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Discreteness of the Steklov Spectrum for Exterior Non-Compact Free-Boundary Minimal Surfaces with Regular Ends of Finite Total Curvature
Abstract
We study the Steklov problem on non-compact exterior free-boundary minimal surfaces. Boundary values need not determine a unique harmonic extension, so the operator also requires a prescription at infinity. For proper surfaces in ℝ3 with compact boundary and finitely many regular ends of finite total curvature, we construct a natural class of such prescriptions. Every resulting operator is self-adjoint with compact resolvent; consequently, its spectrum is discrete, bounded below, and tends to +∞. If the coordinate functions have linearly independent boundary traces, the prescription can be chosen so that these traces are eigenfunctions with eigenvalue −1.
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BibTeX
@misc{kozyrev2026discreteness,
title = {{Discreteness of the Steklov Spectrum for Exterior Non-Compact Free-Boundary Minimal Surfaces with Regular Ends of Finite Total Curvature}},
author = {Igor Kozyrev},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.29318},
eprint = {2609.29318},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.29318},
}
RIS
TY - UNPB
TI - Discreteness of the Steklov Spectrum for Exterior Non-Compact Free-Boundary Minimal Surfaces with Regular Ends of Finite Total Curvature
AU - Igor Kozyrev
PY - 2026
DA - 2026/09/24
PB - arXiv
DO - 10.48550/arxiv.2609.29318
UR - https://arxiv.org/abs/2609.29318
AB - We study the Steklov problem on non-compact exterior free-boundary minimal surfaces. Boundary values need not determine a unique harmonic extension, so the operator also requires a prescription at infinity. For proper surfaces in ℝ^3 with compact boundary and finitely many regular ends of finite total curvature, we construct a natural class of such prescriptions. Every resulting operator is self-adjoint with compact resolvent; consequently, its spectrum is discrete, bounded below, and tends to +∞. If the coordinate functions have linearly independent boundary traces, the prescription can be chosen so that these traces are eigenfunctions with eigenvalue −1.
ER -