Parabolic quaternionic Monge-Amp\`{e}re equation on compact manifolds with a flat hyperK\"ahler metric
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The quaternionic Calabi conjecture was introduced by \linebreak Alesker-Verbitsky, analogous to the K\"ahler case which was raised by Calabi. On a compact connected hypercomplex manifold, when there exists a flat hyperK\"ahler metric which is compatible with the underlying hypercomplex structure, we will consider the parabolic quaternionic \MA~equation. Our goal is to prove the long time existence and $C^{\infty}$ convergence for normalized solutions as $t\ri\infty$. As a consequence, we show that the limit function is exactly the solution of quaternionic \MA~equation, this gives a parabolic proof for the quaternionic Calabi conjecture in this special setting.
Abstract
The quaternionic Calabi conjecture was introduced by \linebreak Alesker-Verbitsky, analogous to the K\"ahler case which was raised by Calabi. On a compact connected hypercomplex manifold, when there exists a flat hyperK\"ahler metric which is compatible with the underlying hypercomplex structure, we will consider the parabolic quaternionic \MA~equation. Our goal is to prove the long time existence and $C^{\infty}$ convergence for normalized solutions as $t\ri\infty$. As a consequence, we show that the limit function is exactly the solution of quaternionic \MA~equation, this gives a parabolic proof for the quaternionic Calabi conjecture in this special setting.
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