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Finite-length property of EVI_λ flows

Determine whether every Evolution Variational Inequality (EVI) gradient flow curve γ:(0,∞)→X with parameter λ>0 in a metric space (X,d) necessarily has finite length; equivalently, establish whether ∫_0^∞ |γ̇|_t dt < ∞ always holds for EVI_λ flows.

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Background

In order to prove that induced submanifolds endowed with the length metric are geodesic spaces, the authors need curves of finite length connecting any two points. They construct such curves by concatenating EVI flows, which requires integrability of the metric derivative.

They show this for starting points in the domain of the energy functional, and in Wasserstein settings for all starting points via stronger short-time regularization. However, whether finite length is automatic for EVI_λ flows with λ>0 in general metric spaces remains unresolved.

References

It appears to be an open question whether or not an $\mathrm{EVI}_\lambda$-flow for $\lambda > 0$ necessarily produces a curve of finite length.

— Geodesic convexity and strengthened functional inequalities in submanifolds of Wasserstein space (2508.13698 - Chaintron et al., 19 Aug 2025) in Section 3.3 (Existence of curves of finite length)