Title
Monotone Sobolev extensions in metric surfaces
Abstract
Every rectifiable Jordan curve in the plane admits a monotone Lipschitz extension over the disc. This fails in general for metric surface targets due to the possibility of 2‑unrectifiable regions. In the Sobolev setting, the situation changes. We show that any monotone $W^{1,2}$ map from $S^{1}$ to the boundary of a Jordan domain in a metric surface with locally finite Hausdorff 2-measure admits a monotone $W^{1,2}$ extension to the disc. Our proof combines energy minimization methods with a collar construction. This is based on joint work with Noa Vikman and Stefan Wenger.
