부산대학교 도서관

The space $\widetilde{\roman{PSL}_2(\Bbb{R})}$ is a homogeneous Riemannian 3-manifold with isometry group of dimension 4. It admits a Riemannian submersion over the hyperbolic plane $\Bbb{H}^2$ with constant bundle curvature, and the fibers of the submersion are the integral curves of a unit Killing vector field. Given a domain $D\subset\Bbb{H}^2$, a minimal graph over $D$ is a minimal surface in $\widetilde{\roman{PSL}_2(\Bbb{R})}$ that is also a section of the submersion over $D$. \par In this paper, the author considers an unbounded domain $D\subset\Bbb{H}^2$ whose boundary is piecewise smooth and consists of geodesic arcs as well as regular curves which are convex with respect to the interior of the domain. The main objective is to establish necessary and sufficient conditions on $D$ for there to exist a minimal graph over $D$ with prescribed (finite or infinite) boundary data. It is proved by a flux argument that if the graph has boundary values $\pm\infty$ along a curve in $\partial D$, then such a curve is a geodesic arc. Hence the components of $\partial D$ are divided into two families of geodesic arcs, $\{A_i\}$ and $\{B_j\}$, where the graph takes values $+\infty$ and $-\infty$, respectively, and a family of convex curves $\{C_k\}$, with prescribed continuous data. No two elements of $\{A_i\}$ (resp. $\{B_j\}$) are allowed to have a common endpoint by the maximum principle. \par A polygon $\scr{P}$ inscribed in $D$ is a geodesic polygon $\scr{P}\subset\overline D$ with finitely many sides, and such that the vertices of $\scr{P}$ are also vertices of $\partial D$ (vertices lying in the ideal boundary $\partial_\infty\Bbb{H}^2$ are also considered). The quantity $\alpha(\scr{P})$ (resp. $\beta(\scr{P})$) is defined as the sum of the lengths of the curves of $\{A_i\}$ (resp. $\{B_j\}$) that are also sides of $\scr{P}$ (removing a small horocycle at each ideal vertex of $\scr{P}$ is necessary so the sum is finite, but the results do not depend on the choice of the horocycle). Likewise, $\ell(\scr{P})$ denotes the sum of the lengths of the sides of $\scr{P}$. \par Under these conditions, the main result (Theorem 2.2) states that there exists a minimal graph over $D$ with $+\infty$ (resp. $-\infty$) values along $\{A_i\}$ (resp. $\{B_j\}$) and prescribed continuous values along $\{C_k\}\neq\emptyset$ if and only if $2\alpha(\scr{P})<\ell(\scr{P})$ and $2\beta(\scr{P})<\ell(\scr{P})$ for all inscribed polygons $\scr P\neq\partial D$. In the case $\{C_k\}=\emptyset$, the condition $\alpha(\scr{P})=\beta(\scr{P})$ is also required (Theorem 2.1). \par The ideas and the results are adapted from those developed by P. Collin and H. Rosenberg in [Ann. of Math. (2) {\bf 172} (2010), no.~3, 1879--1906; MR2726102 (2011i:53004)] for the ambient space $\Bbb{H}^2\times\Bbb{R}$, which is a limit case of $\widetilde{\roman{PSL}_2(\Bbb{R})}$ when the bundle curvature tends to zero. On the other hand, the present paper can be understood as a Jenkins-Serrin-type result [H. Jenkins and J.~B. Serrin Jr., Arch. Rational Mech. Anal. {\bf 21} (1966), 321--342; MR0190811 (32 \#8221)] for unbounded domains of $\widetilde{\roman{PSL}_2(\Bbb{R})}$. Finally, it is worth mentioning that R. Younes previously treated the case where the domain $D$ is bounded in [Illinois J. Math. {\bf 54} (2010), no.~2, 671--712; 2846478 ].