In this addendum we complete the proof of Theorem 7.10 in [1].

The curvature estimates for k curvature equations with general right-hand sides is a longstanding problem. In this paper, we completely solve the problem when k . We also discuss some applications of our estimates.

Let T  (Y, ) be a transverse knot which is the binding of some open book, (T, ), for the ambient contact manifold (Y, ). In this paper, we show that the transverse invariant bT(T ) 2[HFK(−Y,K), defined in [LOSSz09], is nonvanishing for such transverse knots. This is true regardless of whether or not  is tight. We also prove a vanishing theorem for the invariants L and T. As a corollary, we show that if (T, ) is an open book with connected binding, then the complement of T has no Giroux torsion.

We study the collapsing behavior of the Kähler–Ricci flow on a compact Kähler manifold X admitting a holomorphic submersion inherited from its canonical bundle. We show that the flow metric degenerates at exactly the rate of e^{-t} as predicted by the cohomology information, and so the fibres collapse at the optimal rate. Consequently, it leads to some analytic and geometric extensions to the regular case of works by J. Song and G. Tian. Its applicability to general Calabi–Yau fibrations will also be discussed in local settings.

We establish an interior C 2 estimate for k+ 1 convex solutions to Dirichlet problems of k-Hessian equations. We also use such estimate to obtain a rigidity theorem for k+ 1 convex entire solutions of k-Hessian equations in Euclidean space.