In Riemannian geometry, a Bryant surface is a 2-dimensional surface embedded in 3-dimensional hyperbolic space with constant mean curvature equal to 1.[1][2] These surfaces take their name from the geometer Robert Bryant, who proved that every simply-connected minimal surface in 3-dimensional Euclidean space is isometric to a Bryant surface by a holomorphic parameterization analogous to the (Euclidean) Weierstrass–Enneper parameterization.[3]

References

  1. ^ Collin, Pascal; Hauswirth, Laurent; Rosenberg, Harold (2001), "The geometry of finite topology Bryant surfaces", Annals of Mathematics, Second Series, 153 (3): 623–659, arXiv:math/0105265, Bibcode:2001math......5265C, doi:10.2307/2661364, JSTOR 2661364, MR 1836284, S2CID 15020316.
  2. ^ Rosenberg, Harold (2002), "Bryant surfaces", The global theory of minimal surfaces in flat spaces (Martina Franca, 1999), Lecture Notes in Math., vol. 1775, Berlin: Springer, pp. 67–111, doi:10.1007/978-3-540-45609-4_3, MR 1901614.
  3. ^ Bryant, Robert L. (1987), "Surfaces of mean curvature one in hyperbolic space", Astérisque (154–155): 12, 321–347, 353 (1988), MR 0955072.


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