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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 both questions 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?

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 both questions is no. 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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# Length spaces with continuous length functional: is this set Gromov-Hausdorff closed?

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) 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?