The Second Variation Of F-Energy Functions And Its Application

The second variation of F - harmonic map u : M→N has a form as follows:in which V∈Γ(u^-1TN). But it is not sufficient enough to research the uniqueness ofF - harmonic maps. So in a sense it is not the most convenient form.Taking in thought of that,in this thesis,we compute the second variation of F-energy function. Using this method,we analyse and prove the uniqueness of F - harmonic maps,which generalizes the uniqueness of F - harmonic maps.In addition,the second variation can also be used to research the unstability of some kinds of harmonic maps such as the unstability of F- harmonic maps on minimalsubmanifolds with Ric > n/2 . It has been researched very clearly in [8]. In chapter three,wewill show it.The full thesis reads as follows:In chapter one: we will make a general description on the recent researches in our field and show the main results of this thesis.In chapter two: we compute the second variation of F- energy functions,which is the base of the full thesis.In chapter three: using the second variation of F - energy functions,we prove the uniqueness of F - harmonic maps. In addition,we will show a theorem about the unstability of F—harmonic maps, which is another application of the second variation.