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We study area-minimizing hypersurfaces in singular ambient manifolds whose tangent cones have nonnegative scalar curvature on their regular parts. We prove that the singular set of the hypersurface has codimension at least 3 in our settings. We also give an example in which n-dimensional minimizing hypersurface in an ambient space as given above may have singularities of dimension n-3. This shows that our regularity result is sharp.
Yihan Wang (Wed,) studied this question.
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