Several Results In Geometric Analysis

In this thesis, we mainly study several important and interesting problems in geometric analysis and Ricci flow, which is divided into four chapters:In the first chapter, at first, we briefly recall the theory of Ricci flow, which was established to solve the famous Poincare conjecture by R. Hamilton in 1982. Then we introduce the background and motivation of the results in our thesis and present the main results briefly.In the second chapter, we establish a new interpolated curvature invariance between the well known nonnegative and 2-nonnegative curvature invariant along the Ricci flow, which is named as (λ1,λ2)-nonnegative curvature condition. In par-ticular, it includes the nonnegative and 2-nonnegative curvature conditions as its special cases. By using the maximum principle for convex sets established by R. Hamilton, B. Chow and P. Lu [17], we firstly prove that the (λ1,λ2)-nonnegative curvature is invariant along the Ricci flow. Thus as a corollary, we prove the non-negative and 2-nonnegative curvature invariance along the Ricci flow again. Then a related strong maximum principle for the (λ1,λ2)-nonnegativity is also derived along Ricci flow. By using the famous result proved by C. Bohm and B. Wilking [1] that on a compact manifold the normalized Ricci flow evolves a Riemannian metric with 2-positive curvature operator to a limit metric with constant sectional curvature, we derived that on a compact manifold the normalized Ricci flow also evolves a Riemannian metric with (λ1,λ2)-positive curvature operator to a limit metric with constant sectional curvature. Based on these, finally we obtain a rigid-ity property of the scalar curvature for (λ1,λ2)-nonnegative curvature operator.In the third chapter, we deal with f-conformal Killing vector fields, a gen-eralization of Killing vector fields and conformal Killing vector fields involving a function f and two real parametersα,β. Firstly under certain conditions on f and the parameters, some non-existence results for such vector fields are proven. Thus as a corollary, we prove the corresponding non-existence for Killing and conformal Killing vector fields again. We also derive a dimensional estimate for the space of f-conformal Killing vector fields and a non-existence result for the f-conformal Killing vector fields corresponding to harmonic 1-forms with weaker conditions. Furthermore, we generalize the Kazdan-Warner [51] and Bourguignon-Ezin [2] identities for conformal Killing vector fields to f-conformal Killing vector fields. Based on these, finally we consider the f-generalized solutions of Yamabe solitons, which is a generalization of the Yamabe solitons. By using the Kazdan-Warner-type identity for f-conformal Killing vector fields, we prove that when n≥3, an f-generalized solutions of Yamabe solitons on the n-dimensional closed manifold has constant scalar curvature.In the fourth chapter, we deals with isoperimetric-type inequalities for closed convex curves in the Euclidean plane R2. By using Fourier series, we respectively derive the improved versions of two reverse isoperimetric inequalities proved by S. L. Pan, H. Zhang in [69] and S. L. Pan, J. N. Yang in [68]. Our proof is simpler than the approach in [69] and [68]. In fact our result confirms a conjecture by S. L. Pan, X. Y. Tang and X. Y. Wang in [66]. Then we derive a family of parametric isoperimetric-type inequalities involving the following geometric functionals asso-ciated to a given convex curve with a simple Fourier series proof:length, area of the region included by the curve, area of the domain enclosed by the locus of curvature centers and integral of the radius of curvature. As a corollary of our parametric isoperimetric-type inequalities, we obtain some new geometric Bonnesen-type in-equalities. Based on these, finally we investigate stability properties of such in-equalities, and prove that they do have the good stability property with respect to both the Hausdorff distance and the L2-metric.