Lower bounds on Ricci curvature and quantitative behavior of singular sets
Lower bounds on Ricci curvature and quantitative behavior of singular sets
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DOI：
10.1007/s00222-012-0394-3
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发表时间：
2011-03
影响因子：
3.1
通讯作者：
J. Cheeger;A. Naber
中科院分区：
文献类型：
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作者：
J. Cheeger;A. Naber
设Yn表示V-非坍缩黎曼流形的Gromov-Hausdorff极限M^{n}_{i}\stackrel{d_{GH}}{\longrightarrow} Y^{n}$，其中.奇异集是一个分层，如果没有切锥等距地分裂出一个因子λ k+1。这里，我们定义了对所有η>0，0<r≤1，第k个有效奇异层满足。锐化已知的Hausdorff维数界，我们证明了对于ally，的管状邻域的体积满足。这个证明涉及到一个数量微分的论证。这个结果有应用到爱因斯坦流形。记C2-调和半径≤r的点集。如果也是Kähler-Einstein在L ~ 2曲率上有界的，则对于ally。在Kähler-Einstein情形下，在不假设上有任何积分曲率界的情况下，我们得到了一个稍微弱一些的体积界，在这个体积界上，当p <2时，我们得到了一个a priorLp曲率界。本文中开发的方法是新的，适用于许多其他情况。这些包括调和映射，极小超曲面，平均曲率流和椭圆方程的临界解集。
LetYndenote the Gromov-Hausdorff limit $M^{n}_{i}\stackrel{d_{\mathrm{GH}}}{\longrightarrow} Y^{n}$ of v-noncollapsed Riemannian manifolds with. The singular sethas a stratification, whereif no tangent cone atysplits off a factor ℝk+1isometrically. Here, we define for allη>0, 0<r≤1, thek-th effective singular stratumsatisfying. Sharpening the known Hausdorff dimension bound, we prove that for ally, the volume of ther-tubular neighborhood ofsatisfies. The proof involves aquantitative differentiationargument. This result has applications to Einstein manifolds. Letdenote the set of points at which theC2-harmonic radius is ≤r. If also theare Kähler-Einstein withL2curvature bound,, thenfor ally. In the Kähler-Einstein case, without assuming any integral curvature bound on the, we obtain a slightly weaker volume bound onwhich yields an a prioriLpcurvature bound for allp<2. The methodology developed in this paper is new and is applicable in many other contexts. These include harmonic maps, minimal hypersurfaces, mean curvature flow and critical sets of solutions to elliptic equations.