Overdetermined equations and support functions on the hyperbolic and de Sitter planes
Abstract
We study overdetermined equations on the hyperbolic and de Sitter planes by means of the support-function representation of zero mean curvature surfaces in Lorentz--Minkowski three-space. On the hyperbolic plane, solutions of Δu−2u=0 give rise to branched spacelike maximal surfaces. We prove that, on a bounded simply connected domain, nontrivial constant Dirichlet and Neumann data force the domain to be a geodesic disk, provided that the associated support quadric is nonlightlike; moreover, the solution is a multiple of the hyperbolic cosine of the distance from the center. On the de Sitter plane, the corresponding equation is the Klein--Gordon equation Box u+2u=0, and its solutions generate timelike minimal surfaces wherever the support tensor is nondegenerate. We identify the support quadric and the constant-angle condition determined by constant Cauchy data, prove that local rigidity fails along every analytic noncharacteristic curve, and recover rotational symmetry from constant data on a global Cauchy circle. Thus the same support-function formalism reveals a sharp transition from elliptic rigidity to hyperbolic flexibility when the signature changes.
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BibTeX
@misc{batista2026overdetermined,
title = {{Overdetermined equations and support functions on the hyperbolic and de Sitter planes}},
author = {Márcio Batista and Iury Domingos},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.13992},
eprint = {2609.13992},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.13992},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Overdetermined equations and support functions on the hyperbolic and de Sitter planes
AU - Márcio Batista
AU - Iury Domingos
PY - 2026
DA - 2026/09/12
PB - arXiv
DO - 10.48550/arxiv.2609.13992
UR - https://arxiv.org/abs/2609.13992
AB - We study overdetermined equations on the hyperbolic and de Sitter planes by means of the support-function representation of zero mean curvature surfaces in Lorentz--Minkowski three-space. On the hyperbolic plane, solutions of Δu−2u=0 give rise to branched spacelike maximal surfaces. We prove that, on a bounded simply connected domain, nontrivial constant Dirichlet and Neumann data force the domain to be a geodesic disk, provided that the associated support quadric is nonlightlike; moreover, the solution is a multiple of the hyperbolic cosine of the distance from the center. On the de Sitter plane, the corresponding equation is the Klein--Gordon equation Box u+2u=0, and its solutions generate timelike minimal surfaces wherever the support tensor is nondegenerate. We identify the support quadric and the constant-angle condition determined by constant Cauchy data, prove that local rigidity fails along every analytic noncharacteristic curve, and recover rotational symmetry from constant data on a global Cauchy circle. Thus the same support-function formalism reveals a sharp transition from elliptic rigidity to hyperbolic flexibility when the signature changes.
ER -