We prove a Gevrey regularity theorem for the sum of squares of two vector fields in two variables with the assumption that the Hörmander bracket hypothesis is satisfied at length three. In Appendix A, a procedure is discussed for what attains to the possible optimality of the obtained Gevrey order.

We study rotational surfaces in Euclidean 3-space whose Gauss curvature is given as a prescribed function of its Gauss map. By means of a phase plane analysis and under mild assumptions on the prescribed function, we generalize the classification of rotational surfaces of constant Gauss curvature; exhibit examples that cannot exist in the constant Gauss curvature case; and analyze the asymptotic behavior of strictly convex graphs. We also prove the existence of singular radial solutions intersecting orthogonally the axis of rotation.

An LCK manifold is a complex manifold $(M,I)$ equipped with a Hermitian form $\omega$ and a closed 1-form $\theta$, called the Lee form, such that $d\omega=\theta\wedge\omega$. An LCK manifold with potential is an LCK manifold with a positive Kähler potential on its universal cover, such that the deck group multiplies the Kähler potential by a constant. A Lee class of an LCK manifold is the cohomology class of the Lee form. We determine the set of Lee classes on LCK manifolds admitting an LCK structure with potential, showing that it is an open half-space in $H^1(M,{\mathbb R})$. For Vaisman manifolds, this theorem was proven in 1994 by Tsukada; we give a new self-contained proof of his result.

In this article we focus on multiplier Hermitian-Einstein metrics introduced by Mabuchi which include Kähler-Einstein metrics, Kähler-Ricci solitons and Mabuchi solitons as special cases. We also focus on KSM-manifolds, which are introduced by the first author as toric bundles, to establish a criterion for the existence of multiplier Hermitian-Einstein metrics in terms of KSM-data. An explicit example for a KSM-manifold admitting a family of multiplier Hermitian-Einstein metrics is constructed by using a continuous path connecting a Kähler-Ricci soliton with a Mabuchi soliton.