Removable sets and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" id="d1e22" altimg="si3.svg"><mml:msup><mml:mrow><mml:mi>L</mml:mi></mml:mrow><mml:mrow><mml:mi>p</mml:mi></mml:mrow></mml:msup></mml:math>-uniqueness on manifolds and metric measure spaces

We study symmetric diffusion operators on metric measure spaces. Our main question is whether essential self-adjointness or Lp-uniqueness are preserved under the removal of a small closed set from space. provide characterizations critical size removed sets in terms capacities and Hausdorff dimension without any further assumption sets. As key tool we prove non-linear truncation result for potentials nonnegative functions. results robust enough to be applied Laplace general Riemannian manifolds as well sub-Riemannian spaces satisfying curvature-dimension conditions. For non-collapsing Ricci limit with two-sided curvature bounds observe that self-adjoint Laplacian already fully determined by classical regular part.