In previous papers, the author defined a notion of admissible functions for digraphs and studied its properties. The notion of admissible functions naturally comes from the study of mean curvature functions of codimension-one foliations, and admissible functions of foliated manifolds are represented by a divergence formula. In this paper, we show that the similar divergence-like formula characterizes admissible functions of digraphs.

A form of Bernstein theorem states that a complete stable minimal surface in euclidean space is a plane. A generalization of this statement is that there exists no complete stable hypersurface of an $n$-euclidean space with vanishing $(n-1)$-mean curvature and nowhere zero Gauss-Kronecker curvature. We show that this is the case, provided the immersion is proper and the total curvature is finite.

We classify all totally geodesic submanifolds of connected irreducible Riemannian symmetric spaces of noncompact type which arise as a singular orbit of a cohomogeneity one action on the symmetric space.

We classify semi-Riemannian submersions with connected totally geodesic fibres from a real pseudo-hyperbolic space onto a semi-Riemannian manifold under the assumption that the dimension of the fibres is less than or equal to three. Also, we obtain the classification of semi-Riemannian submersions with connected complex totally geodesic fibres from a complex pseudo-hyperbolic space onto a semi-Riemannian manifold under the assumption that the dimension of the fibres is less than or equal to two. We prove that there are no semi-Riemannian submersions with connected quaternionic fibres from a quaternionic pseudo-hyperbolic space onto a Riemannian manifold.

The Kirchhoff elastic rod is one of the mathematical models of thin elastic rods, and is a critical point of the energy functional with the effect of bending and twisting. In this paper, we study Kirchhoff elastic rods in the three-sphere of constant curvature. In particular, we give explicit expressions of Kirchhoff elastic rods in terms of elliptic functions and integrals. In addition, we obtain equivalent conditions for Kirchhoff elastic rods to be closed, and give an example of closed Kirchhoff elastic rods.

Concerning complete orientable minimal surfaces with finite total curvature in Euclidean three-space, we show for any positive genus the existence of noncongruent examples having the same symmetry group and conformal type.

On the setting of bounded smooth domains, we study positive Toeplitz operators between the harmonic Bergman spaces. We give characterizations of bounded and compact Toeplitz operators taking one harmonic Bergman space into another in terms of certain Carleson and vanishing Carleson measures.

We give a new, elementary proof of the global or almost global existence theorem of S. Klainerman. Our result also covers the almost global existence theorem of M. Keel, F. Smith, and C. D. Sogge. The proof is carried out in line with S. Klainerman and T. C. Sideris.

In this article we completely classify Lagrangian $H$-umbilical surfaces of constant curvature in complex Euclidean plane.