We construct gradient Kähler-Ricci solitons on Ricci-flat Kähler cone manifolds and on line bundles over toric Fano manifolds. Certain shrinking and expanding solitons are pasted together to form eternal solutions of the Ricci flow. The method we employ is the Calabi ansatz over Sasaki-Einstein manifolds, and the results generalize constructions of Cao and Feldman-Ilmanen- Knopf.

We consider the local deformation problem of coisotropic submanifolds inside symplectic or Poisson manifolds. To this end the groupoid of coisotropic sections (with respect to some tubular neighbourhood) is introduced. Although the geometric content of this groupoid is evident, it is usually a very intricate object.

We provide a description of the groupoid of coisotropic sections in terms of a differential graded Poisson algebra, called the BFV-complex. This description is achieved by constructing a groupoid from the BFV-complex and a surjective morphism from this groupoid to the groupoid of coisotropic sections. The kernel of this morphism can be easily chracterized.

As a corollary we obtain an isomorphism between the moduli space of coisotropic sections and the moduli space of geometric Maurer–Cartan elements of the BFV-complex. In turn, this also sheds new light on the geometric content of the BFV-complex.

In this paper, we investigate the mean curvature flows having an equifocal submanifold in a symmetric space of compact type and its focal submanifolds as initial data. It is known that an equifocal submanifold of codimension greater than one in an irreducible symmetric space of compact type occurs as a principal orbit of a Hermann action. However, we investigate the flows conceptionally without use of this fact. The investigation is performed by investigating the mean curvature flows having the lifts of the submanifolds to an (infinite dimensional separable) Hilbert space through a Riemannian submersion as initial data.