Strictly convex Wulff shapes and $C^1$ convex integrands
In this paper, it is shown that a Wulff shape is strictly convex if and only if its convex integrand is of class $C^1$. Moreover, applications of this result are given.
math.MG↗
arXiv subjects
Takashi Nishimura
Publications and source records attributed to Takashi Nishimura.
Stability of $C^\infty$ convex integrands
In this paper, it is shown that the set consisting of stable convex integrands $S^n\to \mathbb{R}_+$ is open and dense in the set consisting of $C^\infty$ convex integrands with respect to Whitney $C^\infty$ topology. Moreover, an application of the proof of this result is also shown.
math.GT↗
Self-dual Wulff shapes and spherical convex bodies of constant width $π/{2}$
For any Wulff shape, its dual Wulff shape is naturally defined. A self-dual Wulff shape is a Wulff shape equaling its dual Wulff shape exactly. In this paper, it is shown that a Wulff shape is self-dual if and only if the spherical convex body induced by it is of constant width $π/{2}$.
math.MG↗
Around distance-squared mappings
This is a survey article on distance-squared mappings and related topics.
math.DG↗
Lowerable vector fields for a finitely L-determined multigerm
We show that the module of lowerable vector fields for a finitely L-determined multigerm is finitely generated in a constructive way.
math.DG↗
Aperture of plane curves
For any given $C^\infty$ immersion ${\bf r}: S^1\to \mathbb{R}^2$ such that the set $\mathcal{NS}_{\bf r}=\mathbb{R}^2-\cup_{s\in S^1}\left({\bf r}(s)+d{\bf r}_{s}(T_s(S^1))\right)$ is not empty, a simple geometric model of crystal growth is constructed. It is shown that our geometric model of crystal growth never formulates a polygon while it is growing. Moreover, it is shown also that our model always dissolves to a point.
math.MG↗
Vector fields liftable over finitely determined multigerms of corank at most one
In this paper, we propose one index $i_1(f)-i_2(f)$ which measures how well-behaved a given finitely determined multigerm $f: (\mathbb{K}^n,S)\to (\mathbb{K}^p,0)$ $(n\le p)$ of corank at most one is from the viewpoint of liftable vector fields; and we answer the following problems when the index indicates that the given multigerm $f$ is best-behaved. 1) When is the module of vector fields liftable over $f$ finitely generated? 2) How can we characterize the minimal number of generators when the module of vector fields liftable over $f$ is finitely generated? 3) How can we calculate the minimal number of generators when the module of vector fields liftable over $f$ is finitely generated? 4) How can we construct generators when the module of vector fields liftable over $f$ is finitely generated?
math.DG↗
Recognizable classification of Lorentzian distance-squared mappings
The Lorentzian length, which is one of the most significant functions in Lorentzian geometry, is a complex-valued function. Its square gives a real-valued non-degenerate quadratic function. In this paper, we define naturally extended mappings of Lorentzian distance-squared functions, wherein each component is a Lorentzian distance-squared function; and classify these mappings completely by the likeness of recognition subspaces.
math.GT↗
Distance-squared mappings
A distance-squared function is one of the most significant functions in the application of singularity theory to differential geometry. In this paper, we define naturally extended mappings of distance-squared functions, wherein each component is a distance-squared function. We investigate the properties of these mappings from the viewpoint of differential topology.
math.GT↗
Bi-Lipschitz A-equivalence of K-equivalent map germs
In this paper, two sufficient conditions are provided for given two K-equivalent map-germs to be bi-Lipschitz A-equivalent. These are Lipschitz analogues of the known results on C^r-A-equivalence $(0 \leq r \leq \infty)$ for given two K-equivalent map-germs. As a corollary of one of our results, a Lipschitz version of the well-known Fukuda-Fukuda theorem is provided.
math.AG↗
Wave front evolution and pedal evolution
The calculus correspondence has been known to exist between generic pedal evolutions and generic wave front evolutions. In this paper, we first extend the known results on the calculus correspondence to evolutions with multi-parameters, and then give applications of calculus correspondence. Moreover, we discuss the possibility of generalization of the calculus correspondence to degenerate pedal evolutions and degenerate wave front evolutions.
math.GT↗
Whitney umbrellas and swallowtails
In this paper, we introduce the notions of map-germs of pedal unfolding type and normalized Legendrian map-germs; and then we show that the fundamental theorem of calculus provides a natural one to one correspondence between Whitney umbrellas of pedal unfolding type and normalized swallowtails.
math.GT↗
Multicusps
For a given multicusp $f=c_{(θ_0,..., θ_i)}$ $(1\le i)$, we present a direct sum decomposition theorem of the source space of ${}_i\barωf$, where ${}_i\barωf$ is a higher version of the reduced Kodaira-Spencer-Mather map $\barωf$. As a corollary of our direct sum decomposition theorem, we show that for any $i\in \mathbb{N}$ and any $f=c_{(θ_0,..., θ_i)}$, ${}_i\barωf$ is bijective. The corollary is an affirmative answer to the question raised by M. A. S. Ruas during the 11th International Workshop on Real and Complex Singularities at the University of S${\tilde {\rm a}}$o Paulo in S${\tilde {\rm a}}$o Carlos (2010).
math.DG↗