A Product Principle for Harmonic Schwarz Lemmas: Boxes, Polydiscs, and Metric Geometry
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.