Hyperbolic Carathéodory conjecture. (English) Zbl 1163.53004
Proc. Steklov Inst. Math. 258, 178-193 (2007) and Tr. Mat. Inst. Steklova 258, 185-200 (2007).
Let \(M\) by a compact non-degenerate hyperbolic surface in projective three-space. A surface point \(m \in M\) is called quadratic, if \(M\) is exceptionally amenable to approximation by a quadric surface in \(m\). The authors prove various results related to the conjecture that a generic surface \(M\) possesses at least eight quadratic points. This is similar to a classic conjecture attributed to Carathéodory, stating that a sufficiently smooth closed convex surface in Euclidean three space has at least two umbilic points.
After introducing fundamental concepts and providing basic properties of quadratic surface points, the important notion a “generic surface” is defined.
Subsequently, perturbations of the standard hyperboloid in projective three-space by a smooth periodic function \(f(u,v)\) are studied. The perturbed surface is algebraic of degree two (up to first order) if \(f\) is a linear combination of first harmonics and of degree four (up to first order) if \(f\) is a linear combination of second harmonics. The central result states that a quadratic point \(m\) remains quadratic (up to first order) if
\[ f_{uuu} + f_u \equiv 0,\quad f_{vvv} + f_v \equiv 0 \]
holds at \(m\). The number of solutions to this system equals the number of quadratic points on a small perturbation of the standard hyperboloid. For particular cases the number of solutions can be bounded from below. Examples with at least 16, at least 32, and exactly eight solutions (the conjectured minimal number) are provided.
After introducing fundamental concepts and providing basic properties of quadratic surface points, the important notion a “generic surface” is defined.
Subsequently, perturbations of the standard hyperboloid in projective three-space by a smooth periodic function \(f(u,v)\) are studied. The perturbed surface is algebraic of degree two (up to first order) if \(f\) is a linear combination of first harmonics and of degree four (up to first order) if \(f\) is a linear combination of second harmonics. The central result states that a quadratic point \(m\) remains quadratic (up to first order) if
\[ f_{uuu} + f_u \equiv 0,\quad f_{vvv} + f_v \equiv 0 \]
holds at \(m\). The number of solutions to this system equals the number of quadratic points on a small perturbation of the standard hyperboloid. For particular cases the number of solutions can be bounded from below. Examples with at least 16, at least 32, and exactly eight solutions (the conjectured minimal number) are provided.
Reviewer: Hans-Peter Schröcker (Innsbruck)
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