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It has recently been established byWang and Xia [WX] that local minimizers of perimeter within a ball subject to a volume constraint must be spherical caps or planes through the origin. This verifies a conjecture of the authors and is in contrast to the situation of area minimizing surfaces with prescribed boundary where singularities can be present in high dimensions. This result lends support to the more general conjecture that volume-constrained minimizers in arbitrary convex sets may enjoy better regularity properties than their boundary-prescribed cousins. Here, we show the importance of the convexity condition by exhibiting a simple example, given by the Simons cone, of a singular volume-constrained locally area-minimizing surface within a nonconvex domain that is arbitrarily close to the unit ball.

A singular local minimizer for the volume- constrained minimal surface problem in a nonconvex domain