We study a kind of complex submanifolds in a quaternion projective space, which we call transversally complex submanifolds, from the viewpoint of quaternionic differential geometry. We treat them applying the theory of the quaternionic vector bundles. For a transversally complex immersion, we define a Gauss map whose values are complex structures of a quaternionic vector space. It is a generalization of “the mean curvature sphere” in the theory by Burstall, Ferus, Leschke, Pedit, and Pinkall. The Gauss map is a key notion for our theory. As an application, we show a characterization of complex projective spaces which are transversally complex submanifolds of a quaternion projective space.

We show that the Ricci flat Calabi's metrics on holomorphic line bundles over compact Kähler--Einstein manifolds are not projectively induced. As a byproduct we solve a conjecture addressed in [10] by proving that any multiple of the Eguchi-Hanson metric on the blow-up of $\mathbb{C}^2$ at the origin is not projectively induced.

On a Riemannian manifold, a Liouville transformation is a conformal change which preserves the Ricci tensor. In Finsler geometry, there are various types of Ricci curvature. By adding a Landsberg curvature term to the classical Ricci curvature, we consider the conformal transformations on a Finsler manifold such that this modified Ricci curvature is preserved. We prove that such conformal transformations are homothetic if the space is C-convex. The conformal rigidity for Landsberg surfaces is also obtained.

We consider the Cauchy problem for the nonlinear fourth-order Schrödinger equation in six space dimensions with a quadratic nonlinearity. We prove a global existence and time decay estimates of solutions if the data are small, regular and decay rapidly at infinity.

We establish strong factorization for pairs of smooth fans which are refined by the braid arrangement fan. Our method uses a correspondence between cones and preposets.

In this paper, we propose a new representation of the solution to higher-order dispersive equations (which include the free Schrödinger equation and the Airy equation) by using the short-time Fourier transform. As its application, we give $M^{p,q}$-$M^{p,q}_s$, $M^{p,q}$-$M^{p^\prime,q}$ and Strichartz type estimates for the solutions in the framework of the modulation spaces $M^{p,q}_s$.

We prove the existence and uniqueness of a solution to a double obstacle problem for Musielak-Orlicz Dirichlet energy integral on metric measure spaces supporting a $\Phi$-Poincaré inequality, as an extension of [9, 26]. We also study continuous dependence on obstacles for the double obstacle problem in our setting.

By generalising the notion of a unimodular sequence, we create an expression for the winding number of certain ordered sets of lattice points. Since the winding number of the vertices of a Fano polygon is necessarily one, we use this expression as a restriction to classify all Fano polygons without T-singularities and whose singularities in the basket of residual singularities all have equal Gorenstein index.