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We solve the Börling problem for constant mean curvature one surfaces in hyperbolic three-space and in de Sitter three-space. That is, we show that for any regular, analytic (and spacelike in the case of de Sitter three-space) curve γ and an analytic (timelike in the case of de Sitter three-space) unit vector field $N$ along and orthogonal to γ, there exists a unique (spacelike in the case of de Sitter three-space) surface of constant mean curvature 1 which contains γ and the unit normal of which on γ is $N$. Some of the consequences are the planar reflection principles, and a classification of rotationally invariant CMC 1 surfaces.
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참고문헌 (26)
참고문헌 신청
A. I. Bobenko / 2003 / Hyperbolic constant mean curvature one surfaces: spinor representation and trinoids in hypergeometric functions / Math. Z 245 (1) : 63 ~ 91
D. Brander / 2010 / The Bj¨orling problem for non-minimal constant mean curvature surfaces / Comm. Anal. Geom 18 (1) : 171 ~ 194
D. Brander / 2011 / Singularities of spacelike constant mean curvature surfaces in Lorentz-Minkowski space / Math. Proc. Cambridge Philos. Soc 150 (3) : 527 ~ 556
F. Mercuri / 2009 / On the Bj¨orling problem in a three-dimensional Lie group / Illinois J. Math 53 (2) : 431 ~ 440
H. L. Liu / 2004 / Weingarten rotation surfaces in 3-dimensional de Sitter space / J. Geom 79 (1-2) : 156 ~ 168
D. Brander / 2010 / The Bj¨orling problem for non-minimal constant mean curvature surfaces / Comm. Anal. Geom 18 (1) : 171 ~ 194
D. Brander / 2011 / Singularities of spacelike constant mean curvature surfaces in Lorentz-Minkowski space / Math. Proc. Cambridge Philos. Soc 150 (3) : 527 ~ 556
F. Mercuri / 2009 / On the Bj¨orling problem in a three-dimensional Lie group / Illinois J. Math 53 (2) : 431 ~ 440
H. L. Liu / 2004 / Weingarten rotation surfaces in 3-dimensional de Sitter space / J. Geom 79 (1-2) : 156 ~ 168
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