Some Studies On Biharmonic Maps From Surfaces

It is a well-known fact that there are many interesting examples, important applications in the beautiful theory of harmonic maps from surfaces. As a gen-eralization of harmonic maps, biharmonic maps have been attracted a growing attention from mathematicians around the world and have become a subject of study with many important progresses since2000. In this thesis, we study bi-harmonic maps from surfaces. In particular, we focus on the study of biharmonic maps from a2-sphere, a2-torus and a Riemann2-sphere. The main results include the following:(1) We study biharmonicity of rotationally symmetric maps and linear maps from surfaces with rotationally symmetric metrics. We obtain biharmonic equa-tions for such maps and a result of the existence of locally defined proper bi-harmonic maps from a Riemann surface. We also discuss the relationship be-tween biharmonicity and f-biharmonicity including a method for constructing/-biharmonic maps (biharmonic maps) from given f-biharmonic maps (biharmonic maps).(2) We study biharmonicity of rotationally symmetric maps and linear maps from a2-sphere. We obtain some constructions and classifications of biharmonic maps from a2-sphere which include a classification of biharmonic maps in the family of linear maps including a family of rotationally symmetric maps between2-spheres, and many examples of proper biharmonic maps defined locally on S2including proper biharmonic maps S2â†'Sn(n≥2) from a round sphere with some singular points.(3) We study biharmonicity of rotationally symmetric maps from a Riemann2-sphere. We give some constructions and classifications of biharmonic maps from a Riemann2-sphere. These include a classification of biharmonic maps in a family of rotationally symmetric maps from a Riemann2-sphere to a2-sphere, some existence results of locally defined proper biharmonic maps from a Riemann2-sphere, a method for constructing proper biharmonic maps from given proper biharmonic maps, and many examples of proper biharmonic maps defined locally on S2from a Riemann2-sphere including proper biharmonic maps (S2,f-1g0)â†'Sn(n≥2) from a Riemann2-sphere with some singular points. (4) We study biharmonicity of a family of linear maps from T2into S2. We obtain some classifications on biharmonic maps within family of linear maps from a (non flat) flat T2into S2. In our classifications of the family of maps, no proper biharmonic map of degree±1was found.