Showing papers in "Kodai Mathematical Journal in 2020"

Abstract: Let $(M,\phi,\xi,\eta,g)$ be a three-dimensional Kenmotsu manifold. In this paper, we prove that the triple $(g,V,\lambda)$ on $M$ is a $*$-Ricci soliton if and only if $M$ is locally isometric to the hyperbolic 3-space $\mathbf{H}^3(-1)$ and $\lambda=0$. Moreover, if $g$ is a gradient $*$-Ricci soliton, then the potential vector field coincides with the Reeb vector field. We also show that the metric of a coKahler 3-manifold is a $*$-Ricci soliton if and only if it is a Ricci soliton.

Abstract: It is known that Hirzebruch surfaces of non zero degree do not admit any constant scalar curvature Kahler metric [6, 22, 38]. In this note, we describe how to construct Hermitian metrics of positive constant Chern scalar curvature on Hirzebruch surfaces using Page-Berard-Bergery's ansatz [41, 14]. We also construct the interesting case of Hermitian metrics of zero Chern scalar curvature on some ruled surfaces. Furthermore, we discuss the problem of the existence in a conformal class of critical metrics of the total Chern scalar curvature, studied by Gauduchon in [26, 27].

Abstract: This is the first of two articles in which we define an elliptically degenerating family of hyperbolic Riemann surfaces and study the asymptotic behavior of the associated spectral theory. Our study is motivated by a result which Hejhal attributes to Selberg, proving spectral accumulation for the family of Hecke triangle groups. In this article, we prove various results regarding the asymptotic behavior of heat kernels and traces of heat kernels for both real and complex time. In Garbin et al. (2018) [8], we will use the results from this article and study the asymptotic behavior of numerous spectral functions through elliptic degeneration, including spectral counting functions, Selberg's zeta function, Hurwitz-type zeta functions, determinants of the Laplacian, wave kernels, spectral projections, small eigenfunctions, and small eigenvalues. The method of proof we employ follows the template set in previous articles which study spectral theory on degenerating families of finite volume Riemann surfaces (Huntley et al. (1995) [14] and (1997) [15], Jorgenson et al. (1997) [20] and (1997) [17]) and on degenerating families of finite volume hyperbolic three manifolds (Dodziuk et al. (1998) [4].) Although the types of results developed here and in Garbin et al. (2018) [8], are similar to those in existing articles, it is necessary to thoroughly present all details in the setting of elliptic degeneration in order to uncover all nuances in this setting.

Abstract: We show that the zeroth coefficient of the cables of the HOMFLY polynomial (colored HOMFLY polynomials) does not distinguish mutants. This makes a sharp contrast with the total HOMFLY polynomial whose 3-cables can distinguish mutants.

Abstract: In 2004 Oort studied the foliation on the space of $p$-divisible groups. In his theory, special leaves called central streams play an important role. It is still meaningful to investigate central streams, for example, there remain a lot of unknown things on the boundaries of central streams. In this paper, we classify the boundary components of the central stream for a Newton polygon consisting of two segments, where one slope is less than 1/2 and the other slope is greater than 1/2. Moreover we determine the generic Newton polygon of each boundary component using this classification.

Abstract: The notion of block divisibility naturally leads one to introduce unitary cyclotomic polynomials $\Phi_n^*(x)$. They can be written as certain products of cyclotomic poynomials. We study the case where $n$ has two or three distinct prime factors using numerical semigroups, respectively Bachman's inclusion-exclusion polynomials. Given $m \ge 1$ we show that every integer occurs as a coefficient of $\Phi^*_{mn}(x)$ for some $n\ge 1$ following Ji, Li and Moree [9]. Here $n$ will typically have many different prime factors. We also consider similar questions for the polynomials $(x^n-1)/\Phi_n^*(x)$, the inverse unitary cyclotomic polynomials.

Abstract: We consider the initial-value problem for the equivariant Schrodinger maps near a family of harmonic maps. We provide some supplemental arguments for the proof of local well-posedness result by Gustafson, Kang and Tsai in [Duke Math. J. 145(3) 537-583, 2008]. We also prove that the solution near harmonic maps is unique in $C(I;\dot{H}^1(\mathbf{R}^2)\cap \dot{H}^2(\mathbf{R}^2))$ for time interval $I$. In the proof, we give a justification of the derivation of the modified Schrodinger map equation in low regularity settings without smallness of energy.

