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The geometry of fronts
Kentaro Saji, Masaaki Umehara, Kotaro Yamada
We shall introduce the singular curvature function on cuspidal edges of surfaces, which is related to the Gauss-Bonnet formula and which characterizes the shape of cuspidal edges.…
Maximal surfaces with singularities in Minkowski space
Masaaki Umehara, Kotaro Yamada
We shall investigate maximal surfaces in Minkowski 3-space with singularities. Although the plane is the only complete maximal surface without singular points, there are many other…
Flat fronts in hyperbolic 3-space
Masatoshi Kokubu, Masaaki Umehara, Kotaro Yamada
We shall investigate flat surfaces in hyperbolic 3-space with admissible singularities, called `flat fronts'. An Osserman-type inequality for complete flat fronts is shown. When eq…
An elementary proof of Small's formula for null curves in PSL(2,C) and an analogue for Legendrian curves in PSL(2,C)
Masatoshi Kokubu, Masaaki Umehara, Kotaro Yamada
For null curves in PSL(2,C), there exists a representation formula in terms of two meromorphic functions and their derivatives (Small's formula). In this paper, we give an elementa…
A global theory of flexes of periodic functions
Gudlaugur Thorbergsson, Masaaki Umehara
For a real valued periodic smooth function u on R, , one defines the osculating polynomial (of order 2n+1) at a point to be the unique trigonometric polynomi…
Minimal surfaces that attain equality in the Chern-Osserman inequality
Masatoshi Kokubu, Masaaki Umehara, Kotaro Yamada
In the previous paper, Takahasi and the authors generalized the theory of minimal surfaces in Euclidean n-space to that of surfaces with holomorphic Gauss map in certain class of n…