Some Problems Of The Generalized Φ-harmonic Map

Harmonic mapping is a extension of the concepts of geodesics,minimal submanifolds and harmonic functions in differential geometry.Harmonic mapping originated from physical problems and has been developed for over half a century.It has also been widely used in other disciplines,such as geometric topology and theoretical physics.It is an important research object in Riemannian geometry.In this thesis,we mainly study the related problems of generalized Φ-harmonic maps,including Liouville type results for ΦS,p,ε-harmonic maps and the stability ofΦ(5)-harmonic maps.It consists of the following three chapters.In the first chapter,we introduce the research background and research status of generalized Φ-harmonic maps and main results obtained in the thesis.In the second chapter,we introduce the notion of ΦS,p,ε-harmonic map which is a critical point of the energy functional EΦS,p,ε(u)with respect to any compact supported variation of u.We establish the monotone formula and obtain the Liouville type results for the ΦS,p,ε-harmonic map by using the stress-energy tensor and the asymptotic condition of the map at infinity.In the third chapter,we study Φ(5)-harmonic maps and Φ(5)-SSU manifolds.And we obtain some classical examples for Φ(5)-SSU manifolds.By using the extrinsic average variational method,we prove that stable Φ(5)-harmonic maps from compactΦ(5)-SSU manifolds or into compact Φ(5)-SSU manifolds must be constant.