复双曲空间中的格罗莫夫 - 罗斯猜想
The Gromov--Ros conjecture for rank-one symmetric spaces of noncompact type
AI总结：
研究复双曲空间\(\CH^m\)（\(m\geq2\)）中格罗莫夫 - 罗斯猜想，利用精确体积的径向 - 角向极拉伸等方法，证明有限周长等周区域是测地球，还得出相关不等式无条件成立及精确体积校正等结论。
AI中文摘要：
设\(\CH^m\)，\(m\geq2\)，具有全纯截面曲率\(-4\)。我们证明在每个复维数中，其有限周长等周区域恰好是测地球。从体积几何中位数出发，我们使用精确体积的径向 - 角向极拉伸。这些映射是全局双李普希茨的，且简化边界面积公式通过环境余因子表示其周长。对方向球的坐标函数求二阶导数之和得到测度理论法向量的水平和瑞布分量中一个显式多项式的负值。该多项式在每个实维数\(n = 2m\geq4\)时非负且仅在径向法向量时为零。全局极小性迫使简化边界上几乎处处为径向；通过\(BV U(m)\)不变性论证和加权一维端点比较进而得到单个球。相同的迹恒等式，结合截断、临界黑塞和移动极点论证，还证明每个光滑有界固定体积稳定临界域是测地球。因此，先前有条件的凸加权伯格曼收缩、法伯 - 克拉恩集中和利布 - 韦尔熵不等式在\(\mathbb B^m\)上对于每个\(m\geq2\)及其等式情形变为无条件的。精确体积校正允许无基对数划分公式，其黑塞迹具有精确的高斯协方差解释。在高权重平坦极限下，伯格曼收缩恢复全纯高斯超收缩性。
英文摘要：
We prove the Gromov--Ros conjecture for every rank-one symmetric space of noncompact type: finite-perimeter isoperimetric regions are precisely the geodesic balls, up to ambient isometry and null sets. The proof is organized uniformly in the root multiplicities $(p,q)$. We introduce the volume radius \[ R(r)^n=n\int_0^r\sinh^{n-1}s\,\cosh^q s\,ds, \] which converts the polar volume form into $R^{n-1}dR\,dσ$. Hence radial--angular stretches $R\mapsto e^{tφ(θ)}R$ have the exact Jacobian $e^{ntφ}$. Starting from a volume geometric median, a log-partition correction produces an exactly volume-preserving family of global bi-Lipschitz stretches. The reduced-boundary area formula and the ambient cofactor yield a universal pointwise trace identity depending only on $(p,q)$ and the horizontal and vertical components of the measure-theoretic normal. Its specializations for $q=1,3,7$ admit explicit strict sign certificates, forcing the normal of an isoperimetric region to be radial almost everywhere. Isotropy invariance in BV and a one-dimensional weighted endpoint comparison then force a single ball. For complex hyperbolic space we additionally prove that every smooth bounded fixed-volume stable critical domain is a geodesic ball. The resulting complex-hyperbolic isoperimetric theorem removes the geometric hypothesis in several sharp weighted-Bergman contraction, Faber--Krahn, and Lieb--Wehrl inequalities, and their high-weight scaling limit recovers holomorphic Gaussian hypercontractivity.
