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This is a Quantitative Study study.

October 20, 2025Open Access

Every nonflat conformal minimal surface is homotopic to a proper one

Key Points

Every nonflat conformal minimal immersion can be homotoped to a proper one, enhancing our understanding of geometric structures.
In dimensions five and higher, the proper embedding can be chosen to be injective, expanding the framework of minimal surfaces.
Prescribing the flux allows every immersion to be homotoped to a specific type of embedding known as a holomorphic null embedding.
Results extend to holomorphic immersions directed by Oka cones, illustrating broader applications in complex geometry.

Abstract

Given an open Riemann surface M, we prove that every nonflat conformal minimal immersion Mⁿ (n 3) is homotopic through nonflat conformal minimal immersions Mⁿ to a proper one. If n 5, it may be chosen in addition injective, hence a proper conformal minimal embedding. Prescribing its flux, as a consequence, every nonflat conformal minimal immersion Mⁿ is homotopic to the real part of a proper holomorphic null embedding Mⁿ. We also obtain a result for a more general family of holomorphic immersions from an open Riemann surface into Cⁿ directed by Oka cones in Cⁿ.

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Tjasa Vrhovnik (Wed,) studied this question.

synapsesocial.com/papers/68f5c338e2d8b12842645a96 https://doi.org/https://doi.org/10.48550/arxiv.2505.15352

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