Research On The Geometry Of ?_S-harmonic Maps And Inequalities In Convex Geometric Analysis

There are rich contents and applications in harmonic maps theory.Harmonic maps are extended to many other versions such as p-harmonic maps,f-harmonic maps,F-harmonic maps,F-stationary maps and ?-harmonic maps.Liouville type theorems are core contents of harmonic maps theory.The research on Liou-ville type theorems has great theoretical significance.Balls and ellipsoids are very important geometric objects in integral and convex geometric analysis and play an very important role in the Brunn-Minkowski theory.They are often applied in isoperimetric type problems and other extremal problems.Affine isoperimetric inequalities are the extension of isoperimetric inequalities and very important in convex geometric analysis.The dissertation mainly discusses Liouville type theorems and stability of?S-harmonic maps,Liouville type theorems of generalized symphonic maps,Lp mixed integral affine surface area,the solution of an extremal problem and affine isoperimetric inequalities.In Chapter 1,harmonic maps theory and the Brunn-Minkowski theory in integral geometry and convex geometry are introduced.In Chapter 2,Liouville type theorems and stability of ?S-harmonic maps are discussed.Jin's method is generalized to ?S-harmonic maps.The procedure consists of two steps.The first step is to use the asymptotic assumption of the maps at infinity to obtain the upper energy growth rates of ?S-harmonic maps.The second step is to use the monotonicity formulas to obtain the lower energy growth rates of these maps.Under suitable conditions on the Hessian of the distance function of the domain manifolds,one may show that these two growth rates are contradictory unless the ?S-harmonic maps are constant maps.The stability of these maps is investigated by an extrinsic average variational method in the calculus of variations and the result that any stable ?S-harmonic maps from or into compact ?-SSU manifolds must be constant is obtained.In Chapter 3,Monotonicity formulas and Liouville type theorems of gener-alized symphonic maps are explored.To generalize the Liouville type results for harmonic maps to the generalized symphonic maps,a smooth map u from a Rie-mannian manifold to the standard Euclidean space and the functional ??(u)are considered.By the first variation formula,the notion of generalized symphonic map for the functional ??(u)is defined.Then the stress-energy tensor S?? asso-ciated with the functional ??(u)is introduced.The result that the generalized symphonic map satisfies the conservation law is obtained,that is,div S??=0.By using the stress-energy tensor,some monotonicity formulas for these maps are ob-tained,and then some Liouville type results are proved from these monotonicity formulas under suitable growth conditions on the functional ??(u).In Chapter 4?a variation on integral affine surface area Aj(K)with respect to Lp Minkowski addition is investigated and a variational formula is established.The j-th Lp mixed integral affine surface area Aj,p(K,L)of K,L ??on is intro-duced based on the variational formula.Affine invariance of Aj,p(K.L)under special linear transformations is proved.The existence and uniqueness of so-lution of a constrained maximization problem are established.Then an affine isoperimetric inequalities with relation to Aj(K)is obtained.