A Hamiltonian stable minimal Lagrangian submanifold of projective space with non-parallel second fundamental form

In this note we show that Hamiltonian stable minimal Lagrangian submanifolds of projective space need not have parallel second fundamental form.Comment: 7 page

Actions of vanishing homogeneity rank on quaternionic-Kaehler projective spaces

We classify isometric actions of compact Lie groups on quaternionic-K\"ahler projective spaces with vanishing homogeneity rank. We also show that they are not in general quaternion-coisotropic.Comment: 18 pages. The present version corrects and improves the previous version of the paper entitled "3-coisotropic actions on positive quaternionic-Kaehler manifolds". A key example has been adde

A parabolic flow of balanced metrics

We prove a general criterion to establish existence and uniqueness of a short-time solution to an evolution equation involving "closed" sections of a vector bundle, generalizing a method used recently by Bryant and Xu for studying the Laplacian flow in G_2-geometry. We apply this theorem in balanced geometry introducing a natural extension of the Calabi flow to the balanced case. We show that this flow has always a unique short-time solution belonging to the same Bott-Chern cohomology class of the initial balanced structure and that it preserves the Kaehler condition. Finally we study explicitly the flow on the Iwasawa manifold.Comment: 19 pages. Revised version. To appear in Crelle's Journa

On the stability of the anomaly flow

We prove that the parabolic flow of conformally balanced metrics introduced by Phong, Picard and Zhang in "A flow of conformally balanced metrics with Kaehler fixed points", is stable around Calabi-Yau metrics. The result shows that the flow can converge on a Kaehler manifold even if the initial metric is not conformally Kaehler.Comment: 10 pages. Major revision, to appear in Math. Res. Let

A remark on the Laplacian flow and the modified Laplacian co-flow in G2-Geometry

We observe that the DeTurck Laplacian flow of G2-structures introduced by Bryant and Xu as a gauge fixing of the Laplacian flow can be regarded as a flow of G2-structures (not necessarily closed) which fits in the general framework introduced by Hamilton in [4].Comment: 3 pages, comments are welcom