Convergence of the $J$-flow on toric manifolds

Abstract

We show that on a Kähler manifold whether the J-flow converges or not is independent of the chosen background metric in its Kähler class. On toric manifolds we give a numerical characterization of when the J-flow converges, verifying a conjecture in [18] in this case. We also strengthen existing results on more general inverse σ k equations on Kähler manifolds.

On a Class of Fully Nonlinear Elliptic Equations on Closed Hermitian Manifolds II: L∞ Estimate

“…In view of the results in [6,14], it seems reasonable to expect their results to hold in the general case, and we wish to come back to this in future work. Indeed, it is easy to see that (1.6) implies (1.7).…”

“…For the Donaldson equation, Lejmi and Székelyhidi [14] proposed a numerical criterion for the existence of a cone condition. Lejmi and Székelyhidi [14] verified the criterion in dimension 2, while Collins and Székelyhidi [6] verified it on toric manifolds. Later Székelyhidi [17] posed the following extension for general complex quotient equations.…”

The Deformed Hermitian–Yang–Mills Equation in Geometry and Physics

“…The work of Lejmi-Székelyhidi [22] is based on an extension of K-stability, which plays an important role in the existence of constant scalar curvature Kähler metrics [13,14]. We have Theorem 3.7 (Collins-Székelyhidi [10]…”

Transverse fully nonlinear equations on Sasakian manifolds and applications

“…Let (M, φ, ξ, η, g) be a compact Sasakian manifold with dim R M = 2n+1 (n ≥ 2) and ω † = 1 2 dη = g(φ·, ·) as its transverse Kähler form, and let ω h be a closed strictly transverse k-positive basic (1, 1) form. Then we can find a strictly transverse k-positive basic real (1,1) form ω ′ h ∈ [ω h ] ∈ H 1,1 B (M, R) such that the real basic (n − 1, n − 1) form given in (1.13) with ω ′ h instead of ω h if and only if for all transverse analytic subvarieties V ⊂ M of dimension p = n − ℓ, · · · , n − 1 we have It seems that we can modify the method in [18] to answer the question in the case k = n, ℓ = n−1 on toric Sasakian manifolds.…”