Minimal surfaces and regularity thresholds

An interior stationary surface has zero first variation under compactly supported deformations, corresponding to zero mean curvature on a smooth piece. Stability adds nonnegative second variation, while local mass minimization compares with every admissible compactly supported integral-current change having the same boundary. Minimization implies stability and stationarity where those variations make sense, but the reverse implications are not general. A critical-point equation alone does not certify an absolute minimum.

Calibration gives one direct certificate for special minimizers. A closed k-form with comass at most one bounds the mass of every oriented competitor. If its integral equals the mass of a candidate and is unchanged under the allowed boundary-preserving comparison, the candidate minimizes. Planar patches with their standard orientation provide a concrete example. The argument uses orientation and a declared competitor class, rather than only the visual flatness of the surface.

Let T be a locally mass-minimizing integral m-current in Euclidean $(m+1)$-space. Away from its boundary support, its singular set is empty for m at most six, locally discrete for m equal to seven, and has Hausdorff dimension at most m minus seven for larger m. The regular part is a smooth embedded hypersurface, with locally constant integer multiplicity on its sheets. These are interior results; boundary regularity requires additional assumptions and is not included here.

Equivalently, the ambient threshold is seven dimensions for guaranteed interior smoothness, with the first isolated singular minimizing hypersurface possible in ambient dimension eight. Higher codimension does not inherit this threshold: integral area minimizers there have a different regularity theory and may have lower-dimensional branching phenomena. Nor should the statement be transferred to arbitrary stationary varifolds, nonintegral currents, or anisotropic energies without checking the relevant theorem.

For minimizing currents, rescaling around an interior point produces tangent cones along suitable subsequences. Monotonicity controls normalized mass ratios in boundary-free balls. At a regular point the ratio tends to the local integer multiplicity; a multiplicity-one plane has density one. A density larger than one does not by itself prove singularity, because a smooth sheet with larger multiplicity also has that density. The geometry of the cone must still be considered.

The classical cone $\{|x|=|y|\}\subset\mathbb R^4\times\mathbb R^4$ is the boundary of $\{|x|<|y|\}$. The Bombieri–De Giorgi–Giusti theorem makes it a locally minimizing integral hypersurface, not merely a stationary cone. It is smooth away from the vertex and nonplanar, so it cannot be smooth at its conical vertex: a cone smooth there would equal its tangent plane. Its minimizing property is a substantial theorem; the calculation below only checks its dimension and normalized mass.