Consensus Reaching Process With Minimum Information Loss In Linguistic Group Decision Making

Harmonic maps is one of the most important research topics in differential geometry and it has been widely applied to geometry topology and theoretical physics.A sub-Riemannian manifold（also be called Carnot-Carath’edory space）,roughly speaking,is a manifold endowed with a distribution and a fiber-inner on the distribution.When the distri-bution is the whole tangent bundle,then the sub-Riemannian manifold becomes a Riemannian manifold.The sub-Riemannian manifold has important theoretical and practical applications in non-holonomic mechanics、the theory of geometric control、partial differential equation-s、gauge field theory、quantum mechanics and so on.The main purpose of this thesis is to extend the problem of harmonic maps between Rie-mannian manifolds to the problem of so-called generalized CC harmonic maps from Riemanni-an manifolds into some sub-Riemannian manifolds.The vanishing theorems and their applica-tions for generalized CC harmonic maps which are into sub-Riemannian manifolds、Grushin spaces、Heisenberg groups and Carnot groups are studied respectively by using classical Rie-mannian geometry and analysis ideas.This thesis is composed of six chapters.In Chapter 1,we introduce the background of harmonic maps between Riemannian mani-folds and the research of harmonic maps related to sub-Riemannian manifolds.After that,the main results of this thesis are presented.In Chapter 2,we study vanishing theorems for f-CC harmonic maps with potential H into general sub-Riemannian manifolds.We introduce the energy functional associated with f-CC with potential H and give the concept of f-CC harmonic maps with potential H.Then the first variational formula is established and the stress-energy tensor is introduced,and then the condition satisfying the conservation law is obtained.By using the method of stress-energy tensor,the energy monotonicity formulas are established and then vanishing theorems and their applications for f-CC harmonic maps with potential H under small energy conditions、slowly divergent energy conditions and boundary vanishing conditions respectively are obtained by these monotonicity formulas.In Chapter 3,we investigate vanishing theorems for f-CC stationary maps with potential H into the Grushin space G_α^h+1.By analyzing the geometric structure and properties of the Grushin space G_α^h+1,we introduce the energy functional associated with f-CC with potential H and give the concept of f-CC stationary maps with potential H.Then the stress-energy tensor is introduced and the integral formulas are established,and then the energy monotonicity formulas are obtained.By using energy monotonicity formulas,vanishing theorems for f-CC stationary maps with potential H are obtained under small energy conditions、slowly divergent energy conditions and boundary vanishing conditions respectively,and their applications are also obtained.In Chapter 4,vanishing theorems for（weakly）F-CC stationary maps with potential H into Heisenberg group Hⁿare obtained.We introduce the energy functional associated with F-CC with potential H and give the concept of（weakly）F-CC stationary maps with potential H.Then the stress-energy tensor is introduced and the integral formulas are established,and then the energy monotonicity formulas are obtained.By using energy monotonicity formulas,van-ishing theorems for（weakly）F-CC stationary maps with potential H are obtained under small energy conditions、finite energy conditions、slowly divergent energy conditions and boundary vanishing conditions respectively,and their applications are also obtained.In Chapter 5,we establish vanishing theorems for（weakly）quasi f-CC stationary maps with potential H into Carnot group G.We introduce the energy functional associated with quasi f-CC with potential H and give the concept of（weakly）quasi f-CC stationary maps with po-tential H.Then the stress-energy tensor is introduced and the integral formulas are established.By using these integral formulas,vanishing theorems for（weakly）quasi f-CC stationary map-s with potential H are obtained under finite energy conditions、small energy conditions and boundary vanishing conditions respectively,and their applications are also obtained.In Chapter 6,the summary and prospect of this thesis are presented.