and totally umbilic. So to have an interesting global theory about these surfaces, we need to consider a wider class of surfaces than just complete and immersed ones. Recently, Umehara and Yamada defined such a category of spacelike maximal surfaces with certain kinds of singularities and named them “maxfaces” [UY3]. Then they constructed numerous examples by a transferring method from minimal surfaces in R. Furthermore, Kim and Yang discovered an interesting example of a maxface, which has genus 1 with two embedded ends, even though there does not exist such an example as a complete minimal immersion in R [KY]. In addition, Fernàndez and López and Souam have investigated maximal surfaces with conical singularities [FLS1, FLS2]. The author defined spacelike CMC 1 surfaces with certain kinds of singularities as an analogue of maxfaces, naming them “CMC 1 faces”, and constructed many examples by transferring from reducible CMC 1 surfaces in H [F]. Also, Lee and Yang investigated spacelike CMC 1 surfaces of genus zero with two and three ends [LY]. However, every surface constructed in [F] and [LY] was topologically a sphere with finitely many points removed. Given all of this, it is natural to consider whether or not there exist examples with positive genus. Date: May 19, 2006. 2000 Mathematics Subject Classification. 53A10, 53B30.
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تاریخ انتشار 2006