This book focuses on the study of the volume of vector fields on Riemannian manifolds. Providing a thorough overview of research on vector fields defining minimal submanifolds, and on the existence and characterization of volume minimizers, it includes proofs of the most significant results obtained since the subject’s introduction in 1986. Aiming to inspire further research, it also highlights a selection of intriguing open problems, and exhibits some previously unpublished results. The presentation is direct and deviates substantially from the usual approaches found in the literature, requiring a significant revision of definitions, statements, and proofs. A wide range of topics is covered, including: a discussion on the conditions for a vector field on a Riemannian manifold to determine a minimal submanifold within its tangent bundle with the Sasaki metric; numerous examples of minimal vector fields (including those of constant length on punctured spheres); a thorough analysis of Hopf vector fields on odd-dimensional spheres and their quotients; and a description of volume-minimizing vector fields of constant length on spherical space forms of dimension three. Each chapter concludes with an up-to-date survey which offers supplementary information and provides valuable insights into the material, enhancing the reader's understanding of the subject. Requiring a solid understanding of the fundamental concepts of Riemannian geometry, the book will be useful for researchers and PhD students with an interest in geometric analysis.
Contents
Intro
Preface
Funding Acknowledgements
Contents
1 Introduction
2 Minimal Sections of Tensor Bundles
2.1 Geometry of the Submanifold Determined by a Section of a Tensor Bundle
2.2 Minimal Sections of Tensor Bundles and Sphere Subbundles
2.3 First Variation of the Volume of Vector Fields: Minimal Vector Fields
2.4 Second Variation of the Volume of Vector Fields
2.5 The 2-Dimensional Case
2.6 Notes
2.6.1 Sections That Are Harmonic Maps
2.6.2 Sections That Are Critical Pointsof the Energy Functional
2.6.3 Minimal Oriented Distributions
3 Minimal Vector Fields of Constant Length on the Odd-Dimensional Spheres
3.1 Minimality of the Hopf Vector Fields
3.2 Study of the Stability of the Hopf Vector Fields
3.3 Stability of the Hopf Vector Fields of Odd-Dimensional Space Forms of Positive Curvature
3.4 Notes
3.4.1 Spheres and Their Quotients with Berger Metrics
3.4.2 The Minimality Condition for Unit Killing Vector Fields
3.4.3 Minimality of the Characteristic Vector Field of a Contact Riemannian Manifold
3.4.4 Minimal Invariant Vector Fields on Lie Groups and Homogeneous Spaces
3.4.5 Examples Related with Complex and Quaternionic Structures
4 Vector Fields of Constant Length of Minimum Volume on the Odd-Dimensional Spherical Space Forms
4.1 Hopf Vector Fields as Volume Minimisers in the 3-Dimensional Case
4.2 Hopf Vector Fields on 3-Dimensional Spheres with the Berger Metrics
4.3 Lower Bound of the Volume of Vector Fields of Constant Length
4.4 Asymptotic Behaviour of the Volume Functional
4.5 Notes
4.5.1 Unit Vector Fields on the Two-Dimensional Torus
4.5.2 Lower Bound of the Volume of Unit Vector Fields on Hypersurfaces of Rn+1
4.5.3 Almost Hermitian Structures on S6 That Minimise the Volume
4.5.4 Minimisers of Functionals Related with the Energy.
5 Vector Fields of Constant Length on Punctured Spheres
5.1 The Radial Vector Fields
5.2 Parallel Transport Vector Fields
5.3 The Main Open Problem
5.4 Area Minimising Vector Fields on the 2-Sphere
5.5 Notes
5.5.1 Radial Vector Fields on Riemannian Manifolds
5.5.2 Minimisers of the Volume Among Unit Vector Fields with Singular Points
References.
ISBN
3-031-36857-6
Doi
10.1007/978-3-031-36857-8
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