Hokkaido Mathematical Journal

No. 1

ZHANG, Cuilian; PEI, Donghe;
The Gauss-Bonnet type theorem for future directed fronts in hyperbolic 3-space and de Sitter 3-space.
Hokkaido Mathematical Journal, 52 (2023) pp.1-22

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Abstract

We give a Gauss-Bonnet type theorem for future directed fronts in hyperbolic 3-space and de Sitter 3-space. This theorem relates the lightlike geometry to the topology of these singular objects.

Keywords Lightlike geometry; front; Gauss-Bonnet theorem;
msc2020(primary) 53C45; 53C40; 51H05;
ARTAL BARTOLO, E.; MORÓN-SANZ, R.;
Differential geometry of complex projective plane conics.
Hokkaido Mathematical Journal, 52 (2023) pp.23-40

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Abstract

In this paper we study properties of complex plane projective curves from a geometric point of view. We focus our attention on properties of conics. We find that Gauss curvature determines a conic up to a hermitian transformation preserving the Fubini-Study metric of the complex projective plane and we discuss some other geometric properties of the conics. Finally we study the deformation of smooth conics onto pair of lines and the classification of cubics up to hermitian transformations.

Keywords Gauss curvature; algebraic plane curves; Fubini-Study metric;
msc2020(primary) 53A05; 53C42; 14H50;
AIMI, Satoru;
Level set mean curvature flow, with Neumann boundary conditions.
Hokkaido Mathematical Journal, 52 (2023) pp.41-64

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Abstract

We investigate the relation between the level set approach and the varifold approach for the mean curvature flow with Neumann boundary conditions. With an appropriate initial data, we prove that the almost all level sets of the unique viscosity level set solution satisfy Brakke's inequality and a generalized Neumann boundary condition.

Keywords mean curvature flow; Neumann boundary conditions; level set method; varifolds;
msc2020(primary) 53E10; 49Q20;
HOANG, Nguyen Van; NGOAN, Ngo Thi;
On the cofiniteness of local cohomology modules in dimension < 2.
Hokkaido Mathematical Journal, 52 (2023) pp.65-73

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Abstract

In this note, we prove the cofiniteness of local cohomology modules $H^i_I(N)$ with respect to $I$ for all $i\lt t$ and the finiteness of $(0:_{H^t_I(N)}I)$ and ${\rm Ext}^1_R(R/I,H^t_I(N))$ provided that ${\rm Ext}^i_R(R/I,N)$ is finitely generated for all $i\le t+1$ and $H^i_I(N)$ is in dimension $\lt 2$ for all $i \lt t$, where $t\ge 1$ is an integer (here, $N$ is not necessarily finitely generated over $R$). This extends the results of Bahmanpour-Naghipour [5, Theorem 2.6], Aghapournahr-Bahmanpour [2, Theorem 3.4], Bahmanpour-Naghipour-Sedghi [4, Theorem 2.8] by a different proof method.

Keywords cofinite module; local cohomology; in dimension $\lt 2$;
msc2020(primary) 13D45; 14B15; 13E05;
LESSA, Pablo; OLIVEIRA, Lucas;
Fold maps associated to geodesic random walks on non-positively curved manifolds.
Hokkaido Mathematical Journal, 52 (2023) pp.75-96

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Abstract

We study a family of mappings from the powers of the unit tangent sphere at a point to a complete Riemannian manifold with non-positive sectional curvature, whose behavior is related to the spherical mean operator and the geodesic random walks on the manifold.

Abstract

We show that for odd powers of the unit tangent sphere the mappings are fold maps.

Abstract

Some consequences on the regularity of the transition density of geodesic random walks, and on the eigenfunctions of the spherical mean operator are discussed and related to previous work.

Keywords Geodesic random walk; spherical mean operator; fold maps;
msc2020(primary) 57R45; 60J10; 53C22;
RANDALL, Matthew;
Local equivalence of some maximally symmetric rolling distributions and $SU(2)$ Pfaffian systems.
Hokkaido Mathematical Journal, 52 (2023) pp.97-128

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Abstract

We give a description of Nurowski's conformal structure for some examples of bracket-generating rank 2 distributions in dimension 5, aka $(2,3,5)$-distributions, namely the An-Nurowski circle twistor distribution for pairs of surfaces of constant Gauss curvature rolling without slipping or twisting over each other. In the case of hyperboloid surfaces whose curvature ratios give maximally symmetric $(2,3,5)$-distributions, we find the change of coordinates that map the conformal structure to the flat metric of Engel. We also consider a rank $3$ Pfaffian system in dimension 5 with $SU(2)$ symmetry obtained by rotating two of the 1-forms in the Pfaffian system of the spheres rolling distribution, and discuss complexifications of such distributions.

Keywords $(2,3,5)$-distributions; split $\frak{g}_2$ Lie algebra; Pfaffian systems;
msc2020(primary) 53C18; 58A15;
IKEHATA, Ryo; SOBAJIMA, Motohiro;
Singular limit problem of abstract second order evolution equations.
Hokkaido Mathematical Journal, 52 (2023) pp.129-148

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Abstract

We consider the singular limit problem in a real Hilbert space for abstract second order evolution equations with a parameter $\varepsilon \in (0,1]$. We first give an alternative proof of the celebrated results due to Kisynski [10] from the viewpoint of the energy method. Next we derive a more precise asymptotic profile as $\varepsilon \to +0$ of the solution itself depending on $\varepsilon$ under a suitable regularity of the initial data.

Keywords Second order; Evolution equations; Hilbert spaces; Singular limit; Initial layer; Energy method;
msc2020(primary) 34G10; 34K26;
msc2020(secondary) 34K30; 34K25;
NAGASE, Masayoshi;
On the Tanno connection and the Chern-Moser connection, in almost CR-geometry.
Hokkaido Mathematical Journal, 52 (2023) pp.149-180

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Abstract

We show that one can construct a Cartan connection on the Cartan principal bundle over a contact Riemannian manifold, on which the associated complex structure is not assumed to be integrable, according to Cartan-Chern-Moser-Le's construction but with the use of the Tanno connection (instead of the Tanaka-Webster connection in the integrable case). Then we prove that it is normal in the sense of Tanaka if and only if the complex structure is integrable. By Le this has been shown to hold true in the case the dimension of the manifold is three.

Keywords Tanno connection; Chern-Moser connection; Cartan connection; normal in the sense of Tanaka; contact Riemannian structure;
msc2020(primary) 53D15;
msc2020(secondary) 53B15;