A dual characterization of length spaces with application to Dirichlet metric spaces

We show that under minimal assumptions, the intrinsic metric induced by a strongly local Dirichlet form induces a length space. A main input is a dual characterization of length spaces in terms of the property that the 1-Lipschitz functions form a sheaf.

💡 Research Summary

The paper establishes a new, highly general criterion for when the intrinsic metric associated with a strongly local Dirichlet form turns the underlying space into a length space. A length space (or geodesic metric space) is a metric space in which the distance between any two points equals the infimum of the lengths of continuous curves joining them. Classical results guarantee this property only under fairly restrictive regularity assumptions on the Dirichlet form, the reference measure, or the topology of the space.

The authors’ central contribution is a dual characterization of length spaces that bypasses those geometric constraints. They prove that a metric space ((X,d)) is a length space if and only if the collection of 1‑Lipschitz functions (\mathrm{Lip}_1(X,d)) forms a sheaf: for every open set (U\subset X), every function that is 1‑Lipschitz on (U) can be extended to a global 1‑Lipschitz function on (X). The forward direction (length space ⇒ sheaf property) is well known; the novelty lies in the converse. The proof proceeds by constructing, for any two points (x,y) and any (\varepsilon>0), a 1‑Lipschitz function whose oscillation between (x) and (y) approximates the distance (d(x,y)). The sheaf hypothesis then allows this locally defined function to be extended globally without increasing its Lipschitz constant, which forces the metric to satisfy the length condition.

Having this functional characterization, the paper turns to Dirichlet forms. Let ((\mathcal{E},\mathcal{F})) be a strongly local, regular Dirichlet form on (L^2(X,\mu)). Strong locality means that (\mathcal{E}(f,g)=0) whenever (f) is constant on the support of (g); this property ensures that the associated energy measure (\Gamma(f)) behaves like a pointwise gradient squared. The intrinsic metric is defined by
\