Abstract: Z. Zhou et al. proved that in a Teichmuller equivalence class, there exists an extremal quasiconformal mapping with a weakly non-decreasable dilatation. In this paper, we prove that in an infinitesimal equivalence class, there exists a weakly non-decreasable extremal Beltrami differential.

Abstract: We prove that a projective vertical exact log smooth morphism of fs log analytic spaces with a base of log rank one yields polarized log Hodge structures in the canonical way.

Abstract: We characterize a space-like surface in a pseudo-Riemannian space form with zero mean curvature vector, in terms of complex quadratic differentials on the surface as sections of a holomorphic line bundle. In addition, combining them, we have a holomorphic quartic differential. If the ambient space is $S^4$, then this differential is just one given in [5]. If the space is $S^4_1$, then the differential coincides with a holomorphic quartic differential in [6] on a Willmore surface in $S^3$ corresponding to the original surface through the conformal Gauss map. We define the conformal Gauss maps of surfaces in $E^3$ and $H^3$, and space-like surfaces in $S^3_1$, $E^3_1$, $H^3_1$ and the cone of future-directed light-like vectors of $E^4_1$, and have results which are analogous to those for the conformal Gauss map of a surface in $S^3$.

Abstract: We classify up to conjugacy the group generated by a commuting pair of a periodic diffeomorphism and a hyperelliptic involution on an oriented closed surface. This result can be viewed as a refinement of Ishizaka's result on classification of the mapping classes of hyperelliptic periodic diffeomorphisms. As an application, we obtain the Dehn twist presentations of hyperelliptic periodic mapping classes, which are closely related to the ones obtained by Ishizaka.

Abstract: We show that the normalized Ricci flow $g(t)$ on a smooth closed manifold $M$ existing for all $t \geq 0$ with scalar curvature converging to constant in $L^2$ norm should satisfy $$\liminf_{t\rightarrow \infty} \int_M|\stackrel{\circ}{r}|_{g(t)}^2d\mu_{g(t)} =0,$$ where $\stackrel{\circ}{r}$ is the trace-free part of Ricci tensor. Using this, we give topological obstructions to the existence of such a Ricci flow (even with positive scalar curvature tending to $\infty$ in sup norm) on 4-manifolds.

Abstract: Combinatorial properties of some ideals related to bipartite graphs are examined. A description of the integral closure expressed through the log set of edge ideals of complete bipartite graphs is illustrated together with the fact that nontrivial generalized graph ideals of a strong complete quasi-bipartite graph are integrally closed.

Abstract: In this paper, we investigate simultaneous properties of a convex integrand $γ$ and its dual $δ$. The main results are the following three. (1) For a $C^\infty$ convex integrand $\gamma: S^n \to \mathbf{R}_+$, its dual convex integrand $\delta: S^n \to \mathbf{R}_+$ is of class $C^\infty$ if and only if $γ$ is a strictly convex integrand. (2) Let $\gamma: S^n \to \mathbf{R}_+$ be a $C^\infty$ strictly convex integrand. Then, $γ$ is stable if and only if its dual convex integrand $\delta: S^n \to \mathbf{R}_+$ is stable. (3) Let $\gamma: S^n \to \mathbf{R}_+$ be a $C^\infty$ strictly convex integrand. Suppose that $γ$ is stable. Then, for any $i$ ($0 \le i \le n$), a point $\theta_0 \in S^n$ is a non-degenerate critical point of $γ$ with Morse index $i$ if and only if its antipodal point $-\theta_0 \in S^n$ is a non-degenerate critical point of the dual convex integrand $δ$ with Morse index ($n-i$).

Abstract: In this article, we study the Heller relative $K_{0}$ group of the map $\mathbf{P}_{X}^{r} \to \mathbf{P}_{S}^{r}$, where $X$ and $S$ are quasi-projective schemes over a commutative ring. More precisely, we prove that the projective bundle formula holds for Heller's relative $K_{0}$, provided $X$ is flat over $S$. As a corollary, we get a description of the relative group $K_{0}(\mathbf{P}_{X}^{r} \to \mathbf{P}_{S}^{r})$ in terms of generators and relations, provided $X$ is affine and flat over $S$.

Abstract: We show that for an arbitrarily given closed Riemannian manifold $M$ admitting a point $p \in M$ with a single cut point, every closed Riemannian manifold $N$ admitting a point $q \in N$ with a single cut point is diffeomorphic to $M$ if the radial curvatures of $N$ at $q$ are sufficiently close in the sense of $L^1$-norm to those of $M$ at $p$.