Change search

On the geometry of calibrated manifolds: with applications to electrodynamics
Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
2013 (English)Independent thesis Advanced level (degree of Master (Two Years)), 20 credits / 30 HE creditsStudent thesisAlternative title
Kalibrerade mångfalders geometri : med tillämpningar inom elektrodynamik (Swedish)
##### Abstract [en]

In this master thesis we study calibrated geometries, a family of Riemannian or Hermitian manifolds with an associated differential form, φ. We show that it isuseful to introduce the concept of proper calibrated manifolds, which are in asense calibrated manifolds where the geometry is derived from the calibration. In particular, the φ-Grassmannian is considered in the case of proper calibratedmanifolds. The impact of proper calibrated manifolds as a model is studied, aswell as the usefulness of pluripotential theory as tools for the model. The specialLagrangian calibration is an example of an important calibration introduced byHarvey and Lawson, which leads to the definition of the special Lagrangian differentialequation. This partial differential equation can be formulated in threeand four dimensions as det(H(u)) = Δu, where H(u) is the Hessian matrix of some potential u. We prove the existence of solutions and some other propertiesof this nonlinear differential equation and present the resulting 6- and 8-dimensional manifolds defined by the graph {x + i$\nabla$u(x)}. We also considerthe physical applications of calibrated geometry, which have so far largely beenrestricted to string theory. However, we consider the manifold (M,g,F), whichis calibrated by the scaled Maxwell 2-form. Some geometrical properties of relativisticand classical electrodynamics are translated into calibrated geometry.

2013.
##### Keyword [en]
Calibrations, geometry, plurisubharmonic functions, special Lagrangian, electrodynamics, manifolds.
##### National Category
Geometry Mathematical Analysis Physical Sciences
##### Identifiers
OAI: oai:DiVA.org:umu-80675DiVA: diva2:650840
##### Examiners
Available from: 2013-10-23 Created: 2013-09-23 Last updated: 2015-06-09Bibliographically approved

#### Open Access in DiVA

##### File information
File name FULLTEXT01.pdfFile size 533 kBChecksum SHA-512
d80fdefe727e0f6035afc3a36776e06fc5719d327d84d7ee60b87c810a2ce099f7de017a93e728b16ff8ef1aa4ebc54f0ef310747b90e67970d19851bd19f7ab
Type fulltextMimetype application/pdf
##### By organisation
Department of Mathematics and Mathematical Statistics
##### On the subject
GeometryMathematical AnalysisPhysical Sciences