论文标题
非紧密组合赫尔米利亚对称空间的呼吸函数和度量压实
Horofunctions and metric compactification of noncompact Hermitian symmetric spaces
论文作者
论文摘要
考虑到Hermitian对称空间$ m $ nongapact类型，我们通过实现$ m $作为$ m $的$ m $公制压实中的呼吸函数，作为banach space $的开放单位$ d $ d $（v，\ | \ cdot \ |），称为Jordan $ d $ d $ d $ d $ d $ d $ d $ $ \ mathrm {jb}^*$ - 三重。 $ d $上的carathéodory距离$ρ$具有鳍结构。它是carathéodory差异度量的集成距离，而实现的规范$ \ | \ cdot \ | $是carathéodory规范相对于d $中的原点$ 0 \。我们还确定了$（v，\ | \ cdot \ |）$的度量压实的呼吸功能，并将其几何形状和全球拓扑与封闭的双单元球（即$ d $的极性）联系起来。此外，我们表明，指数映射$ \ exp_0 \ colon v \ longrightArrow d $ at d $ in d $ in d $ in d $延伸至$（v，v，\ | \ cdot \ |）$和$（d，ρ）$之间的标准压实之间的同态形态。因此，$ m $的度量压实将具体的实现作为$（v，\ | \ cdot \ |）$的封闭双单位球。
Given a Hermitian symmetric space $M$ of noncompact type, we give a complete description of the horofunctions in the metric compactification of $M$ with respect to the Carathéodory distance, via the realisation of $M$ as the open unit ball $D$ of a Banach space $(V,\|\cdot\|)$ equipped with a Jordan structure, called a $\mathrm{JB}^*$-triple. The Carathéodory distance $ρ$ on $D$ has a Finsler structure. It is the integrated distance of the Carathéodory differential metric, and the norm $\|\cdot\|$ in the realisation is the Carathéodory norm with respect to the origin $0\in D$. We also identify the horofunctions of the metric compactification of $(V,\|\cdot\|)$ and relate its geometry and global topology to the closed dual unit ball (i.e., the polar of $D$). Moreover, we show that the exponential map $\exp_0 \colon V \longrightarrow D$ at $0\in D$ extends to a homeomorphism between the metric compactifications of $(V,\|\cdot\|)$ and $(D,ρ)$, preserving the geometric structure. Consequently, the metric compactification of $M$ admits a concrete realisation as the closed dual unit ball of $(V,\|\cdot\|)$.
