论文标题
cmc图与平面边界$ \ mathbb {h}^{2} \ times \ times \ mathbb {r} $
CMC Graphs With Planar Boundary in $\mathbb{H}^{2}\times \mathbb{R}$
论文作者
论文摘要
众所周知，以$ω\ subset \ mathbb {r}^{2} $无限制的凸域和$ h> 0 $，存在图$ g \ subset \ subset \ mathbb {r}^{r}^{3} $ contand curvature curvature curvature curvature curvature curvature $ h $ a $ a $ $ $ $的$ $ y in c = $ y in c = $ $ f =宽度$ 1/h $。在本文中，我们以$ \ mathbb {h}^{2} \ times \ mathbb {r} $在相同方向上以$ h \ in \ left（0,1/2 \ right）$，如果$ h \ in \ yathbb {0,1/2 \ right）$在$ \ mathbb {h} $ pertiand coult（0,1/2 \ right）$中，则包括$ h \。等距Hypercycles $ \ ell（h）$相距，我们表明，如果$ \partialΩ$的地球曲率从下面的$ -1限制为$ -1，则$ h $ -graph $ g $ ive $ g $ yo $ \ $ \ partial g = \ partial g = \ partial partial optial o部分。我们还提供了更多精致的存在结果，涉及$ \partialΩ的曲率，$也可能小于$ -1。
It is known that for $Ω\subset \mathbb{R}^{2}$ an unbounded convex domain and $H>0$, there exists a graph $G\subset \mathbb{R}^{3}$ of constant mean curvature $H$ over $Ω$ with $\partial G=$ $\partial Ω$ if and only if $Ω$ is included in a strip of width $1/H$. In this paper we obtain results in $\mathbb{H}^{2}\times \mathbb{R}$ in the same direction: given $H\in \left( 0,1/2\right) $, if $Ω$ is included in a region of $\mathbb{ H}^{2}\times \left\{ 0\right\} $ bounded by two equidistant hypercycles $\ell(H)$ apart, we show that, if the geodesic curvature of $\partial Ω$ is bounded from below by $-1,$ then there is an $H$-graph $G$ over $Ω$ with $\partial G=\partial Ω$. We also present more refined existence results involving the curvature of $\partialΩ,$ which can also be less than $-1.$
