Some Geometric Inequalities And Cuvature Flows

This report is composed of two main parts, one concerns some geometric inequalities about curves and an application of the Minkowski's support function, the other deals with the perimeter-preserving flow of closed convex curves in the plane and an application of the curve shortening flow on surfaces.In the first part, we will first deal with the strong Bonnesen-style inequality (2.1.3) for closed convex curves in the plane (the numbers of formulae and references are those of them in the context below). Bonnesen had first proved the weaker inequality (2.1.2) in [12] and several years later, he outlined in his monograph [13] various Bonnesen-style inequalities including (2.1.3), he considered, however, (2.1.3) as a direct consequence of Kritikos' theorem for convex bodies in higher dimensional Euclidean spaces,. Here, we will give an independent proof of the existence for inequality (2.1.3), and by the way, give an estimate on the width of the bi-enclosing annulus of closed convex curves in the plane. Secondly, in this part, we will introduce the notation of average geodesic curvature for curves in the hyperbolic plane, and investigate the relationship between the embeddedness of the curve and its average geodesic curvature. Finally, we will employ the Minkowski's support function to construct a new kind of non-circular smooth constant breadth curves in order to attack some open problems on the constant width curves ( for example, whether there is a non-circular polynomial curve of constant width, etc.)In the second part, we will first follow the ideas of Gage-Hamilton [28], Gage [26] and the author's dissertation [47] to present a perimeter-preserving closed convex curve flow in the plane, which is from physical phenomena. Under this flow, the convex initial curve will preserve its perimeter, enlarge the enclosed area and make its curvature to be positive definitely. And as the time lasts, it will become more and more circular, and finally, as the time goes to infinity, the curve will converge to a circle in the Hausdorff metric. And then, in this part, following the idea of Topping [51, 52], we will show an isoperimetric inequality (6.3.1) on surfaces of non-positive Gaussian curvature by means of the curve shortening flow on surfaces, this inequality can be considered as a generalization of the Banchoff-Pohl inequality in the Euclidean plane. It is worth noting that (6.3.1) has been obtained in Howard [34] which can be considered as a direct consequence of a special Sobolev's inequality.