Affiliation:
1. Shanghai Center for Mathematical Sciences , Fudan University , Shanghai 200438 , P. R. China
Abstract
Abstract
In this paper, we study the angle estimate of distance functions from minimal hypersurfaces in manifolds of Ricci curvature bounded from below using Colding’s method in [T. H. Colding,
Ricci curvature and volume convergence,
Ann. of Math. (2) 145 1997, 3, 477–501].
With Cheeger–Colding theory, we obtain the Laplacian comparison for limits of distance functions from minimal hypersurfaces in the version of Ricci limit space.
As an application, if a sequence of minimal hypersurfaces converges to a metric cone
C
Y
×
ℝ
n
-
k
{CY\times\mathbb{R}^{n-k}}
(
2
≤
k
≤
n
{2\leq k\leq n}
) in a non-collapsing metric cone
C
X
×
ℝ
n
-
k
{CX\times\mathbb{R}^{n-k}}
obtained from ambient manifolds of almost nonnegative Ricci curvature, then we can prove a Frankel property for the cross section Y of CY.
Namely, Y has only one connected component in X.
Funder
National Natural Science Foundation of China
Subject
Applied Mathematics,General Mathematics
Reference30 articles.
1. U. Abresch and D. Gromoll,
On complete manifolds with nonnegative Ricci curvature,
J. Amer. Math. Soc. 3 (1990), no. 2, 355–374.
2. W. K. Allard,
On the first variation of a varifold,
Ann. of Math. (2) 95 (1972), 417–491.
3. L. Ambrosio, N. Fusco and D. Pallara,
Functions of bounded variation and free discontinuity problems,
Oxford Math. Monogr.,
The Clarendon, Oxford 2000.
4. L. Ambrosio, N. Gigli and G. Savaré,
Metric measure spaces with Riemannian Ricci curvature bounded from below,
Duke Math. J. 163 (2014), no. 7, 1405–1490.
5. K. Bacher and K.-T. Sturm,
Localization and tensorization properties of the curvature-dimension condition for metric measure spaces,
J. Funct. Anal. 259 (2010), no. 1, 28–56.