相关论文: A parabolic approach to the Calabi-Yau problem in …

We show that the parabolic quaternionic Monge-Amp\`ere equation on a compact hyperk\"ahler manifold has always a long-time solution which once normalized converges smoothly to a solution of the quaternionic Monge-Amp\`ere equation. This is…

We solve the quaternionic Monge-Amp\`ere equation on hyperK\"ahler manifolds. In this way we prove the ansatz for the conjecture raised by Alesker and Verbitsky claiming that this equation should be solvable on any hyperK\"ahler with…

A quaternionic version of the Calabi problem was recently formulated by M. Verbitsky and the author. It conjectures a solvability of a quaternionic Monge-Ampere equation on a compact HKT manifold (HKT stays for HyperKaehler with Torsion).…

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 show that the quaternionic Monge-Amp\`ere equation on these…

A quaternionic version of the Calabi problem on Monge-Ampere equation is introduced. It is a quaternionic Monge-Ampere equation on a compact hypercomplex manifold with an HKT-metric. The equation is non-linear elliptic of second order. For…

The quaternionic Calabi conjecture, posed by Alesker and Verbitsky \cite{Alesker-Verbitsky (2010)}, predicts that the quaternionic Monge-Amp\`ere equation can always be solved on any compact HKT manifold. Motivated by this conjecture, we…

We prove the long time existence and uniqueness of solution to a parabolic quaternionic Monge-Amp\`{e}re type equation on a compact hyperK\"{a}hler manifold. We also show that after normalization, the solution converges smoothly to the…

We prove the long time existence and uniqueness of solution to a parabolic Monge-Amp\`ere type equation on compact Hermitian manifolds. We also show that the normalization of the solution converges to a smooth function in the smooth…

We prove the long-time existence and convergence of solutions to a general class of parabolic equations, not necessarily concave in the Hessian of the unknown function, on a compact Hermitian manifold. The limiting function is identified as…

We establish a C^0 a priori bound on the solutions of the quaternionic Calabi-Yau equation (of Monge-Ampere type) on compact HKT manifolds with a locally flat hypercomplex structure. As an intermediate step, we prove a quaternionic version…

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 approach of Sz\'{e}kelyhidi to the hypercomplex setting, we…

We prove the long time existence and uniqueness of solutions to the parabolic Monge-Amp\`ere equation on compact almost Hermitian manifolds. We also show that the normalization of solution converges to a smooth function in $C^{\infty}$…

We consider a general class of elliptic equations on hypercomplex manifolds which includes the quaternionic Monge-Amp\`ere equation, the quaternionic Hessian equation and the Monge-Amp\`ere equation for quaternionic $(n-1)$-plurisubharmonic…

A new class of Riemannian metrics, called octonionic K\"ahler, is introduced and studied on a certain class of 16-dimensional manifolds. It is an octonionic analogue of K\"ahler metrics on complex manifolds and of HKT-metrics of…

We study the real Monge-Amp\`ere equation in two and three dimensions, both from the point of view of the SYZ conjecture, where solutions give rise to semi-flat Calabi-Yau's and in affine differential geometry, where solutions yield…

In the neighborhood of a regular point, generalized Kahler geometry admits a description in terms of a single real function, the generalized Kahler potential. We study the local conditions for a generalized Kahler manifold to be a…

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 Sz\'ekelyhidi to the hypercomplex setting, we prove some a priori estimates…

We study the long-time existence and convergence of general parabolic complex Monge-Ampere type equations whose second order operator is not necessarily convex or concave in the Hessian matrix of the unknown solution.

We prove $C^\infty$ convergence for suitably normalized solutions of the parabolic complex Monge-Amp\`ere equation on compact Hermitian manifolds. This provides a parabolic proof of a recent result of Tosatti and Weinkove.

A complex Monge-Amp\`ere equation for differential $(p,p)$-forms is introduced on compact K\"ahler manifolds. For any $1 \leq p < n$, we show the existence of smooth solutions unique up to adding constants. For $p=1$, this corresponds to…