Bibliographic Information: El Emam, C., & Sagman, N. (2024). Holomorphic Dependence for the Beltrami Equation in Sobolev Spaces. arXiv preprint arXiv:2410.06175v1.
Research Objective: This paper aims to extend the classical result of Ahlfors and Bers on the holomorphic dependence of Beltrami equation solutions to higher-order Sobolev spaces. This extension is motivated by applications in quasi-Fuchsian geometry, particularly in studying the properties of Bers metrics.
Methodology: The authors employ techniques from complex analysis and the theory of elliptic partial differential equations. They establish a family of elliptic estimates for the Beltrami operator and utilize a bootstrapping argument based on the work of Ahlfors and Bers to prove their main results.
Key Findings:
Main Conclusions: The holomorphic dependence of Beltrami equation solutions and Bers metrics on complex structures in Sobolev spaces has significant implications for the study of Teichmüller theory and quasi-Fuchsian geometry. This result provides a powerful tool for analyzing the properties of these objects and their applications in related fields.
Significance: This research contributes to a deeper understanding of the interplay between complex analysis, geometric function theory, and the theory of partial differential equations. The results have potential applications in areas such as the study of equivariant immersions of surfaces and higher Teichmüller theory.
Limitations and Future Research: The paper focuses on the holomorphic dependence of solutions for Beltrami differentials with bounded sup-norm. Exploring similar dependence properties for Beltrami differentials in other function spaces or with weaker regularity assumptions could be an interesting avenue for future research. Additionally, investigating the applications of these results to specific problems in quasi-Fuchsian geometry and related areas could lead to further advancements in the field.
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by Christian El... at arxiv.org 10-10-2024
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