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Non-unique conical and non-conical tangents to rectifiable stationary varifolds in \(\mathbb R^4\)

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Abstract

We construct a rectifiable stationary 2-varifold in \({\mathbb {R}}^4\) 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 \({\mathbb {R}}^{4}\) that has more than one conical tangent varifold at a point.

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Notes

  1. Though, in different context the tangent cone is sometimes defined to be what we call the unique tangent cone, see for example [17, p. 159].

  2. See for example [25, p. 591]: “... but it is far from obvious (and an open question) whether or not \({{\mathrm{Tan}}}_X M\) can contain more than one cone C if \(X\in {\text {sing}} M\)”. The same paper contains a result on the uniqueness of tangents m-almost everywhere in the singular set [25, p. 650, (2), (1)], where m is the ‘top dimension’ (e.g., \(m=\dim M - 2\), depending on the context).

  3. Since \(C\ne 0\), we have \(\theta ^m(\mu _V,x_0) \in (0,\infty )\) from the monotonicity formula for stationary varifolds, cf. [23, 40.5]. Therefore the first assumption of Corollary 42.6 (namely 42.1) is satisfied.

  4. Although there are several possible definition of approximate tangent plane (see [20], [1, p. 428, (3) and (b)] and [23, 11.2]), they agree \(\mu \)-almost everywhere. The definitions of rectifiable varifolds in [1, 20] essentially agree with that of [23], cf. footnote on [23, p. 77].

  5. The surface is neither a linear space nor a convex set: it contains points (1, 0, 0, 0) (\(a=c=1\), \(b=d=0\)) and (0, 0, 0, 1) (\(a=c=0\), \(b=d=1\)) but does not contain (1 / 2, 0, 0, 1 / 2). Indeed, \((t,0,0,t)=(ac,bc,ad,bd)\), \(t\ne 0\) leads to \(a\ne 0\), \(c=t/a\), \(b\ne 0\), \(d=t/b\), then \(bt/a=0\), \(at/b=0\) and finally \(b=0=a\), a contradiction.

  6. The surface is actually a copy of the three-dimensional cone generated by \({\mathbb {S}}^1\times {\mathbb {S}}^1\) as can be seen from the relation \((\cos \gamma ,\sin \gamma ,\cos \delta ,\sin \delta )=(x+w,y-z,x-w,z+y)\) where \((x,y,z,w)=F((\cos \alpha ,\sin \alpha ),(\cos \beta ,\sin \beta ))\), \(\gamma =\alpha -\beta \),\(\delta =\alpha +\beta \).

    The surface was the first known nontrivial minimal cone in \(\mathbb {R}^4\) [21, p. 1113]. \({\mathbb {S}}^1\times {\mathbb {S}}^1\) is so called Clifford torus. Recently, Simon Brendle announced that (up to a congruence) it is the only embedded minimal torus in \(\mathbb S^3\) [9].

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Correspondence to Jan Kolář.

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Communicated by L. Simon.

Research supported by Grants P201/12/0290 of GA ČR, IAA100190903 of GA AV and RVO: 67985840.

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Kolář, J. Non-unique conical and non-conical tangents to rectifiable stationary varifolds in \(\mathbb R^4\) . Calc. Var. 54, 1875–1909 (2015). https://doi.org/10.1007/s00526-015-0847-9

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