In this paper, we parametrize the space of isometric immersions of the hyperbolic plane into the hyperbolic 3-space in terms of null-causal curves in the space of oriented geodesics. Moreover, we characterize “ideal cones” (i.e., cones whose vertices are on the ideal boundary) by behavior of their mean curvature.

We find the characterization of maximum dimensional proper-biharmonic integral $\mathcal{C}$-parallel submanifolds of a Sasakian space form and then we classify such submanifolds in a 7-dimensional Sasakian space form. Working in the sphere $\boldsymbol{S}^7$ we explicitly find all 3-dimensional proper-biharmonic integral $\mathcal{C}$-parallel submanifolds. We also determine the proper-biharmonic parallel Lagrangian submanifolds of $\boldsymbol{C}P^3$.

We propose a new representation of the Schrödinger operator of a free particle by using the short-time Fourier transform and give its applications.

We show rather unexpectedly and surprisingly the existence of a hyperbolic Riemann surface $W$ enjoying the following two properties: firstly, the converse of the celebrated Parreau inclusion relation that the harmonic Hardy space $HM_{2}(W)$ with exponent 2 consisting of square mean bounded harmonic functions on $W$ includes the space $HD(W)$ of Dirichlet finite harmonic functions on $W$, and a fortiori $HM_{2}(W)=HD(W)$, is valid; secondly, the linear dimension of $HM_{2}(W)$, hence also that of $HD(W)$, is infinite.

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.

We give two generalizations of the Clifford theorem to algebraic surfaces. As an application, we obtain some bounds for the number of moduli of surfaces of general type.

We study property of Nevanlinna's deficiency as functions on linear systems in smooth complex projective algebraic varieties. We give a structure theorem for the set of deficient divisors. This structure theorem yields that the set of values of deficiency is at most countable. Moreover, we have a correspondence between the deficiencies and the linear systems.