论文标题
没有抛物线的域，最小的亚策略和弱双曲线域
Domains without parabolic minimal submanifolds and weakly hyperbolic domains
论文作者
论文摘要
我们表明，如果$ω$是$ \ Mathbb r^n $中的$ M $ -Convex域，则可能是$ 2 \ le m <n $，其边界$bΩ$具有正半径为正半径的管状邻域，而不是$ m $ flat附近，则$ω$不包含任何浸泡的寄生虫副质子paraboric minimal submanifold of demeension $ $ $ \ ge m。特别是，如果$ m $是$ \ mathbb r^n $的适当嵌入的非纤维最小的超浮雕，则带有正半径为正半径的邻居，则每个沉浸式抛物线抛物性hypersurface in $ \ mathbb r^n $相交$ m $。在尺寸$ n = 3 $中，如果$ m $具有高斯曲率的限制。我们还介绍了$ \ Mathbb r^n $中的一类弱双曲线域$ω$，其特征是该属性的特征是每个保形和谐波映射$ \ mathbb c \toΩ$都是恒定的，我们阐明了它们与双曲线域和域的关系，而无需抛物性最小的表面。
We show that if $Ω$ is an $m$-convex domain in $\mathbb R^n$ for some $2\le m<n$ whose boundary $bΩ$ has a tubular neighbourhood of positive radius and is not $m$-flat near infinity, then $Ω$ does not contain any immersed parabolic minimal submanifold of dimension $\ge m$. In particular, if $M$ is a properly embedded nonflat minimal hypersurface in $\mathbb R^n$ with a tubular neighbourhood of positive radius then every immersed parabolic hypersurface in $\mathbb R^n$ intersects $M$. In dimension $n=3$ this holds if $M$ has bounded Gaussian curvature. We also introduce the class of weakly hyperbolic domains $Ω$ in $\mathbb R^n$ characterised by the property that every conformal harmonic map $\mathbb C\toΩ$ is constant, and we elucidate their relationship with hyperbolic domains and domains without parabolic minimal surfaces.
