发表机构
Faculty of Mathematics, University of Belgrade（贝尔格莱德大学数学学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文建立调和映射微分范数的乘积原理，证明盒子与多圆盘上的精确算子范数估计及等号情形，并引入调和乘积度量，揭示其与Kobayashi-Royden度量的关系。
AI 中文摘要
我们建立了调和映射微分的欧几里得算子范数的精确乘积原理。对于有界域 \\(G\subset\R^m\\) 和 \\(p\in G\\)，令 \\(M_G(p)\\) 表示所有满足 \\(F(0)=p\\) 的调和映射 \\(F:\D\to G\\) 上 \\(\\|dF_0\\|\\) 的上确界。对于有界域 \\(G_j\subset\R^{m_j}\\)，我们证明 \\[ M_{G_1\times\cdots\times G_N}(p_1,\ldots,p_N)^2 =\sum_{j=1}^N M_{G_j}(p_j)^2. \\] 该定理将各因子的几何与乘积的欧几里得几何分离开来：因子极值常数通过平方和定律组合，而等号成立由一个相容性条件控制，即分量微分存在公共的最大化方向。既不要求因子凸性，也不要求因子极值可达。将乘积原理与区间和圆盘的尖锐因子问题相结合，可得到调和映射到盒子与多圆盘的精确算子范数估计及所有等号情形。在这两类族中，每个极值映射在原点的实微分具有一维像。相同的因子常数还定义了坐标方向的度量，当源圆盘配备曲率为 \\(-1\\) 的Poincaré度量时，调和压缩估计是尖锐的。对于盒子，所得度量是完备的，且等于垂直条带乘积的Kobayashi-Royden度量的两倍的限制。对于多圆盘，调和乘积度量在形如 \\(\max_j a_j(p)|v_j|\\)（其中 \\(a_j(p)>0\\)）的压缩度量中逐点最大。它在每个非零切向量上严格小于Kobayashi-Royden度量的两倍，且其诱导路径度量是不完备的。
英文摘要
We establish an exact product principle for the Euclidean operator norm of differentials of harmonic maps. For a bounded domain \(G\subset\R^m\) and \(p\in G\), let \(M_G(p)\) denote the supremum of \(\|dF_0\|\) over harmonic maps \(F:\D\to G\) with \(F(0)=p\). For bounded domains \(G_j\subset\R^{m_j}\), we prove \[ M_{G_1\times\cdots\times G_N}(p_1,\ldots,p_N)^2 =\sum_{j=1}^N M_{G_j}(p_j)^2. \] The theorem separates the geometry of the individual factors from the Euclidean geometry of the product: the factor extremal constants combine by a sum-of-squares law, while equality is governed by a single compatibility condition, namely a common maximizing direction for the component differentials. Neither convexity nor attainment of the factor suprema is required. Combining the product principle with the sharp interval and disk factor problems yields exact operator-norm estimates and all equality cases for harmonic maps into boxes and polydiscs. In both families, for every extremal map, the real differential at the origin has one-dimensional image. The same factor constants also define coordinatewise metrics for which the harmonic contraction estimate is sharp when the source disk is equipped with its Poincaré metric of curvature \(-1\). For boxes, the resulting metric is complete and equals twice the restriction of the Kobayashi-Royden metric of the product of vertical strips. For polydiscs, the harmonic product metric is pointwise maximal among contracting metrics of the form \(\max_j a_j(p)|v_j|\), with \(a_j(p)>0\). It is strictly smaller than twice the Kobayashi-Royden metric on every nonzero tangent vector, and its induced path metric is incomplete.
Comments30 pages
