We investigate special lcs and twisted Hamiltonian torus actions on strict lcs manifolds and characterize them geometrically in terms of the minimal presentation. We prove a convexity theorem for the corresponding twisted moment map, establishing thus an analog of the symplectic convexity theorem of Atiyah and Guillemin-Sternberg. We also prove similar results for the symplectic moment map (defined on the minimal presentation) whose image is then a convex cone. In the special case of a compact toric Vaisman manifold, we obtain a structure theorem.
Preliminaries on lcs, lcK and Vaisman manifoldsLet (M, ω) be an almost symplectic manifold of real dimension greater than 2, where ω is a non-degenerate 2-form. Often ω := g(J·, ·) will be the fundamental form of an (almost) Hermitian metric g on (M, J), where J : T M → T M is an (almost) complex structure on M . We will usually consider the complex case, when J is integrable.If every point of M admits a neighborhood U and a smooth function f U : U → R such that the two-form e −fU ω| U is closed, we call (M, ω) a locally conformally symplectic manifold (lcs). If ω is the fundamental (or Kähler) form of a Hermitian manifold (M, g, J), then we call it a locally conformally Kähler manifold (lcK).From the definition, it follows that the local 1-forms df U glue together to a global 1-form θ, called the Lee form, satisfying on M
Abstract. In this note we give a characterization of Kähler metrics which are both Calabi extremal and Kähler-Ricci solitons in terms of complex Hessians and the Riemann curvature tensor. We apply it to prove that, under the assumption of positivity of the holomorphic sectional curvature, these metrics are Einstein.
We extend to the Sasakian setting a result of Tian and Zhu about the decomposition of the Lie algebra of holomorphic vector fields on a Kähler manifold in the presence of a Kähler-Ricci soliton. Furthermore we apply known deformations of Sasakian structures to a Sasaki-Ricci soliton to obtain a stability result concerning generalized Sasaki-Ricci solitons, generalizing Li in the Kähler setting and also He and Song by relaxing some of their assumptions.2010 Mathematics Subject Classification. 53C25.
For a given minimal Legendrian submanifold L of a Sasaki-Einstein manifold we construct two families of eigenfunctions of the Laplacian of L and we give a lower bound for the dimension of the corresponding eigenspace. Moreover, in the case the lower bound is attained, we prove that L is totally geodesic and a rigidity result about the ambient manifold. This is a generalization of a result for the standard Sasakian sphere done by Lê and Wang.
We apply the concept of castling transform of prehomogeneous vector spaces to produce new examples of minimal homogeneous Lagrangian submanifolds in the complex projective space. Furthermore we verify the Hamiltonian stability of a low dimensional example that can be obtained in this way.2010 Mathematics Subject Classification. 32J27, 53D12, 57S25.
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