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Topological uniqueness of negatively curved surfaces

Published online by Cambridge University Press:  11 January 2016

Hsungrow Chan
Hsungrow Chan*
Affiliation:
National Pingtung University of Education, Pingtung 900-03, Taiwan, hchan@mail.npue.edu.tw
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Abstract

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In this paper we consider complete, noncompact, negatively curved surfaces that are twice continuously differentiably embedded in Euclidean three-space, showing that if such surfaces have square integrable second fundamental form, then their topology must, by the index method, be an annulus. We then show how this relates to some minimal surface theorems and has a corollary on minimal surfaces with finite total curvature. In addition, we discuss, by the index method, the relation between the topology and asymptotic curves. Finally, we apply the results yielded to the problem of isometrical immersions into Euclidean three-space of black hole models.

Keywords

0240G0240M0240E0470B
Type
Research Article
Information
Nagoya Mathematical Journal , Volume 199 , September 2010 , pp. 137 - 149
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 2010

References

[1] Chan, H., Embedding Misner and Brill-Lindquist initial data for black-hole collisions, Math. Phys. Anal. Geom. 6 (2003), 9–27.Google Scholar
[2] Chan, H., Embedding negatively curved initial data of black-hole collisions in R3 , Classical Quantum Gravity 23 (2006), 225–234.Google Scholar
[3] Chan, H., Simply connected nonpositively curved surfaces in R3 , Pacific J. Math. 233 (2006), 1–4.Google Scholar
[4] Chan, H. and Treibergs, A., Nonpositively curved surfaces, J. Differential Geom. 57 (2001), 389–407.CrossRefGoogle Scholar
[5] Connell, C. and Ghomi, M., Topology of negatively curved real affine algebraic surfaces, J. Reine Angew. Math. 624 (2008), 1–26.Google Scholar
[6] Hadamard, J., Les surfaces á courboures opposées et leurs lignes gé odésiques, J. Math. Pures Appl. 4 (1898), 27–73.Google Scholar
[7] Hartman, P., Ordinary Differential Equations, New York, Wiley, 1964, 27–73.Google Scholar
[8] Lopez, F. and Ros, A., On embedded complete minimal surfaces of genus zero, J. Differential Geom. 33 (1991), 293–300.Google Scholar
[9] Price, R. and Romano, J., Embedding initial data for black-hole collisions, Classical Quantum Gravity 12 (1995), 875–893.Google Scholar
[10] Rozendorn, E. R., Weakly irregular surfaces of negative curvature, Russian Math. Surveys 21 (1966), 57–112.Google Scholar
[11] Schoen, R., Uniqueness, symmetry, and embeddedness of minimal surfaces, J. Differential Geom. 18 (1983), 791–809.Google Scholar
[12] Verner, A., Topological structure of complete surfaces with nonpositive curvature which have one to one spherical mappings (in Russian), Vestnik Leningrad Univ. 20 (1965), 16–29.Google Scholar
[13] Verner, A., Tapering saddle surfaces, Sib. Mat. Z. 11 (1968), 567–581.Google Scholar
[14] White, B., Complete surface of finite total curvature, J. Differential Geom. 26 (1987), 315–326.Google Scholar