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^∞ convex integrand γ: Sⁿ → R₊, its dual convex integrand δ: Sⁿ → R₊ is of class C^∞ if and only if γ is a strictly convex integrand.

(2) Let γ: Sⁿ → R₊ be a C^∞ strictly convex integrand. Then, γ is stable if and only if its dual convex integrand δ: Sⁿ → R₊ is stable.

(3) Let γ: Sⁿ → R₊ be a C^∞ strictly convex integrand. Suppose that γ is stable. Then, for any i (0 ≤ i ≤ n), a point θ₀ ∈ Sⁿ is a non-degenerate critical point of γ with Morse index i if and only if its antipodal point -θ₀ ∈Sⁿ is a non-degenerate critical point of the dual convex integrand δ with Morse index (n-i).

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.

Let (M,φ,ζ,η,g) be a three-dimensional Kenmotsu manifold. In this paper, we prove that the triple (g,V,λ) on M is a ＊-Ricci soliton if and only if M is locally isometric to the hyperbolic 3-space H³(-1) and λ=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 coKähler 3-manifold is a ＊-Ricci soliton if and only if it is a Ricci soliton.

We show that the normalized Ricci flow g(t) on a smooth closed manifold M existing for all t ≥ 0 with scalar curvature converging to constant in L² norm should satisfy

where 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 ∞ in sup norm) on 4-manifolds.

We consider the initial-value problem for the equivariant Schrödinger 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 ; (R²) ∩ (R²)) for time interval I. In the proof, we give a justification of the derivation of the modified Schrödinger map equation in low regularity settings without smallness of energy.

Z. Zhou et al. proved that in a Teichmüller 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.

We show that for an arbitrarily given closed Riemannian manifold M admitting a point p ∈ M with a single cut point, every closed Riemannian manifold N admitting a point q ∈ 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 L¹-norm to those of M at p.

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.