微分几何中几个不等式及其推广

Several Inequalities in Differential Geometry and Their Generalizations

研究了微分几何中的几个不等式 ,提出了几个相关的不等式 .( 1 )对平面上的Schur定理 ,给出了一种解析的证法 ,它比已知的一些 (几何的 )证法显得简洁、明快 ,进而还用积分几何方法作了些讨论 .( 2 )对欧氏空间中闭曲线的F偄ｒｙ不等式 ,用活动标架法 ,将其推广到了球面 (正常高斯曲率曲面 )中 .( 3)对三维欧氏空间中闭曲面的F偄ｒｙ不等式 ,用活动标架法 ,将其中积分式前的常系数 4 π进一步改进为 1 ;此外 ,还将其推广到四维的欧氏空间中 .这一不等式可能推广于更高维或一般的欧氏空间中 ,有待进一步研究 .

Several inequalities are discussed. (1)For Schur's inequality on convex curves of plane, we give a new analytic proof for it, which maybe is simpler or clearer than known ones; we make further discussions by means of integral geometry and get more results. Moreover several related inequalities are put forward and proved. We also propose a conjecture which is generalization of Schur's inequality in case of spherical surface. (2)For Fáry's inequality on closed curves of Euclidean space E3, we generalize it into spherical surface (i.e. surface with positive constant Gauss curvature) using method of moving frame. (3)For Fáry's inequality on closed surface of Euclidean space E3:A≤4πR2∫ Σ|K|dσ, we enhance it to A≤R2∫ Σ|K|dσ using method of moving frame. Moreover this inequality has been also generalized into 4-dimension case:A≤R2∫ Σ(|K 3|+|K 4|)dσ.Furthermore, a conjecture on further generalization to higher dimension case or general Euclidean space is proposed which requires further study.