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BIRS Workshop Lecture Videos

A geometric flow of Balanced metrics Vezzoni, Luigi


An Hermitian metric $g$ on a complex manifold $M$ is called {\em balanced} if its fundamental form $\omega$ is co-closed. Typically examples of balanced manifolds are given by modifications of K\"ahler manifolds, twistor spaces over anti-self-dual oriented Riemannian $4$-manifolds and nilmanifolds. In the talk it will be discussed a geometric flow of balanced metrics. The flow was introduced in \cite{BFV1} and consists in a generalisation of the Calabi flow to the balanced context. The flow preserves the Bott-Chern cohomology class of the initial metric and in the K\"ahler case reduces to the classical Calabi flow. It will be showed that the flow is well-posedn and stable around Ricci flat K\"ahler metrics. Furthermore, the talk focuses on a rencent problem in balanced geometry proposed in \cite{FV}. \begin{thebibliography}{12} \bibitem{BFV1} {\sc L. Bedulli, L. Vezzoni}, A parabolic flow of balanced metrics. {\em Journal f\"ur die reine und angewandte Mathematik (Crelle)}. {\bf 723} (2017), 79--99. \bibitem{BFV2} {\sc L.~Bedulli and L. Vezzoni}, A scalar Calabi-type flow in Hermitian Geometry, to appear in {\em Ann. Sc. Norm. Super. Pisa Cl. Sci. (5)}. \bibitem{BFV3} {\sc L.~Bedulli and L. Vezzoni}, Stability of geometric flows of closed forms. {\tt\, arXiv:1811.09416}. \bibitem{FGV} {\sc A. Fino, G. Grantcharov, L. Vezzoni}, Astheno-K\"ahler and balanced structures on fibrations. {\tt arXiv:1608.06743}, to appear in {\em IMRN}. \bibitem{FV} {\sc A. Fino, L. Vezzoni,} Special Hermitian metrics on compact solvmanifolds, {\em J. Geom. Phys.} {\bf 91} (Special Issue) (2015), 40--53. \end{thebibliography}

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