Premium
Boundary value problems for surfaces of constant Gauss Curvature
Author(s) -
Hoffman David,
Rosenberg Harold,
Spruck Joel
Publication year - 1992
Publication title -
communications on pure and applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 3.12
H-Index - 115
eISSN - 1097-0312
pISSN - 0010-3640
DOI - 10.1002/cpa.3160450807
Subject(s) - mathematics , dirichlet problem , order (exchange) , gauss , dirichlet distribution , constant (computer programming) , boundary value problem , mathematical analysis , calculus (dental) , algebra over a field , pure mathematics , computer science , physics , medicine , dentistry , programming language , finance , quantum mechanics , economics
The compact smooth surfaces in 3 with constant positive Gauss Curvature (Ksurfaces) form a natural class. A K-surface without boundary is itself the boundary of a convex body, so it must be embedded. The surfaces of interest to us have non-empty boundary and so are not necessarily embedded. A fundamental question is this: given a collection γ = {C1 . . . , Cn} of Jordan curves in , what are the K-surfaces with boundary γ? For example, if γ is a single Jordan curve with no inflection points, does γ bound a K-surface? When γ is a single planar Jordan curve, the simplest case, a great deal can be said. We begin our discussion of this case by recalling several elementary facts. Let P be a plane and S a smooth surface in . Let γ be a component of S ∩ P that is not a point. The normal curvature (up to sign) of S ∩ P is the projection of the curvature vector of S ∩P , considered as a curve in P , onto the normal line of S at P . This means that an inflection point of γ ⊂ P ∩ S will have an asymptotic direction Research supported in part by NSF grants DMS-900083, DMS-8802858 and DOE grant DEFG02-86ER250125.