The Isoperimetric Problems In Three Sub-Riemannian Manifolds

The isoperimetric problem has its origin in the more ancient Dido's problem which is to find the figure with maximal area and a fixed perimeter.Dido's problem is a classical isoperi-metric problem in the Euclidean plane.Its equivalent formulation is:given a fixed area,find the region with this area and minimal perimeter.The isoperimetric problem is the premier exemplar of a problem in the calculus of variations.The sharp isoperimetric inequality is also one of the most powerful tools of modern mathematics.In general the isoperimetric problem concerns the existence,characterization and regularity of isoperimetric sets,The isoperimetric problem has been associated with substantial mathematical research in numerous areas such as geometric measure theory,differential geometry,PDE,functional analysis,geometric function theory and so on.A sub-Riemannian manifold(also known as Carnot-Caratheodory space),roughly speak-ing,is a smooth manifold endowed with a distribution and a fiber-inner product on it.When the distribution is the whole tangent bundle,then the sub-Riemannian manifold reduces to a Riemannian manifold.The sub-Riemannian manifold has many important applications to dif-ferential geometry,non holomorphic mechanics,partial differential equations,control theory and so on.In the past more than thirty years,problems of sub-Riemannian geometric analy-sis,such as the optimal transportation problem,differential geometry problem and so on,have aroused people's wide attention and research.Particularly Pansu's research on the Heisenberg isoperimetric problem has stimulated many mathematicians such as Ambrosio to make an in-tensive study of calculus of variation and geometric measure theory in sub-Riemannian space.Since the sub-Riemannian isoperimetric problem is closely related to geometric measure theory,calculus of variations,optimal transportation and so on in sub-Riemannian manifold,the research on the sub-Riemannian isoperimetric problem has great significance in theory and application.In this dissertation we study the isoperimetric problems in three sub-Riemannian mani-folds.First,we establish the weighted isoperimetric and Sobolev inequalities for non-characteristic compact hypersurfaces of C2 class in Carnot groups with a smooth density ef;Next,we consider the isoperimetric problem for axially symmetric sets in Heisenberg groups Hn with density |z|p;Finally,we study the quantitative isoperimetric inequalities for sets satisfying some conditions in Grushin spaces Rh+1 with density |x|p.In Chapter 1,we describe the developments of the isoperimetric problem in Euclidean and sub-Riemannian spaces and the main results of this dissertation.In Chapter 2,we study the the weighted isoperimetric and Sobolev inequalities for non-characteristic compact hypersurfaces of C2 class in Carnot groups Gn with a smooth density ef.First,we give the first variational formula of the weighted H-perimeter measure along this hypersurface;Next,we establish the weighted coarea formula,the weighted linear isoperimet-ric inequality and a monotonicity formula about the weighted H-perimeter;Then,we prove the weighted isoperimetric inequality for non-characteristic compact hypersurfaces of C2 class;Fi-nally,using the the weighted isoperimetric inequality we obtain the weighted Sobolev inequality for non-characteristic compact hypersurfaces of C2 class without boundary.In Chapter 3,we consider the isoperimetric problem for axially symmetric sets in Heisen-berg group Hn with density |z|p.First,we give reduction formulas of weighted H-perimeter and volume for axially symmetric sets.In such a way,the weighted axially symmetric isoperimet-ric problem can be turned into Q-isoperimetric problem in the half plane = R+ x R;Next,we prove the existence of Q-isoperimetric sets and give the characterization of Q-isoperimetric sets;Then,using variational method we deduce that up to a dilation,a vertical translation and a negligible set,any weighted isoperimetric set is the Pansu ball,i.e.Eisop = {(z,t)? H1:|t|<arccos |z| + |z|(?),|z|<1};At last,we prove that up to a constant multiplicator,the horizontal radial density is only |z|p if Pansu ball Eisop is a weighted isoperimetric set in Hn with a horizontal radial density.In Chapter 4,first we study the isoperimetric problem for x-spherically symmetric sets in Grushin spaces Rh+1 with density |x|p using the same method in Chapter 3.We obtain any weighted isoperimetric set is E? = {(x,y)? Rh+1:|y|<(?)sina+?+1(t)dt,|x|<1},up to a dila-tion,a vertical translation and a negligible set.Then we establish the quantitative isoperimetric inequalities for sets satisfing some conditions.i.e.they have the same weighted volume of Ea and the symmetric difference between these sets and Ea is in given half-cylinders.