The Related Problems And The Geometry Of Φ_S,F-Harmonic Map

Harmonic mapping theory is one of the core theories of differential geometry,which spans many fields such as physics,partial differential equation theory,finite element theory,numerical calculation and differential approximation.In this thesis,we mainly study the geometry of ΦS,F-harmonic maps and related problems including the stability of ΦS,F-harmonic maps,Liouville type results ofΦS,F,ε-harmonic maps and ΦS,H-harmonic maps.There is four chapters in the thesis.In the first chapter,we introduce the research background and status of the geometry of harmonic maps and related problems,the preliminary knowledge used in the subsequent proof and the main results obtained in the thesis.In the second chapter,we mainly study an energy functional EΦS,F associated to smooth maps between Riemannian manifolds.We derive the first variation formula and the second variation formula of EΦS,F.By using F-stress-energy tensor,we obtain some Liouville theorems of ΦS,F-harmonic maps.Then we prove that ΦS,F-harmonic map which is from the sphere Sm（m≥ 5）or into the sphere Sn（n≥ 5）is unstable.In the third chapter,we introduce the concept of the ΦS,F,ε-harmonic maps with respect to the Ginzburg-Landau type energy functional EΦS,F,ε.And we deduce the first variation formula of this energy functional EΦS,F,ε.By assuming an asymptotic condition at infinity,we estimate the upper and lower bounds of the energy functional EΦS,F,ε and obtain some Liouville type results for ΦS,F,ε-harmonic maps under some certain conditions.In the last chapter,we consider the ΦS,H-harmonic maps from Riemannian manifolds to pseudo-Hermitian manifolds.And we derive the variational formulas of energy functional EΦS,H related to ΦS,H-harmonic maps.Next,we obtain Liouville type theorem of ΦS,H-harmonic maps and obtain that ΦS,H-harmonic maps which are fromΦH-SSU manifolds is unstable through the second variational formula.At the same time,we obtain some geometric results of ΦS,H-harmonic maps when the target manifolds are Sasakian manifolds.