Target Geometry in Prescribed-Value Schwarz Lemmas for Harmonic Maps
Abstract
We study harmonic maps F:D→ G into bounded domains in real Hilbert spaces, prescribing F(0) and dF0(R2) when dF0 is nonzero and conformal. We prove a target-independent identity for the Möbius-weighted Hilbert pairing. After normalization, the pairing is affine in |dF0| with positive slope, yielding an exact equivalence between the derivative extremal problem and a boundary-pairing problem. For a round affine section, this gives a necessary and sufficient integral criterion for the Möbius parametrization to be extremal. A supporting-hyperplane condition guarantees the required integral inequality and characterizes equality. Examples show that roundness alone is insufficient and that the supporting condition is not necessary. For unit balls of real Hilbert spaces of dimension at least two, including infinite-dimensional spaces, we obtain the sharp prescribed-value Schwarz-Pick estimate, all equality cases, and quantitative L2 boundary stability. We also prove a Cayley-Klein contraction under pointwise distortion. For K≥1, let MK denote the supremum of LF(0) over harmonic maps F:D→BH satisfying F(0)=0 and 0<ℓF(0)≤ LF(0)≤ KℓF(0), where LF(0) and ℓF(0) are the maximal and minimal stretchings at the origin, respectively. We determine MK, identify the unique optimizing parameter, and characterize all extremals. The function K↦ MK is strictly increasing, with M1=1 and MK→4/π as K→∞.
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BibTeX
@misc{knezevic2026target,
title = {{Target Geometry in Prescribed-Value Schwarz Lemmas for Harmonic Maps}},
author = {Miljan Knežević and Miodrag Mateljević},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.19609},
eprint = {2609.19609},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.19609},
}
RIS
TY - UNPB
TI - Target Geometry in Prescribed-Value Schwarz Lemmas for Harmonic Maps
AU - Miljan Knežević
AU - Miodrag Mateljević
PY - 2026
DA - 2026/09/17
PB - arXiv
DO - 10.48550/arxiv.2609.19609
UR - https://arxiv.org/abs/2609.19609
AB - We study harmonic maps F:D→ G into bounded domains in real Hilbert spaces, prescribing F(0) and dF_0(R^2) when dF_0 is nonzero and conformal. We prove a target-independent identity for the Möbius-weighted Hilbert pairing. After normalization, the pairing is affine in |dF_0| with positive slope, yielding an exact equivalence between the derivative extremal problem and a boundary-pairing problem. For a round affine section, this gives a necessary and sufficient integral criterion for the Möbius parametrization to be extremal. A supporting-hyperplane condition guarantees the required integral inequality and characterizes equality. Examples show that roundness alone is insufficient and that the supporting condition is not necessary. For unit balls of real Hilbert spaces of dimension at least two, including infinite-dimensional spaces, we obtain the sharp prescribed-value Schwarz-Pick estimate, all equality cases, and quantitative L^2 boundary stability. We also prove a Cayley-Klein contraction under pointwise distortion. For K≥1, let M_K denote the supremum of L_F(0) over harmonic maps F:D→B_H satisfying F(0)=0 and 0<ℓ_F(0)≤ L_F(0)≤ Kℓ_F(0), where L_F(0) and ℓ_F(0) are the maximal and minimal stretchings at the origin, respectively. We determine M_K, identify the unique optimizing parameter, and characterize all extremals. The function K↦ M_K is strictly increasing, with M_1=1 and M_K→4/π as K→∞.
ER -