Results on separating currents, mass growth, resolvent regularization, densities and tangent measures

The course proves the statements below; each link leads to the statement with its assumptions and its proof. The lessons are self-checked by the writing AI; independent mathematical review is not complete.

• Two-forms on an almost Hermitian four-manifold — Lemma 1.1. In a unitary frame, F = e12 + e34 is self-dual of norm squared 2; the anti-invariant two-forms are self-dual; every unit complex-line bivector l is invariant with F(l) = 1, l^+ = F/2 and *l = F - l.
• Invariant bivectors as Hermitian matrices — Lemma 1.2. Invariant bivectors correspond to Hermitian 2 x 2 matrices with the trace inner product; v ^ Jv corresponds to z z*; those of nonnegative matrix and trace one are convex combinations t1 l1 + t2 l2 of complex lines of a unitary frame, of norm at most one with equality exactly for complex lines, and of rank two exactly in that case.
• A closed lift of anti-invariant forms — Lemma 3.4. With the projected elliptic operator R d d^* on anti-invariant forms, its kernel projection H and generalized inverse G, the order-zero operator B = d d^* G + H satisfies dB = 0 and RB = id.
• Measure representation of positive functionals — Proposition 4.1. A linear functional on two-forms that vanishes on anti-invariant forms, is nonnegative on semidefinite invariant forms and equals one on F is integration of alpha(l) against a probability measure on the unit complex-line bivectors, with trace measure and a polarization field.
• The separating current — Proposition 5.1. If no closed form is compatible with J, there are a positive current P annihilating closed invariant forms and a self-dual anti-invariant correction Q such that P + Q annihilates all closed forms.
• Order zero on concentrated inputs — Lemma 1.2. An operator of order zero on a closed four-manifold has kernel O(d^-4) off the diagonal, and maps a form supported in a ball of radius a with k-th derivatives at most M a^-k to a form bounded by CM.
• Quadratic mass growth — Proposition 4.1. The trace measure of the separating current satisfies mu(B_s(x)) <= C s^2 for all centres and radii.
• The zero pairing — Lemma 2.1. If T annihilates all closed forms and T' is a closed current, then <S_r T, * S_r T'> = 0 for the resolvent powers S_r = (1 + r^2 Delta)^-3.
• The angular identity — Lemma 4.1. For unit complex-line bivectors l at x and m at y and parallel transport tau, <l, * tau m> = |l - tau m|^2/2 - |F_x - tau F_y|^2/4.
• The relative projection estimate — Lemma 5.1. If a smooth orthogonal projection Pi annihilates a distribution W, then ||Pi S_r W|| <= C r ||S_r W||.
• Square integrability and the angular bound — Proposition 6.1. Under the heat kernel hypothesis (HK), the correction Q is square integrable, the angular integral is O(r^4), and the cross pairings of Q with every subcurrent of P tend to zero.
• Closedness of the positive-density part — Theorem 1.1. For a positive current on unit complex-line bivectors with a square integrable anti-invariant correction making it closed, quadratic mass growth and an O(r^4) angular bound imply that the density exists everywhere and that the restriction of the current to the points of positive density is closed.
• Existence of the two-dimensional density — Proposition 2.1. Under the hypotheses of the closedness theorem, r^-2 mu(B_r(p)) converges at every point, by an almost monotonicity formula of Lelong type.
• Planar tangent measures and the transverse moment — Proposition 4.1. At almost every point of positive density the polarization is a complex line and every tangent measure is a multiple of area on it; the mean-square transverse displacement is o(r^2).
• Tangential derivatives of density cutoffs — Proposition 3.2. The derivatives of the smoothed densities along the complex lines of the current tend to zero in the mean, because closure converts them into the product of angular and transverse errors.
• Taming implies compatibility — Theorem 5.1. Under the heat kernel hypothesis (HK): an almost complex structure on a closed four-manifold that is tamed by a closed two-form admits a compatible closed two-form.

Reading order

Five lessons, to be read in order. The theorem is proved from one named hypothesis on the heat kernel of the Hodge Laplacian, stated in the third lesson; elliptic operators, Riemannian geometry, measure differentiation and functional analysis are taken from other courses of the programme and cited where they are used.