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 Simon (Lectures on geometric measure theory, p 243, 1983) and provides a new example for a related question of Allard (Ann Math (2) 95(3): 417-491, 1972, p 460). There is also a (rectifiable) stationary 2-varifold in R4 that has more than one conical tangent varifold at a point.
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