부산대학교 도서관

The space $\widetilde{\roman{PSL}_2(\Bbb{R})}$ is a homogeneousRiemannian 3-manifold with isometry group of dimension 4. It admits aRiemannian submersion over the hyperbolic plane $\Bbb{H}^2$ withconstant bundle curvature, and the fibers of the submersion are theintegral 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 ofthe submersion over $D$.\parIn this paper, the author considers an unbounded domain$D\subset\Bbb{H}^2$ whose boundary is piecewise smooth and consists ofgeodesic arcs as well as regular curves which are convex with respectto the interior of the domain. The main objective is to establishnecessary and sufficient conditions on $D$ for there to exist a minimalgraph over $D$ with prescribed (finite or infinite) boundary data. Itis 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 ageodesic arc. Hence the components of $\partial D$ are divided into twofamilies of geodesic arcs, $\{A_i\}$ and $\{B_j\}$, where the graphtakes values $+\infty$ and $-\infty$, respectively, and a family ofconvex curves $\{C_k\}$, with prescribed continuous data. No two elements of$\{A_i\}$ (resp. $\{B_j\}$) are allowed tohave a common endpoint by the maximum principle.\parA polygon $\scr{P}$ inscribed in $D$ is a geodesic polygon$\scr{P}\subset\overline D$ with finitely many sides, and such that thevertices of $\scr{P}$ are also vertices of $\partial D$ (vertices lyingin the ideal boundary $\partial_\infty\Bbb{H}^2$ are also considered).The quantity $\alpha(\scr{P})$ (resp. $\beta(\scr{P})$) is defined asthe 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 eachideal vertex of $\scr{P}$ is necessary so the sum is finite, but theresults do not depend on the choice of the horocycle). Likewise,$\ell(\scr{P})$ denotes the sum of the lengths of the sides of$\scr{P}$.\parUnder these conditions, the main result (Theorem 2.2) states that thereexists a minimal graph over $D$ with $+\infty$ (resp. $-\infty$) valuesalong $\{A_i\}$ (resp. $\{B_j\}$) and prescribed continuous valuesalong $\{C_k\}\neq\emptyset$ 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 $\alpha(\scr{P})=\beta(\scr{P})$ isalso required (Theorem 2.1).\parThe ideas and the results are adapted from thosedeveloped by P. Collin and H. Rosenberg in [Ann. of Math. (2) {\bf 172}(2010), no.~3, 1879--1906; MR2726102 (2011i:53004)] for theambient space$\Bbb{H}^2\times\Bbb{R}$, which is a limit case of$\widetilde{\roman{PSL}_2(\Bbb{R})}$ when the bundle curvature tends tozero. On the other hand, the present paper can be understood as aJenkins-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 mentioningthat R. Younes previously treated the case where the domain $D$ isbounded in [Illinois J. Math. {\bf 54} (2010), no.~2, 671--712; 2846478 ].