Quaternionic holomorphic geometry: Pluecker formula, Dirac eigenvalue estimates and energy estimates of harmonic 2-tori

Abstract

The paper develops the fundamentals of quaternionic holomorphic curve theory. The holomorphic functions in this theory are conformal maps from a Riemann surface into the 4-sphere, i.e., the quaternionic projective line. Basic results such as the Riemann-Roch Theorem for quaternionic holomorphic vector bundles, the Kodaira embedding and the Pluecker relations for linear systems are proven. Interpretations of these results in terms of the differential geometry of surfaces in 3- and 4-space are hinted at throughout the paper. Applications to estimates of the Willmore functional on constant mean curvature tori, respectively energy estimates of harmonic 2-tori, and to Dirac eigenvalue estimates on Riemannian spin bundles in dimension 2 are given.

70 pages, 1 figure

References in corpus (2)

• Conformal Geometry of Surfaces in the 4-Sphere and Quaternions
• The Weierstrass representation of spheres in $R^3$, the Willmore numbers, and soliton spheres

Cited by in corpus (32)

• The geometric complexity of special Lagrangian $T^2$-cones
• On organizing principles of Discrete Differential Geometry. Geometry of spheres
• A proof of the Willmore conjecture
• Surfaces in the four-space and the Davey--Stewartson equations
• Conformal maps from a 2-torus to the 4-sphere
• Dressing transformations of constrained Willmore surfaces
• The spectral curve of a quaternionic holomorphic line bundle over a 2-torus
• Soliton Spheres
• Darboux transforms and simple factor dressing of constant mean curvature surfaces
• Loop Group Methods for Constant Mean Curvature Surfaces
• Generalised Bianchi permutability for isothermic surfaces
• Constrained Willmore and CMC tori in the 3-sphere
• Energy of sections of the Deligne-Hitchin twistor space
• Infinitesimal Darboux transformations of the spectral curves of tori in the four-space
• Equivariant constrained Willmore tori in the 3-sphere
• Minimal Tori in $S^3$
• Surfaces in three-dimensional Lie groups in terms of spinors
• Willmore spheres in quaternionic projective space
• On Konopelchenko's representation for surfaces in 4 dimensions
• Higher solutions of Hitchin's self-duality equations
• Bryant Surfaces with Smooth Ends
• The quaternionic KP hierarchy and conformally immersed 2-tori in the 4-sphere
• Discrete holomorphic geometry I. Darboux transformations and spectral curves
• A factorization of a super-conformal map
• Conformal deformations of immersed discs in $R^3$ and elliptic boundary value problems
• Simple factor dressing and the Lopez-Ros deformation of minimal surfaces in Euclidean 3-space
• Constrained Willmore Tori in the 4--Sphere
• Rectangular constrained Willmore minimizers and the Willmore conjecture
• Euclidean minimal tori with planar ends and elliptic solitons
• Envelopes and osculates of Willmore surfaces
• The Painlevé III equation of type (0,0,4,-4), its associated vector bundles with isomonodromic connections, and the geometry of the movable poles
• Twistor lifts and factorization for conformal maps from a surface to the Euclidean four-space