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 Poincaré inequality and an essential assumption in the theory of Lipschitz-Volume rigidity.
From: Emanuele Caputo [view email]
[v1]
Tue, 23 Jun 2026 10:07:17 UTC (25 KB)
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