This is a survey article on the spherical method for studying Wulff shapes and related topics. The spherical method, which seems less common, is a powerful tool to study Wulff shapes and their related topics. It is verified how powerful the spherical method is by various results which seem difficult to be obtained without using the method. In this survey, the spherical method is explained in detail, and results obtained by using the spherical method until April 2016 are explained as well.

This is a half survey article on the recent development of the theory of graph-like Legendrian unfoldings which is the sequel to the previous surveys. The notion of big Legendrian submanifolds was introduced by Zakalyukin for describing the wave front propagations. Graph-like Legendrian unfoldings belong to a special class of big Legendrian submanifolds. In particular, natural equivalence relations among graph-like Legendrian unfoldings are introduced and geometric properties of these equivalence relations are investigated. Although this is a survey article, some new original results and proofs for some implicitly known results are given.

In this paper we introduce the notion of asymptotic directions of a surface in a 4-dimensional riemannian manifold, and study the special case of a surface in a 4-dimensional metric Lie group. It appears that this notion depends on the left invariant metric in general.

We have gotten singularity classifications of lightlike hypersurfaces of a pseudo null curve in $\mathbb{R}_2^4$^[6]. This paper is to characterize singularities of lightlike hypersurfaces of a partially null curve in the same space and give an example of such curves.

We construct triply periodic zero mean curvature surfaces of mixed type in the Lorentz-Minkowski 3-space $\mathbb{L}^3$, with the same topology as the triply periodic minimal surfaces in the Euclidean 3-space $\mathbb{R}^3$, called Schwarz rPD surfaces.

A constraint submanifold in a symplectic space after P.A.M. Dirac is determined locally by geometric restriction of the symplectic form to the constraint. The natural symplectic invariant associated to this restriction is the space of Hamiltonian vector fields which uniquely restrict to the solvable Hamiltonian ones on a constraint. By investigation of solvability of generalized Hamiltonian systems we characterize the constraint invariants and find them explicitly in the generic cases. Moreover the Poisson-Lie algebra on submanifold is described and an example of the Hamiltonian vector fields on the 2-sphere in symplectic space was constructed.

We consider the local geometry of a generic 1-parameter family of smooth curves in the real plane for which one member of the family has parallel tangents at two inflexion points. We study the equidistants of this family, that is the loci of points at a fixed ratio along chords joining points with parallel tangents, as a 2-parameter family depending on the value of the fixed ratio and on the parameter in the family of curves. Codimension 2 singularities of type ‘gull’ arise in this way and are in general versally unfolded by the two parameters. We also calculate the family of duals of the equidistants; here it is necessary to view them as bifurcation sets of bigerms and they evolve through ‘moth’ and ‘nib’ singularities also encountered in 1-parameter families of symmetry sets in the plane. Finally we show that certain sub-families of the 2-parameter family of equidistants can be classified by reduction to a normal form.

We study the rectifying developable surface of a framed base curve and a framed helix in the Euclidean space. A framed base curve is a smooth curve with a moving frame which may have singular points. By using the curvature of a framed base curve, we investigate the rectifying developable surface and a framed helix. Moreover, we introduce two new invariants of a framed base curve, which characterize singularities of the rectifying developable surface and a framed helix.

In this paper, we introduce a new concept of strongly keen for Heegaard splittings, and show that, for any integers $n\geq 2$ and $g\geq 3$, there exists a strongly keen Heegaard splitting of genus $g$ whose Hempel distance is $n$.

We can use a moving frame, as in the case of regular plane curves in the Euclidean plane, in order to define the arc-length parameter and the Frenet formula for non-lightlike regular curves in the Lorentz-Minkowski plane. This leads naturally to a well defined evolute associated to non-lightlike regular curves without inflection points in the Lorentz-Minkowski plane. However, at a lightlike point the curve shifts between a spacelike and a timelike region and the evolute cannot be defined by using this moving frame. In this paper, we introduce an alternative frame, the lightcone frame, that will allow us to associate an evolute to regular curves without inflection points in the Lorentz-Minkowski plane. Moreover, under appropriate conditions, we shall also be able to obtain globally defined evolutes of regular curves with inflection points. We investigate here the geometric properties of the evolute at lightlike points and inflection points.

Limiting tangent hyperplanes associated with isolated hypersurface singularities are considered in the context of symbolic computation. A new effective method is proposed to compute the limiting tangent space of a given hypersurface. The key of the method is the concept of parametric local cohomology systems. The proposed method can provide the decomposition of the limiting tangent space by Milnor numbers of hyperplane sections of a given hypersurface. The resulting algorithm has been implemented in the computer algebra system $\mathsf{Risa/Asir}$. Examples of the computation for some typical cases are given.

We study $D_4$-singularities of constant mean curvature (CMC) surfaces in Riemannian and semi-Riemannian spaceforms. We will give criteria for $D_4^{-}$-singularities, which are related to the Hopf differential. We also show the non-existence of (spacelike) CMC surfaces with $D_4^+$-singularities.

We study the singularities of secant maps associated to pairs of plane curves providing their geometrical interpretation up to codimension 2. We show that for most pairs of closed plane curves the secant map is a stable map from the torus to the plane. We determine the isotopy type of the singular set of the secant map associated to pairs of convex closed curves in terms of their Whitney indices.

In this paper we define and analyze singularities of discrete linear Weingarten surfaces with Weierstrass-type representations in 3-dimensional Riemannian and Lorentzian spaceforms. In particular, we discuss singularities of discrete surfaces with non-zero constant Gaussian curvature, and parallel surfaces of discrete minimal and maximal surfaces, and discrete constant mean curvature 1 surfaces in de Sitter 3-space, including comparisons with different previously known definitions of such singularities.

We study the topological configurations of the lines of principal curvature, the asymptotic and characteristic curves on a cuspidal edge, in the domain of a parametrization of this surface as well as on the surface itself. Such configurations are determined by the 3-jets of a parametrization of the surface.

In this paper we give the Weierstrass-type representation for Lorentz conformal minimal surfaces in Minkowski 3-space that was derived by Konderak, and a new one for Lorentz conformal constant mean curvature 1 surfaces in anti de Sitter 3-space, using integrable systems techniques. As an application, we analyze their singularities. Finally, we describe first steps toward discretization of these timelike surfaces.