A parabolic approach to the Calabi-Yau problem in HKT geometry

Abstract

We consider the natural generalization of the parabolic Monge-Ampère equation to HKT geometry. We prove that in the compact case the equation has always a short-time solution and when the hypercomplex manifold is locally flat and admits a hyperkähler metric, then the equation has a long-time solution whose normalization converges to a solution of the quaternionic Monge-Ampère equation first introduced in [4]. The result gives an alternative proof of a theorem of Alesker in [1].

A remark on the quaternionic Monge-Ampère equation on foliated manifolds

“…From the previous lemma it follows that under our assumptions for a basic function F equation ( 4) reduces to (6) ∆ϕ + Q(∇ϕ, ∇ϕ)…”

“…In order to prove existence of solutions to (6) we need to show some a priori estimates. For the C 0 bound we use the Alexandrov-Bakelman-Pucci estimate in the following version developed by Székelyhidi.…”

“…Hence the conjecture of Alesker and Verbitsky is the natural counterpart of the classical Calabi conjecture in the realm of HKT geometry. Even if the conjecture is still open there are some partial results suggesting that it should be true (see [1,2,4,6,8,10,18] and the references therein).…”

Fully non-linear elliptic equations on compact hyperhermitian manifolds with a flat hyperkähler metric

“…In particular f r ≥ f s anytime λ r ≤ λ s . Finally, we observe that by [49, Lemma 9 (b)] for any fixed x ∈ M there is a constant τ > 0 depending on h(x) such that (10)…”

“…Animated by the study of "canonical" HKT metrics, in analogy to the Calabi conjecture [15] proved by Yau in [56], Alesker and Verbitsky proposed in [6] to study the quaternionic Monge-Ampère equation: on a compact HKT manifold, where H ∈ C ∞ (M, R) is given, while (ϕ, b) ∈ C ∞ (M, R) × R + is the unknown. Even if the solvability of the quaternionic Monge-Ampère equation is still an open problem in its general form, several partial results are available in the literature [2,3,4,6,10,19,22,23,46,58]. A nice geometric application of the solvability of equation ( 1) is the existence of a unique balanced metric g on a compact HKT manifold (M, I, J, K, g) with holomorphically trivial canonical bundle with respect to I such that the form Ω induced by g belongs to the class {Ω + ∂∂ J ϕ} (see [54]).…”