Non-unique conical and non-conical tangents to rectifiable stationary varifolds in R^4

TL;DR

This paper constructs examples of rectifiable stationary 2-varifolds in R^4 with non-unique tangent varifolds, including non-conical ones, answering longstanding questions in geometric measure theory.

Contribution

It provides the first known examples of rectifiable stationary varifolds with non-conical, non-unique tangent varifolds in R^4, addressing open problems in the field.

Findings

Existence of rectifiable stationary 2-varifolds with non-conical tangent varifolds.

Existence of rectifiable stationary 2-varifolds with multiple conical tangent varifolds.

Answers to questions posed by L. Simon and W.K. Allard.

Abstract

We construct a rectifiable stationary 2-varifold in R^4 with non-conical, and hence non-unique, tangent varifold at a point. This answers a question of L. Simon (Lectures on geometric measure theory, 1983, p. 243) and provides a new example for a related question of W.K. Allard (On the first variation of a varifold, Ann. of Math., 1972, p. 460). There is also a (rectifiable) stationary 2-varifold in R^4 that has more than one conical tangent varifold at a point. keywords: stationary varifold, varifold tangent, tangent cone, non-unique, non-conical, minimal surface, regularity