Geometry of Biharmonic Mappings

Differential Geometry of Variational Methods

The author describes harmonic maps which are critical points of the energy functional, and biharmonic maps which are critical points of the bienergy functional. Also given are fundamental materials of the variational methods in differential geometry, and also fundamental materials of differential geometry.

Contents:

• Fundamental Materials on the Theory of Harmonic Maps and Biharmonic Maps:

  • Fundamental Materials of Riemannian Geometry
  • The First and Second Variational Formulas of the Energy

• Rigidity and Abundance of Biharmonic Maps:

  • Biharmonic Maps into a Riemannian Manifold of Non-positive Curvature
  • Biharmonic Submanifolds in a Riemannian Manifold with Non-positive Curvature
  • Biharmonic Hypersurfaces in a Riemannian Manifold with Non-positive Ricci Curvature
  • Note on Biharmonic Map Equations
  • Harmonic Maps into Compact Lie Groups and Integrable Systems
  • Biharmonic Maps into Symmetric Spaces and Integrable Systems
  • Bubbling of Harmonic Maps and Biharmonic Maps
  • Conformal Change of Riemannian Metrics and Biharmonic Maps

• Biharmonic Submanifolds:

  • Biharmonic Submanifolds in a Riemannian Manifold
  • Sasaki Manifolds, Kähler Cone Manifolds and Biharmonic Submanifolds
  • Biharmonic Lagrangian Submanifolds in Kähler Manifolds

• Further Developments on Biharmonic Maps:

  • Rigidity of Transversally Biharmonic Maps between Foliated Riemannian Manifolds
  • CR-Rigidity of Pseudo Harmonic Maps and Pseudo Biharmonic Maps
  • Harmonic Maps and Biharmonic Maps on the Principal Bundles and Warped Products

Readership: Researchers in global analysis, geometry, calculus of variations and partial differential equations.
Key Features:

• New theory and complete theory are described in detail