VOLUME ABOVE DISTANCE BELOW

被引:0
作者
Allen, Brian [1 ]
Perales, Raquel [2 ]
Sormani, Christina [3 ,4 ]
机构
[1] Lehman Coll, 250 Bedford Pk Blvd W, Bronx, NY 10468 USA
[2] Invest Matemat, AC Jalisco S-N,Col Valenciana, Mexico City 36023, Mexico
[3] CUNYGC, 365 Fifth Ave, New York, NY 10016 USA
[4] Lehman Coll, 365 Fifth Ave, New York, NY 10016 USA
关键词
CONVERGENCE; MANIFOLDS;
D O I
暂无
中图分类号
O1 [数学];
学科分类号
0701 ; 070101 ;
摘要
Given a pair of metric tensors g(1) >= g(0) on a Riemannian manifold, M, it is well known that Vol(1) (M) >= Vol(0) (M). Furthermore one has rigidity: the volumes are equal if and only if the metric tensors are the same g(1) = g(0) . Here we prove that if g(j) >= g(0) and Vol(j) (M) -> Vol(0) (M) then (M, g(j) ) converge to (M, g(0) ) in the volume preserving intrinsic flat sense. Well known examples demonstrate that one need not obtain smooth, C-0 , Lipschitz, or even Gromov-Hausdorff convergence in this setting. Our theorem may also be applied as a tool towards proving other open conjectures concerning the geometric stability of a variety of rigidity theorems in Riemannian geometry. To complete our proof, we provide a novel way of estimating the intrinsic flat distance between Riemannian manifolds which is interesting in its own right.
引用
收藏
页码:837 / 874
页数:38
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