The Related Problems And The Geometry Of ?_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 ?S,F-harmonic maps and related problems including the stability of ?S,F-harmonic maps,Liouville type results of?S,F,?-harmonic maps and ?S,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 E?S,F associated to smooth maps between Riemannian manifolds.We derive the first variation formula and the second variation formula of E?S,F.By using F-stress-energy tensor,we obtain some Liouville theorems of ?S,F-harmonic maps.Then we prove that ?S,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 ?S,F,?-harmonic maps with respect to the Ginzburg-Landau type energy functional E?S,F,?.And we deduce the first variation formula of this energy functional E?S,F,?.By assuming an asymptotic condition at infinity,we estimate the upper and lower bounds of the energy functional E?S,F,? and obtain some Liouville type results for ?S,F,?-harmonic maps under some certain conditions.In the last chapter,we consider the ?S,H-harmonic maps from Riemannian manifolds to pseudo-Hermitian manifolds.And we derive the variational formulas of energy functional E?S,H related to ?S,H-harmonic maps.Next,we obtain Liouville type theorem of ?S,H-harmonic maps and obtain that ?S,H-harmonic maps which are from?H-SSU manifolds is unstable through the second variational formula.At the same time,we obtain some geometric results of ?S,H-harmonic maps when the target manifolds are Sasakian manifolds.