Rate of decay for the mass ratio of pseudo-holomorphic integral 2-cycles

We consider any pseudo holomorphic integral 2-cycle in an arbitrary almost complex manifold and perform a blow up analysis at an arbitrary point. Building upon a pseudo algebraic blow up (previously introduced by the author) we prove a geometric rate of decay for the mass ratio towards the limiting density, with an explicit exponent of decay expressed in terms of the density of the current at the point.Comment: in Calc. Var. published online (2015

Tangent cones to positive-(1,1) De Rham currents

We consider positive-(1,1) De Rham currents in arbitrary almost complex manifolds and prove the uniqueness of the tangent cone at any point where the density does not have a jump with respect to all of its values in a neighbourhood. Without this assumption, counterexamples to the uniqueness of tangent cones can be produced already in C^n, hence our result is optimal. The key idea is an implementation, for currents in an almost complex setting, of the classical blow up of curves in algebraic or symplectic geometry. Unlike the classical approach in C^n, we cannot rely on plurisubharmonic potentials.Comment: 37 pages, 2 figure

Embeddedness of liquid-vapour interfaces in stable equilibrium

We consider a classical (capillary) model for a one-phase liquid in equilibrium. The liquid (e.g. water) is subject to a volume constraint, it does not mix with the surrounding vapour (e.g. air), it may come into contact with solid supports (e.g. a container), and is subject to the action of an analytic potential field (e.g. gravity). The region occupied by the liquid is described as a set of locally finite perimeter (Caccioppoli set) in $\mathbb{R}^3$; no a priori regularity assumption is made on its boundary. The (twofold) scope in this note is to propose a weakest possible set of mathematical assumptions that sensibly describe a condition of stable equilibrium for the liquid-vapour interface (the capillary surface), and to infer from those that this interface is a smoothly embedded analytic surface. (The liquid-solid-vapour junction, or free boundary, can be present but is not analysed here.) The result relies fundamentally on the recent varifold regularity theory developed by Wickramasekera and the author, and on the identification of a suitable formulation of the stability condition

Uniqueness of Tangent Cones to Positive-(p,p) Integral Cycles

Let (M, \om) be a symplectic manifold, endowed with a compatible almost complex structure J and the associated metric g . For any p \in {1, 2, ... (dim M)/2} the form \Om := \frac{\om^p}{p!} is a calibration. More generally, dropping the closedness assumption on \om, we get an almost hermitian manifold (M, \om, J, g) and then \Om is a so-called semi-calibration. We prove that integral cycles of dimension 2p (semi-)calibrated by \Om possess at every point a unique tangent cone. The argument relies on an algebraic blow up perturbed in order to face the analysis issues of this problem in the almost complex setting.Comment: 22 page

Almost complex structures and calibratedintegralcycles in contact5-manifolds

In a contact manifold , we consider almost complex structures J that satisfy, for any vector v in the horizontal distribution, . We prove that two-dimensional integral cycles whose approximate tangent planes have the property of being J-invariant and positively oriented are in fact smooth Legendrian curves except possibly at isolated points and we investigate how such structures J are related to calibration

Generic existence of multiplicity-1 minmax minimal hypersurfaces via Allen - Cahn

In Guaraco (J. Differential Geom. 108(1):91–133, 2018) a new proof was given of the existence of a closed minimal hypersurface in a compact Riemannian manifold Nn+1 with n≥2. This was achieved by employing an Allen–Cahn approximation scheme and a one-parameter minmax for the Allen–Cahn energy (relying on works by Hutchinson, Tonegawa, Wickramasekera to pass to the limit as the Allen-Cahn parameter tends to 0). The minimal hypersurface obtained may a priori carry a locally constant integer multiplicity. Here we modify the minmax construction of Guaraco (J. Differential Geom. 108(1):91–133, 2018), by allowing an initial freedom on the choice of the valley points between which the mountain pass construction is carried out, and then optimising over said choice. We then prove that, when 2≤n≤6 and the metric is bumpy, this minmax leads to a (smooth closed) minimal hypersurface with multiplicity 1. (When n=2 this conclusion also follows from Chodosh and Mantoulidis (Ann. Math. 191(1):213–328, 2020).) As immediate corollary we obtain that every compact Riemannian manifold of dimension n+1, 2≤n≤6, endowed with a bumpy metric, admits a two-sided smooth closed minimal hypersurface (this existence conclusion also follows from Zhou X (Ann. Math. (2), 192(3):767–820, 2020) for minmax constructions via Almgren–Pitts theory)