Construct analogous singular stationary varifolds in Euclidean or real-analytic ambient spaces

Construct stationary integral varifolds with a flat tangent plane of multiplicity two at an isolated singular point, smooth immersed minimal regular part away from that point, and infinite topology in every neighborhood of the singular point, when the ambient space is Euclidean or is equipped with a real-analytic metric.

Background

The paper constructs, for every dimension m≥3, a stationary integral m-varifold in a smooth ambient Riemannian metric whose regular part is a properly immersed smooth minimal surface, whose tangent varifold at the singular point is a multiplicity-two plane, and whose topology is infinite in every neighborhood of that point. The authors identify the ambient smoothness and immersion assumptions as limitations of the construction.

The first explicitly identified unresolved direction is to determine whether analogous examples can be realized in Euclidean space itself or with a real-analytic ambient metric. Such examples would show that the topological obstruction exhibited by the paper persists under substantially more rigid ambient geometries.