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.
Similar content being viewed by others
Notes
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].
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).
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.
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.
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].
References
Allard, W.K.: On the first variation of a varifold. Ann. Math. (2) 95(3), 417–491 (1972)
Allard, W.K., Almgren, F.J.: The structure of stationary one dimensional varifolds with positive density. Invent. Math. 34(2), 83–97 (1976). doi:10.1007/BF01425476
Allard, W.K., Almgren Jr, F.J.: On the radial behavior of minimal surfaces and the uniqueness of their tangent cones. Ann. Math. 113, 215–265 (1981)
Bellettini, C.: Tangent cones to positive-(1, 1) De Rham currents. J. Reine Angew. Math
Bellettini, C.: Uniqueness of tangent cones to positive-\((p, p)\) integral cycles. Duke Math. J. 163, 705–732 (2014)
Bellettini, C.: Semi-calibrated 2-currents are pseudoholomorphic, with applications. Bull. Lond. Math. Soc. 46(4), 881–888 (2014)
Blel, M.: Sur le cône tangent à un courant positif fermé. J. Math. Pures Appl. 72, 517–536 (1993)
Brakke, K.A.: The motion of a surface by its mean curvature. In: Mathematical Notes, vol. 20. Princeton University Press, Princeton (1978)
Brendle, S.: Embedded minimal tori in \(S^3\) and the Lawson conjecture. arXiv:1203.6597
Chang, S.X.-D.: Two-dimensional area minimizing integral currents are classical minimal surfaces. J. Am. Math. Soc. 1(4), 699–778 (1988)
Černý, R., Kolář, J., Rokyta, M.: Concentrated monotone measures with non-unique tangential behavior in \(R^3\). Czechoslov. Math. J. 61(4), 1141–1167 (2011)
Černý, R., Kolář, J., Rokyta, M.: Monotone measures with bad tangential behavior in the plane. Comment. Math. Univ. Carol. 52(3), 317–339 (2011)
De Lellis, C., Spadaro, E.N.: Q-valued functions revisited. Mem. Am. Math. Soc. 211(991), vi+79 (2011). ISBN 978-0-8218-4914-9. arXiv:0803.0060v4
Federer, H.: Geometric Measure Theory. Springer, New York (1969)
Harvey, R., Lawson Jr, H.B.: Calibrated geometries. Acta Math. 148, 47–157 (1982)
Hutchinson, J.E., Meier, M.: A remark on the nonuniqueness of tangent cones. Proc. Am. Math. Soc. 97(1), 184–185 (1986)
Kiselman, C.O.: Tangents of plurisubharmonic functions. In: International Symposium in Memory of Hua Loo Keng (August, 1988), vol. II, pp. 157–167. Science Press and Springer, New York (1991)
Kolář, J.: Non-regular tangential behaviour of a monotone measure. Bull. Lond. Math. Soc. 38, 657–666 (2006)
Lawlor, G.: The angle criterion. Invent. math. 95, 437–446 (1989)
O’Neil, T.C.: Geometric measure theory, online at Encyclopedia of Mathematics. http://www.encyclopediaofmath.org/index.php?title=Geometric_measure_theory&oldid=28204. (First appeared in Supplement III of the Encyclopedia of Mathematics. Kluwer Academic Publishers, Kluwer, 2002)
Osserman, R.: Minimal varieties. Bull. Am. Math. Soc. 75(6), 1092–1120 (1969)
Pumberger, D., Riviére, T.: Uniqueness of tangent cones for semi-calibrated 2-cycles. Duke Math. J. 152(3), 441–480 (2010)
Simon, L.: Lectures on geometric measure theory. Proceedings of the Centre for Mathematical Analysis, vol. 3. Australian National University, Canberra (1983)
Simon, L.: Asymptotics for a class of non-linear evolution equations, with applications to geometric problems. Ann. Math. Second Ser. 118(3), 525–571 (1983)
Simon, L.: Cylindrical tangent cones and the singular set of minimal submanifolds. J. Differ. Geom. 38, 585–652 (1993)
Simon, L.: Uniqueness of some cylindrical tangent cones. Commun. Anal. Geom. 2(1), 1–33 (1994)
Siu, Y.T.: Analyticity of sets associated to Lelong numbers and the extension of closed positive currents. Invent. Math. 27, 53–156 (1974)
Taylor, J.E.: The structure of singularities in soap-bubble-like and soap-film-like minimal surfaces. Ann. Math. (2) 103(3), 489–539 (1976)
White, B.: Tangent cones to two-dimensional area-minimizing integral currents are unique. Duke Math. J. 50(1), 143–160 (1983). doi:10.1215/S0012-7094-83-05005-6
White, B.: The mathematics of F. J. Almgren, Jr. J. Geom. Anal 8(5), 681–702 (1998)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by L. Simon.
Research supported by Grants P201/12/0290 of GA ČR, IAA100190903 of GA AV and RVO: 67985840.
Rights and permissions
About this article
Cite this article
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
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00526-015-0847-9