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Register a new userThe Calabi-Yau equation on the Kodaira-Thurston manifold, viewed as an S^1-bundle over a 3-torus
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We prove that the Calabi-Yau equation on the Kodaira-Thurston manifold has a unique solution for every $S^1$-invariant initial datum.
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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 manifolds can always be solved for every data which is invariant by the action of a 3-dimensional torus.
We find a new class of invariant metrics existing on the tangent bundle of any given almost-Hermitian manifold. We focus here on the case of Riemannian surfaces, which yield new examples of Kahlerian Ricci-flat manifolds in four real dimensions.
We formulate a Calabi-Yau type conjecture in generalized Kahler geometry, focusing on the case of nondegenerate Poisson structure. After defining natural Hamiltonian deformation spaces for generalized Kahler structures generalizing the notion of Kahler class, we conjecture unique solvability of Gualtieris Calabi-Yau equation within this class. We establish the uniqueness, and moreover show that all such solutions are actually hyper-Kahler metrics. We furthermore establish a GIT framework for this problem, interpreting solutions of this equation as zeros of a moment map associated to a Hamiltonian action and finding a Kempf-Ness functional. Lastly we indicate the naturality of generalized Kahler-Ricci flow in this setting, showing that it evolves within the given Hamiltonian deformation class, and that the Kempf-Ness functional is monotone, so that the only possible fixed points for the flow are hyper-Kahler metrics. On a hyper-Kahler background, we establish global existence and weak convergence of the flow.
We construct balanced metrics on the family of non-Kahler Calabi-Yau threefolds that are obtained by smoothing after contracting $(-1,-1)$-rational curves on Kahler Calabi-Yau threefold. As an application, we construct balanced metrics on complex manifolds diffeomorphic to connected sum of $kgeq 2$ copies of $S^3times S^3$.
We consider the natural generalization of the parabolic Monge-Amp`ere 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 hyperkahler metric, then the equation has a long-time solution whose normalization converges to a solution of the quaternionic Monge-Amp`ere equation introduced by Alesker and Verbitsky. The result gives an alternative proof of a theorem of Alesker.
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