Conformal embedding of a disc with a Lorentz metric into the plane (Q1374724)

Given an open disc \({\overset \circ D}{^2}\) with a Lorentz metric \(g\), the authors seek a conformal embedding of \(({\overset \circ D}{^2},g)\) into \(E^{1,1}\) \((\mathbb{R}^2\) with coordinates \((x,y))\). The work is based on the classification theory of foliations of open discs developed by \textit{W. Kaplan} [Duke Math. J. 8, 11-46 (1941; Zbl 0025.09301)], \textit{A. Haefliger} and \textit{G. Reeb} [Enseign. Math., II. Sér. 3, 107-125 (1957; Zbl 0079.17101)]. Basic facts and definitions concerning the classification theory are recalled in Section 2 in order to establish the following result: if \(({\overset \circ D}{^2}; F_h,F_v)\) is an open disc with two transverse foliations and no short leaves, then there is a topological embedding of \({\overset \circ D}{^2}\) into the plane sending leaves of \(F_h\) (resp. \(F_v)\) to horizontal (resp. vertical) lines. A stronger version of this result is proven in Section 3. In Section 4 some examples of twice-foliated discs which cannot be embedded in the plane are given. The paper concludes with the conjecture that a twice-foliated disc \((\overset \circ {D}^2;F_h,F_v)\) embeds in the plane if and only if it has no subdomain \(U\subset {\overset\circ D}{^2}\) such that \((U;F_h |_U,f_v |_U)\) is isomorphic to one of the given examples.