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arXiv · 2606.24374

Sobolev-to-Lipschitz property of geodesically complete spaces with curvature bounded from above

Emanuele Caputo·Nicola Cavallucci·Toni Ikonen
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Abstract

We prove that every length space with curvature bounded from above that is geodesically complete has the Sobolev-to-Lipschitz property with exponent infinity. That is, every Sobolev map in the $W^{1,\infty}$-space has a Lipschitz representative so that the Lipschitz constant coincides with the infinity energy of the map. The proof is geometric and relies on arbitrarily small perturbations of geodesics to a curve that has zero length on the singular set. The motivation is to develop the analytic theory of such spaces; in particular, our result implies that GCBA spaces satisfy the infinity Poincar\'e inequality and an essential assumption in the theory of Lipschitz-Volume rigidity.

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Emanuele Caputo, Nicola Cavallucci, Toni Ikonen. 2026-06-23. Sobolev-to-Lipschitz property of geodesically complete spaces with curvature bounded from above. https://arxiv.org/abs/2606.24374

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