##
**Uniqueness of tangent cones to positive-\((p,p)\) integral cycles.**
*(English)*
Zbl 1298.53044

The regularity question raised by Harvey and Lawson in their seminal paper on calibrations in 1981, whether calibrated integral currents admit unique tangent cones was only solved for 2-currents.

The author considers an almost-complex manifold \((M, J)\) of dimension \(2n+2\). A positive-\((p,p)\) integral cycle \(T\) is a \(2p\)-dimensional cycle with almost all approximate tangents \(J\)-invariant and positively oriented. He proves that these cycles have unique tangent cones and in case of the known integrable case the present proof provides a new, simpler and direct one. The idea is to consider (locally) some suitable Riemannian metric \(g\) and a \(2p\)-semicalibration \(\Omega=(1/p!)\omega^p\) (not necessarily closed form), where \(\omega(\cdot, \cdot)=g(J\cdot, \cdot)\), that calibrates the positive-\((p,p)\) integral cycles, that is, on any open set of \(U\subset M\), \(T(\Omega\llcorner U)=M(T\llcorner U)\) holds. These cycles verify an almost monotonicity property, namely at every point \(x_0\), \(M(T\llcorner B_{r}(x_0))/r^{2p}\) is weakly monotonically increasing function of \(r\) up to an infinitesimal function \(O(r)\) when \(r\to 0\), and an upper semicontinuous density function of \(T\) bounded by one from below can be defined. Using geodesic balls and normal coordinates centered at \(x_0\), an algebraic pseudoholomorphic blow-up of \(T\) modeled in the almost-complex case is taken inspired in the case \(p=1\) worked by the author [C. R., Math., Acad. Sci. Paris 349, No. 19–20, 1025–1029 (2011; Zbl 1235.32022)] defining a sequence of integral semicalibrated cycles for a suitable sequence of semicalibrations, with uniformly bounded masses, and that converges weakly to a tangent cone.

The author proves that this tangent cone is unique for any dimension \(2p\) and codimension, extending the known cases \(p=1\) obtained by D. Pumberger and T. Rivière [Duke Math. J. 152, No. 3, 441–480 (2010; Zbl 1195.53064)]. For that he shows that the space of tangent cones to \(T\) at each point \(x_0\) is a connected and closed subspace of the space of \(2p\)-integral cycles and that the density function, that is integer-valued, is continuous acting on calibrated integral cycles with same support, and so must be constant.

The author considers an almost-complex manifold \((M, J)\) of dimension \(2n+2\). A positive-\((p,p)\) integral cycle \(T\) is a \(2p\)-dimensional cycle with almost all approximate tangents \(J\)-invariant and positively oriented. He proves that these cycles have unique tangent cones and in case of the known integrable case the present proof provides a new, simpler and direct one. The idea is to consider (locally) some suitable Riemannian metric \(g\) and a \(2p\)-semicalibration \(\Omega=(1/p!)\omega^p\) (not necessarily closed form), where \(\omega(\cdot, \cdot)=g(J\cdot, \cdot)\), that calibrates the positive-\((p,p)\) integral cycles, that is, on any open set of \(U\subset M\), \(T(\Omega\llcorner U)=M(T\llcorner U)\) holds. These cycles verify an almost monotonicity property, namely at every point \(x_0\), \(M(T\llcorner B_{r}(x_0))/r^{2p}\) is weakly monotonically increasing function of \(r\) up to an infinitesimal function \(O(r)\) when \(r\to 0\), and an upper semicontinuous density function of \(T\) bounded by one from below can be defined. Using geodesic balls and normal coordinates centered at \(x_0\), an algebraic pseudoholomorphic blow-up of \(T\) modeled in the almost-complex case is taken inspired in the case \(p=1\) worked by the author [C. R., Math., Acad. Sci. Paris 349, No. 19–20, 1025–1029 (2011; Zbl 1235.32022)] defining a sequence of integral semicalibrated cycles for a suitable sequence of semicalibrations, with uniformly bounded masses, and that converges weakly to a tangent cone.

The author proves that this tangent cone is unique for any dimension \(2p\) and codimension, extending the known cases \(p=1\) obtained by D. Pumberger and T. Rivière [Duke Math. J. 152, No. 3, 441–480 (2010; Zbl 1195.53064)]. For that he shows that the space of tangent cones to \(T\) at each point \(x_0\) is a connected and closed subspace of the space of \(2p\)-integral cycles and that the density function, that is integer-valued, is continuous acting on calibrated integral cycles with same support, and so must be constant.

Reviewer: Isabel Salavessa (Lisboa)

### MSC:

53C38 | Calibrations and calibrated geometries |

49Q15 | Geometric measure and integration theory, integral and normal currents in optimization |

49Q20 | Variational problems in a geometric measure-theoretic setting |

### Keywords:

integral cycle; almost-complex manifold; regularity; tangent cone; uniqueness; semicalibration; pseudoholomorphic blowup
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\textit{C. Bellettini}, Duke Math. J. 163, No. 4, 705--732 (2014; Zbl 1298.53044)

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