부산대학교 도서관

This work deals with graphs in $\Bbb H\times\Bbb R$ which have constant mean curvature $H$ defined over an unbounded domain in $\Bbb H$. It is well known that there is no entire graph for $H>1/2$ in $\Bbb H\times\Bbb R$; moreover, in [L. Hauswirth, H. Rosenberg and J. Spruck, Comm. Anal. Geom. {\bf 16} (2008), no.~5, 989--1005; MR2471365 (2010d:53009)] it was proved that a complete graph with $H=1/2$ in $\Bbb H\times\Bbb R$ is an entire graph. Hence, the authors consider in this work values of $H>0$ less than $1/2$. They take a convex domain $\scr D$ whose boundary $\partial\scr D$ is composed of ideal arcs $\{A_i\}$, $\{B_j\}$ and $\{C_k\}$ such that the curvatures of the arcs with respect to the domain are $\kappa(A_i)=2H$, $\kappa(B_j)=-2H$ and $\kappa(C_k)\ge 2H$. They give necessary and sufficient conditions on the geometry of the domain $\scr D$ which assure the existence of a function $u$ defined in $\scr D$, whose graph has constant mean curvature and $u$ assumes the value $+\infty$ on each $A_i$, $-\infty$ on each $B_j$ and prescribed continuous data on each $C_k$. The condition, as in [H. Jenkins and J.~B. Serrin Jr., Arch. Rational Mech. Anal. {\bf 21} (1966), 321--342; MR0190811 (32 \#8221)] for graphs over domains $D\subset\Bbb R^2$, is considered in terms of the length and areas of inscribed polygons. Since these quantities are infinite in general, the formulation of the conditions is somewhat delicate. In order to control lengths the authors use the ideas in [P. Collin and H. Rosenberg, Ann. of Math. (2) {\bf 172} (2010), no.~3, 1879--1906; MR2726102 (2011i:53004)]; however, the new and key idea appears when they consider the area and they split it into two parts, one finite and the other infinite (see Section 3). \par Section 4 contains general maximum principles. Section 5 contains the flux formulas, which give the requirement on the curvature of the boundary arcs of admissible domains. In Section 6, the authors study some characteristics of the sets where a sequence of solutions in a domain $D$ converges or diverges. An important conclusion of this lemma is that the divergence set is given by $\scr V=\bigcup_{i\in I}L_i$, where $L_i$ is a curve, called a divergence line, having curvature $2H$.