Hausdorff measure bounds for density-Q flat singularities of minimizing integral currents

2025/04/27 by Gianmarco Caldini, Anna Skorobogatova, Caldini, Gianmarco +1
Mathematics · #Geometric Analysis and Curvature Flows #Nonlinear Partial Differential Equations #Numerical methods in inverse problems

paper · pdf · doi:10.48550/arxiv.2504.19234

Abstract

In this article we prove that the set of flat singular points of locally highest density of area-minimizing integral currents of dimension m and general codimension in a smooth Riemannian manifold Σ has locally finite (m-2)-dimensional Hausdorff measure. In fact, the set of such flat singular points can be split into a union of two sets, one of which we show is locally H^m-2-negligible, while for the other we obtain local (m-2)-dimensional Minkowski content bounds.

Citations

• Rectifiability of the singular set and uniqueness of tangent cones for semicalibrated currents
• Interior regularity of area minimizing currents within a C^2,α-submanifold
• The fine structure of the singular set of area-minimizing integral currents II: rectifiability of flat singular points with singularity degree larger than 1
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• Analysis of singularities of area-minimizing currents, Part II: a uniform height bound, estimates away from branch points of rapid decay, and uniqueness of tangent cones
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• Almgren's type regularity for Semicalibrated Currents
• Regularity theory for 2-dimensional almost minimal currents III: blowup
• Regularity theory for 2-dimensional almost minimal currents I: Lipschitz approximation
• Regularity theory for 2-dimensional almost minimal currents II: branched center manifold
• Regularity of area minimizing currents I: gradient L^p estimates
• Regularity of area minimizing currents II: center manifold
• Regularity of area minimizing currents III: blow-up
• Multiple valued functions and integral currents
• Q-valued functions revisited
• Minimal cones and the Bernstein problem
• Minimal Varieties in Riemannian Manifolds

Related

• An upper Minkowski bound for the interior singular set of area minimizing currents
• Minkowski content estimates for generic area minimizing hypersurfaces
• On the boundary behavior of mass-minimizing integral currents
• Interior regularity of area minimizing currents within a C^2,α-submanifold
• Regularity of area minimizing currents II: center manifold