Mean curvature and sharp Willmore inequalities in metric spaces

What the math gives to ML

The paper supplies a weak, pointwise-compatible notion of tangential divergence for level sets of Sobolev functions in spaces where classical surface smoothness is unavailable. The transferable asset is the projection of ambient vector-field divergence onto directions tangent to a learned scalar field's level sets, giving a coordinate-free geometric signal without explicitly constructing a mesh. A practical adaptation is a curvature-aware regularizer for neural implicit surfaces, signed-distance functions, or scalar decision boundaries. The extracted material does not include enough of the sharp Willmore inequality to transfer its constants, so the strongest concrete proposal is the tangential-divergence construction itself.