# zbMATH — the first resource for mathematics

Topological uniqueness of negatively curved surfaces. (English) Zbl 1201.53002
Summary: 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 that of 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.
##### MSC:
 53A05 Surfaces in Euclidean and related spaces 53A10 Minimal surfaces in differential geometry, surfaces with prescribed mean curvature
##### Keywords:
index method; annulus; total curvature; black hole models
Full Text:
##### References:
 [1] H. Chan, Embedding Misner and Brill-Lindquist initial data for black-hole collisions , Math. Phys. Anal. Geom. 6 (2003), 9-27. · Zbl 1070.53030 [2] H. Chan, Embedding negatively curved initial data of black-hole collisions in R 3 , Classical Quantum Gravity 23 (2006), 225-234. · Zbl 1087.83041 [3] H. Chan, Simply connected nonpositively curved surfaces in R 3 , Pacific J. Math. 233 (2006), 1-4. · Zbl 1120.53001 [4] H. Chan and A. Treibergs, Nonpositively curved surfaces , J. Differential Geom. 57 (2001), 389-407. · Zbl 1041.53001 [5] C. Connell and M. Ghomi, Topology of negatively curved real affine algebraic surfaces , J. Reine Angew. Math. 624 (2008), 1-26. · Zbl 1167.53005 [6] J. Hadamard, Les surfaces á courboures opposées et leurs lignes gé odésiques , J. Math. Pures Appl. 4 (1898), 27-73. · JFM 29.0522.01 [7] P. Hartman, Ordinary Differential Equations , New York, Wiley, 1964, 27-73. · Zbl 0125.32102 [8] F. Lopez and A. Ros, On embedded complete minimal surfaces of genus zero , J. Differential Geom. 33 (1991), 293-300. · Zbl 0719.53004 [9] R. Price and J. Romano, Embedding initial data for black-hole collisions , Classical Quantum Gravity 12 (1995), 875-893. · Zbl 0821.53067 [10] E. R. Rozendorn, Weakly irregular surfaces of negative curvature , Russian Math. Surveys 21 (1966), 57-112. · Zbl 0173.23201 [11] R. Schoen, Uniqueness, symmetry, and embeddedness of minimal surfaces , J. Differential Geom. 18 (1983), 791-809. · Zbl 0575.53037 [12] A. Verner, Topological structure of complete surfaces with nonpositive curvature which have one to one spherical mappings (in Russian), Vestnik Leningrad Univ. 20 (1965), 16-29. [13] A. Verner, Tapering saddle surfaces , Sib. Mat. Z. 11 (1968), 567-581. · Zbl 0219.53052 [14] B. White, Complete surface of finite total curvature , J. Differential Geom. 26 (1987), 315-326. · Zbl 0631.53007
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.