In this paper, we shall study the behavior of the principal distributions on the graph of a homogeneous polynomial in two variables such that the set of its umbilical points is finite. In particular, we shall present a method of describing the indices of the umbilical points and the point at infinity.

Imagine a thin elastic rod like a piano wire. We consider the situation that the elastic rod is bent and twisted and both ends are welded together to form a smooth loop. Then, does there exist a stable equilibrium? In this paper, we generalize the energy of uniform symmetric Kirchhoff elastic rods in the $3$-dimensional Euclidean space to consider such a variational problem in a Riemannian manifold. We give the existence and regularity of minimizers of the energy in a compact or homogeneous Riemannian manifold.

We classify singular fibers of curves of genus three and determine their topological monodromies and the strata of their moduli points.

The pre-Tango structure is a certain invertible sheaf of locally exact differentials on a curve in positive characteristic. On any curve of sufficiently high genus, there necessarily exist pre-Tango structures. Meanwhile, by using the notion of pre-Tango structure, we can construct a form of the affine line over the curve. The completions of the forms are regarded as a generalization of Raynaud's counter-example to the Kodaira vanishing theorem. This suggests that we may have certain pathological phenomena on the completions of all such forms. For the time being, we consider whether every curve of genus greater than one has a pre-Tango structure which brings certain pathological phenomena. In the present article, we give a sufficient condition for the completion of the form which is induced from a pre-Tango structure to have non-closed global differential 1-forms. Moreover, we give a lower bound for the dimension of the locus of the curves which have pre-Tango structures satisfying that sufficient condition, in the moduli space of curves.

For asymptotically almost periodic functional differential equations with infinite delay in a Banach space, some stability properties of a bounded solution are deduced from stabilities in a certain limiting equation which is obtained by employing the Bohr topology.

Using a standard technique of Li and Yau, we study heat kernel estimates for a special type of compact conformally Kähler manifold, called a multiplier Hermitian manifold of type $\sigma$, which we derive from a Hamiltonian holomorphic vector field on the manifold. In particular, we obtain a lower bound estimate for the Green function averaged by the associated group action. For a fixed $\sigma$, such an estimate is known to play a crucial role in the proof of the uniqueness, modulo a group action, of Einstein multiplier Hermitian structures on a given Fano manifold.

We present an extension of the well-known $3/2$-stability criterion by Yorke for two term functional differential equations. We prove the exact nature of the obtained stability region which coincides with the Yorke result in the special case when the decay term is absent. Moreover, we reveal some interesting links existing between the Yorke conditions, Halanay inequalities and differential equations with maxima, all of them essentially involving the maximum functionals.

In Y. Hamada, Tohoku Math. J., 52 (2000), 79-94, we have given a remark on the domain of holomorphy of the solution of the Cauchy problem for the differential operator with polynomial coefficients. In this article, we give some complements to the results of Y. Hamada, Tohoku Math. J., 52 (2000), 79-94.

Dans Y. Hamada, Tohoku Math. J., 52 (2000), 79-94, nous avons donné une remarque sur le domaine d'holomorphie de la solution du problème de Cauchy pour l'opérateur différentiel à coefficients polynomiaux. Dans cet article, nous donnons quelques compléments des résultats de Y. Hamada, Tohoku Math. J., 52 (2000), 79-94.

We give a result on strong unique continuation property for a certain elliptic system of first order in the two dimensional space. Two coefficient matrices are normal and commutative with each other. We assume, further, that their components are Hölder continuous and have continuous first order derivatives except at one point. Without any regularity assumptions on the eigenvalues, we can show the strong unique continuation property for a class of such systems under certain quantitative conditions on the first order derivatives. This result gives an improvement of a work by G. N. Hile and M. H. Protter in a special case.

We show that a codimension-one minimal foliation with growth at most 2 of a complete Riemannian manifold with non-negative Ricci curvature is totally geodesic. We present some foliated versions of the result given by Alencar and do Carmo, and of minimal graphs by Miranda. Further, we simplify the proof of Meeks' result concerning constant mean curvature foliations of 3-dimensional Euclidean space.