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As far as I can tell, a major motivation for the study of length spaces is that they arise as Gromov-Hausdorff limits of Riemannian manifolds. Specifically,

  1. A complete connected Riemannian manifold is a complete length space.
  2. A Gromov-Hausdorff limit of complete length spaces is a complete length space.

But of course there are stronger metric properties of Riemannian manifolds that one might hope would carry over to their limits. One that I have been wondering about is the continuity (say in the compact-open topology EDIT (see below): some other topology) of the length functional. After a couple of days' thought I've decided I have absolutely no intuition for this. So, I'd be very glad to hear:

  1. Is the length functional of a complete connected Riemannian manifold indeed continuous? (Proof in some special cases: if $\Gamma:[0,1]\times (-\epsilon,\epsilon)\to M$ is continuously differentiable, then $$ \lim_{t\to 0}\int_0^1|\frac{\partial\Gamma}{\partial s}(s,t)|ds = \int_0^1|\frac{\partial\Gamma}{\partial s}(s,0)|ds $$ by limit-swapping.)
  2. Is a Gromov-Hausdorff limit of complete-length-spaces-with-continuous-length-functional also a complete-length-space-with-continuous-length-functional?

EDIT: It was quickly pointed out by Anton Petrunin, Pietro Majer and Vitali Kapovitch that for the compact-open topology, the answer to the first question is no (and that the second question is vacuous). Is it possible that there is some finer topology on (perhaps some subspace of) the space of curves in a length space, for which the answer to these questions is yes?

For instance, consider the following property that a length space $X$ (with length functional $\mathcal{L}$) might possess:

For any Lipschitz map $\Gamma:[0,1]\times(-\epsilon,\epsilon)\to X$, $$ \lim_{t\to 0}\ \mathcal{L}(\Gamma(\cdot,t))=\mathcal{L}(\Gamma(\cdot,0)). $$

It seems plausible to me that this would be true of complete connected Riemannian manifolds and that it would not be true of arbitrary length spaces. Is this so? And if so, is the set of length spaces which do have this property Gromov-Hausdorff closed?

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On the space of curves equipped with uniform convergence, the length functional is upper semicontinuous, but not continuous. Is it you question? –  Anton Petrunin Dec 20 '11 at 17:14
    
I think you mean lower semicontinuous, wrto uniform convergence. indeed it's a supremum of continuous functionals, namely finite sums on partitions. (and, being l.s.c., it attains a minimum on compact sets, which makes everybody happy). –  Pietro Majer Dec 20 '11 at 17:40
    
Yes, the length functional of any complete length space is lower semicontinuous. My (first) question is whether for Riemannian manifolds it is actually continuous. –  macbeth Dec 20 '11 at 18:11
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the length functional on any fixed length space is obviously only semicontinuos but not continuous. For example you can always approximate a constant point curve by non-constant ones of fixed length. –  Vitali Kapovitch Dec 20 '11 at 18:25
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@macbeth Your new property fails for $\mathbb R$. The counterexample is basically the same as before: take a sequence of oscillating curves of constant length with oscillation going to zero. it's easy to extend it to a Lipschitz family using straight line homotopy. since it fails for $\mathbb R$ it fails for any inner metric space other than a point. Nothing like this is going to work in such generality. You need higher regularity on the family of curves which means that you need stronger convergence than Gromov-Hausdorff if you want a property like this to be preserved under limits. –  Vitali Kapovitch Dec 22 '11 at 1:02
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1 Answer

In a metric space $(X,d)$, you may look at the length functional as a FUNCTIONAL defined over the space of Lipschitz functions from $[0,1]$ to $(X,d)$. One of the natural notions of convergence of functionals is the gamma-convergence.

For a sequence of metric spaces $(X_{n},d_{n})$ which converges (in the Gromov-Hausdorff distance) to a metric space $(X,d)$, we know that there is a metric space $(Y,D)$, a sequence of isometric embeddings $F_{n}: (X,d_{n}) \rightarrow (Y,D)$ and another isometric embedding $F:(X,d) \rightarrow (Y,D)$, such that the sequence $F_{n}(X_{n})$ Hausdorff converges to $F(X)$.

A natural question would be: is there a sequence of isometries $f_{n}: (X,d_{n}) \rightarrow (Y,D_{n})$ and an isometry $f:(X,d) \rightarrow (Y,D)$, such that

  • $D_{n}$ and $D$ generate the same uniformity on $Y$,

  • $D_{n}$ converges uniformly (on compact sets, on bounded sets, etc, pick your choice) to $D$,

  • pick the topology of uniform convergence on the space $\mathcal{C}([0,1],Y)$. Then the sequence of length functionals associated to $(f_{n}(X_{n}), D_{n})$ gamma converges to the length functional of the limit $(f(X), D)$.

If true then any sequence of $D_{n}$ geodesics (length minimizers) converges to a geodesic of $D$, for example.

This is true in a Riemannian manifold (seen as the metric space $(X,d)$), if you take $(X_{n},d_{n})$ as $(B(x,1/n), \frac{1}{n}d)$, which GH converges to the (unit ball in the) tangent space at $x \in X$, for every point $x$. (This follows from results from the paper G. Buttazzo, L. De Pascale, I. Fragala, Topological equivalence of some variational problems involving distances, Discrete Contin. Dynam. Systems 7 (2001), no. 2, 247-258).

It is also true for sub-riemannian manifolds, see the paper M. Buliga, A characterization of sub-riemannian spaces as length dilation structures constructed via coherent projections, Commun. Math. Anal. 11 (2011), No. 2, pp. 70-111, arxiv link.

All this is related to the metric characterization of riemannian (and sub-riemannian) spaces, but in a different way than the great paper I.G. Nikolaev, A metric characterization of riemannian spaces, Siberian Advances in Mathematics, 1999, v. 9, N4, 1-58, MathSciNet link.

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Thanks, this is interesting! You discuss rescaling sequences; is it known whether a general GH-convergent sequence of Riemannian manifolds has convergent length functionals in the sense you describe? (That is, given $(M_n, g_n)$ GH−converging to $(X,d)$, is there a sequence of isometric embeddings into a common metric space which generate the same uniformity etc and such that the length functionals gamma-converge?) –  macbeth Dec 25 '11 at 20:49
    
The cited paper by Buttazzo, De Pascale, Fragala contains results concerning general sequences of distances on the same space. On one side, these results apply for situations more general than rescaling sequences (but look very carefully to the hypotheses, some of them apply only for distances of riemannian type). On the other side, the only place I know where this notion of gamma-convergence of length functionals associated to a sequence of GH-converging distances is formulated (and solved for rescaling distances) is the second cited paper. –  Marius Buliga Dec 31 '11 at 16:48
    
Maybe here are two questions: 1. given a sequence of GH-converging metric spaces, what more do we need to suppose in order to have a sequence of isometric embeddings which satisfy the first two items from the answer? 2. Once we have such a sequence, what more do we need to suppose in order to have gamma convergence of length functionals? –  Marius Buliga Dec 31 '11 at 16:48
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