Poincaré index and the volume functional of unit vector fields on punctured spheres

2017/08/04 by Fabiano G. B. Brito, Fabiano Brito, Brito, Fabiano G. B. +5
Mathematics · #53C20 #53C43 #57R20 #57R25 #Advanced Differential Equations and Dynamical Systems #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Meromorphic and Entire Functions #math.DG #msc:53C20 #msc:53C43 #msc:57R20 #msc:57R25

paper · pdf · doi:10.48550/arxiv.1708.01575

Revised version. 15 pages, 2 figures

openalex publication_date 2017/08/04 · arxiv created 2017/09/19 · arxiv updated 2017/09/21 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

For n≥ 1, we exhibit a lower bound for the volume of a unit vector field on \mathbbS²ⁿ⁺¹\backslash\± p\ depending on the absolute values of its Poincaré indices around ± p. We determine which vector fields achieve this volume, and discuss the idea of having multiple isolated singularities of arbitrary configurations.

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