学海网

毕业论文-三个带拓扑约束的变分问题，共62页，
In this article we briefly review some important functionals in modern physics
by the language of differential geometry: Chern-Simons functional, Ginzburg-Landau
functional and static knot energy. We present some properties of the solutions to the
corresponding Euler-Lagrange equations obtained from the variation of the functionals.
After that we calculate Chern-Simons functional on some 3-manifolds (e.g., Berger
sphere, warped product 3-manifolds) which obtain particular features in physics. We
also try to generalize the potential term in Ginzburg-Landau functional and get a reduced
form of this functional, involving the Chern class, on conformal Riemannian
manifolds. Finally we generalize the static knot energy in Faddeev model and get an
estimate between the Faddeev energy and the Hopf invariant, basing on which the existence
of the solution to the minimization problem can be proved.