Totally Real Pseudo-Umbilical Submanifold, Wulff-Ros Isoperimetric Inequality And Orlicz-Brunn-Minkowski Type Inequalities

In this thesis, Pinching problem about submanifold, Wulff flows, Lp-Brunn-Minkowski theory and dual Lp-Brunn-Minkowski theory, Orlicz-Brunn-Mink-owski theory and dual Orlicz-Brunn-Minkowski theory are investigated. We obtain some intrinsic rigidity theorems, sufficient and necessary condition of equality sign in the Wulff-Gage isoperimetric inequality, the isoperimetric inequalitiy of Wulff-Ros. We chalk up the definition of the dual Orlicz mixed quermassintegrals. We prove the dual Orlicz-Minkowski inequality and the dual Orlicz-Brunn-Minkowski inequality. All these areas are attracted increased interest by more and more researchers.In the second chapter, let Mn be a compact totally real pseudo-umbilical sub-manifold of a complex space form, we study the position of the parallel umbilical normal vector field of Mn in the normal bundle. By estimating the Laplacian for the length square of the second fundamental form, choosing a suitable frame field, using Hopf maximum principle and Stokes theorem, through the length square of the sec-ond fundamental form, the sectional curvature restrictions and the Ricci curvature, obtain some rigidity theorems such that Mn becomes totally umbilical submanifold.In the third chapter, first gives a strengthened form of Wulff-Gage isoperimetric inequality about bounded convex domain K and W in Euclidean plane R2, i.e., it is proved that the equality sign in Wulff-Gage isoperimetric inequality holds if and only if plane convex domain K and W are homothetic, get a veritable Wulff-Gage isoperimetric inequality. In particular, when W as disc, we can get the strengthened of Gage isoperimetric inequality. And then, by making use of the Minkowski support function, it is rewrited Wulff-Gage isoperimetric inequality in an integral inequality of 2π-periodic functions, which can be looked upon as the analytic version of Wulff-Gage isoperimetric inequality. We also proved that the equality sign in Wulff entropy isoperimetric inequality holds if and only if plane convex domain K and W are homothetic.In the fourth chapter, on the basis of the properties of support function about plane ovaliod domain K+tW, prove the Wulff-Ros isoperimetric deficit is an invari-ant of the outward Wulff flows about W. We investigate the Wulff-Ros isoperimetric inequality and seek out the relationship between the Wulff-Ros isoperimetric deficit and the Wulff isoperimetric deficit. We obtain some lower bounds about Wulff-Ros isoperimetric inequality. Finally, we prove the circulate inequality for the sequence of the Wulff curvature.In the fifth chapter, we study the dual Orlicz-Brunn-Minkowski theory. Through discussion radial Orlicz linear combination, we extend the classical dual mixed quer-massintegrals to the Orlicz setting, obtain the definition of the dual Orlicz mixed quermassintegrals. By calculating the Orlicz first variations of dual quermassinte-grals, receive the integral formula of the dual Orlicz mixed quermassintegrals. We establish the dual Orlicz-Minkowski inequality for dual Orlicz mixed quermassin-tegrals and the dual Orlicz-Brunn-Minkowski inequality for radial Orlicz combina-tion. The equivalence between the dual Orlicz-Minkowski inequality and the dual Orlicz-Brunn-Minkowski inequality is demonstrated. We generalize the dual Orlicz-Cauchy-Kubota formula. We shows the continuity and the uniqueness result of the dual Orlicz mixed quermassintegrls, prove a property of the dual Orlicz mixed quer-massintegrls under linear transformation. Further, the circulate inequality for the dual Orlicz mixed quermassintegrls is obtained.In the sixth chapter, based on the notions of Blaschke-Minkowski homomor-phisms in valuation theory of convex bodies and radial Blaschke-Minkowski homo-morphisms in valuation theory of star bodies, the Orlicz-Brunn-Minkowski type in-equalities for Blaschke-Minkowski homomorphisms with respect to Orlicz linear com-bination and the dual Orlicz-Brunn-Minkowski type inequalities for radial Blaschke-Minkowski homomorphisms with respect to radial Orlicz linear combination are es-tablished. The aim of this chapter is to establish the (dual) Lp-Brunn-Minkowski type inequalities for the (dual) quermassintegral differences functions of (radial) Blaschke-Minkowski homomorphisms, the differences functions about the quermass- integral of Blaschke-Minkowski homomorphisms and the dual quermassintegral of radial Blaschke-Minkowski homomorphisms.