数学年刊A辑
數學年刊A輯
수학년간A집
CHINESE ANNALS OF MATHEMATICS,SERIES A
2009年
6期
787-792
,共6页
直径%体积比较定理%Hausdorff收敛
直徑%體積比較定理%Hausdorff收斂
직경%체적비교정리%Hausdorff수렴
Diameter%Volume comparison theorem%Hausdorff convergence
设M~(2n)是2n维紧致无边单连通的Riemannian流形, S~(2n)为欧氏空间R~(2n+1)中的单位球面.探讨了满足截面曲率K_M∈(0,1),体积0<V(M)≤2(1+η)V(B_(3/4π))的流形M~(2n)的直径估计,这里η是某个仅依赖于n的正数,B_(3/4π)是S~(2n)上半径为3/4π的测地球,并且给出了这类流形上的一个gap现象及流形上Laplacian算子第一特征值的一个下界估计.
設M~(2n)是2n維緊緻無邊單連通的Riemannian流形, S~(2n)為歐氏空間R~(2n+1)中的單位毬麵.探討瞭滿足截麵麯率K_M∈(0,1),體積0<V(M)≤2(1+η)V(B_(3/4π))的流形M~(2n)的直徑估計,這裏η是某箇僅依賴于n的正數,B_(3/4π)是S~(2n)上半徑為3/4π的測地毬,併且給齣瞭這類流形上的一箇gap現象及流形上Laplacian算子第一特徵值的一箇下界估計.
설M~(2n)시2n유긴치무변단련통적Riemannian류형, S~(2n)위구씨공간R~(2n+1)중적단위구면.탐토료만족절면곡솔K_M∈(0,1),체적0<V(M)≤2(1+η)V(B_(3/4π))적류형M~(2n)적직경고계,저리η시모개부의뢰우n적정수,B_(3/4π)시S~(2n)상반경위3/4π적측지구,병차급출료저류류형상적일개gap현상급류형상Laplacian산자제일특정치적일개하계고계.
Let M~(2n) be a 2n-dimensional compact, simply connected Riemannian manifold without boundary and S~(2n) be the unit sphere in Euclidean space R~(2n+1) . The authors derive an estimate of the diameter in this note whenever the manifold concerned satisfies that the sectional curvature K_M varies in (0, 1]and the volume V(M) is not larger than 2(1+η)V(B_(3/4π)) for some positive number η depending only on n, where B_(3/4π) is the geodesic ball on S~(2n) with radius 3/4π A gap phenomenon of the manifold concerned is given and finally a lower bound of the first eigenvalue of Laplacian operator on manifold M is obtained.
