TITLE: Hausdorff convergence of submanifolds in Riemannian manifolds
QUESTION [10 upvotes]: Let $(M^n,g)$ be a smooth compact Riemannian manifold. It is well known that any sequence $\{X_i\}$ of compact subsets of $M$ has a subsequence which converges in the Hausdorff metric to a compact subset $X\subset M$.
Assume now that $\{X_i\}$ are, in addition, smooth connected submanifolds with a uniform lower bound on the sectional curvature and a uniform upper bound on the diameter with respect to the induced intrinsic (!) metric.
Question. What is known about a limit space $X$?
E.g. should $X$ have an integer Hausdorff dimension?
Remark. Under the above assumptions, the Gromov compactness theorem implies that there is a subsequence converging to a compact Alexandrov space $Y$ in the Gromov-Hausdorff sense (thus $Y$ is not a subset of $M$ a priori). Is there any relation between $X$ and $Y$? (I believe there is a 1-Lipschitz map onto $Y\to X$.)

REPLY [8 votes]: You are right about 1-Lipschitz map, but that is about all you can expect.
In particular the dimension of $X$ might be not integer.
Assume your $X_i$ all isometric to to a flat torus $\mathbb{T}$ and
$M=\mathbb{E}^N$; that is your manifold is the Euclidean space of high enough dimensions $N$.
By Nash and Whitney, any short map $\mathbb{T}\to \mathbb{E}^N$ can be arbitrary well approximated by an isometric embedding.
It reamins to construct a short map $\mathbb{T}\to \mathbb{E}^N$ with non-integer Hausdorff dimension. 
An image of that type is shown on the picture.
It can be arranged that the Cantor set on the top has Hausdorff higher than 1 and lower than 2. 
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