13 papers · 1 filter
HKT geometry on locally conformally hyperkähler manifolds
Giovanni Gentili, Luigi Vezzoni
We study the HKT geometry of compact locally conformally hyperkähler manifolds. We prove that on these manifolds the quaternionic Monge-Ampère equation is always solvable, extendin…
Continuous solutions to the quaternionic Monge-Ampère equation on locally flat HKT manifolds
Giovanni Gentili, Antonio Trusiani, Luigi Vezzoni
We prove that on a compact locally flat HKT manifold the quaternionic Monge--Ampère equation always has a unique continuous solution. This provides new evidence for the Conjecture…
A remark on the second order estimates for the quaternionic Calabi-Yau problem on hyperkähler manifolds
Giovanni Gentili, Luigi Vezzoni
We revisit the second order estimate for solutions to the quaternionic Calabi-Yau problem on hyperkähler manifolds, originally established by Dinew and Sroka. In this note, we pres…
The holonomy of the Obata connection on Joyce hypercomplex manifolds
Beatrice Brienza, Udhav Fowdar, Giovanni Gentili +1
We study the holonomy of the Obata connection on Joyce hypercomplex manifolds. For all such group manifolds except , we show that the holonomy group is strictly…
Fully non-linear elliptic equations on compact hyperkähler manifolds
Giovanni Gentili, Luigi Vezzoni
We consider a general class of elliptic equations on hypercomplex manifolds which includes the quaternionic Monge-Ampère equation, the quaternionic Hessian equation and the Monge-A…
The parabolic quaternionic Calabi-Yau equation on hyperkähler manifolds
Lucio Bedulli, Giovanni Gentili, Luigi Vezzoni
We show that the parabolic quaternionic Monge-Ampère equation on a compact hyperkähler manifold has always a long-time solution which once normalized converges smoothly to a soluti…
HKT manifolds: Hodge theory, formality and balanced metrics
Giovanni Gentili, Nicoletta Tardini
Let be a compact HKT manifold and denote with the conjugate Dolbeault operator with respect to , , $\partial^Λ:=[…
Fully non-linear parabolic equations on compact manifolds with a flat hyperkähler metric
Giovanni Gentili, Jiaogen Zhang
Our recent work about fully non-linear elliptic equations on compact manifolds with a flat hyperkähler metric is hereby extended to the parabolic setting. This approach will help u…
Fully non-linear elliptic equations on compact hyperhermitian manifolds with a flat hyperkähler metric
Giovanni Gentili
Mainly motivated by a conjecture of Alesker and Verbitsky, we study a class of elliptic equations on compact hyperhermitian manifolds. By adapting the approach of Székelyhidi to th…
Fully nonlinear elliptic equations on compact manifolds with a flat hyperKähler metric
Giovanni Gentili, Jiaogen Zhang
Mainly motivated by a conjecture of Alesker and Verbitsky, we study a class of fully non-linear elliptic equations on certain compact hyperhermitian manifolds. By adapting the appr…
A remark on the quaternionic Monge-Ampère equation on foliated manifolds
Giovanni Gentili, Luigi Vezzoni
We study the quaternionic Monge-Ampère equation on HKT manifolds admitting an HKT foliation having corank 4. We show that in this setting the quaternionic Monge-Ampère equation has…
A parabolic approach to the Calabi-Yau problem in HKT geometry
Lucio Bedulli, Giovanni Gentili, Luigi Vezzoni
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 w…
The quaternionic Calabi conjecture on abelian hypercomplex nilmanifolds viewed as tori fibrations
Giovanni Gentili, Luigi Vezzoni
We study the quaternionic Calabi-Yau problem in HKT geometry introduced by Alesker and Verbitsky on 8-dimensional 2-step nilmanifolds with an abelian hypercomplex structure. We sho…