In this article we classify the free involutions of every torus semi-bundle Sol 3-manifold. Moreover, we classify all the triples $M, \tau, \mathbf{R}^n$, where $M$ is as above, $\tau$ is a free involution on $M$, and $n$ is a positive integer, for which the Borsuk-Ulam Property holds.

Let $\delta_{g,n}$ be the minimal dilation of pseudo-Anosovs defined on an orientable surface of genus $g$ with $n$ punctures. It is proved by Tsai that for any fixed $g\ge2$, there exists a constant $c_g$ depending on $g$ such that \[ \frac{1}{c_g}\cdot \frac{\log n}{n} \lt \log \delta_{g,n} \lt c_g \cdot \frac{\log n}{n} \qquad \text{for any }n\ge3 \] This means that the logarithm of the minimal dilatation $\log \delta_{g, n}$ is on the order of $\log n/n$. We prove that if $2g + 1$ is relatively prime to $s$ or $s + 1$ for each $0\le s\le g$, then \[ \limsup_{n\to\infty}\frac{n(\log \delta_{g,n})}{\log n}\le 2 \] holds. In particular, if $2g + 1$ is prime, then the above inequality on $\delta_{g,n}$ holds. Our examples of pseudo-Anosovs $\phi$’s which provide the upper bound above have the following property: The mapping torus $M_\phi$ of $\phi$ is a single hyperbolic 3-manifold $N$ called the magic manifold, or the fibration of $M_\phi$ comes from a fibration of $N$ by Dehn filling cusps along the boundary slopes of a fiber.

We give a description of confluence for the general Schlesinger systems (GSS) from the view point of twistor theory. GSS is a system of nonlinear di¤erential equations on the Grassmannian manifold $G_{2,N}(\mathbf{C}$ which is obtained, for any partition $\lambda$ of $N$, as the integrability condition of a connection $\nabla_\lambda$ on $\mathbf{P}^1\times G_{2,N}$ constructed using the twistor-theoretic point of view and is known to describe isomonodromic deformation of linear differential equations on the projective space $\mathbf{P}^1$. For a pair of partitions $\lambda, \mu$ of $N$ such that m is obtained from $\lambda$ by making two parts into on parts and leaving other parts unchanged, we construct the limit process $\nabla_\lambda\to \nabla_\mu$ and as a result the confluence for GSS.

Let $N\ge 1, 1 \lt p \lt \infty$ and $p^*=\max(1,p-1)$. Let $\Omega$ be a bounded domain of $\mathbf{R}^N$. We establish the strong maximum principle for the $p$-Laplace operator with a nonlinear potential term. More precisely, we show that every super-solution $u \in \Omega^{1, p^*}_{\mathrm{loc}}(\Omega)$ vanishes identically in $\Omega$, if $u$ is admissible and $u = 0$ a.e on a set of positive $p$-capacity relative to $\Omega$.

We obtain conditions for stable extendibility of some complex vector bundles over the $(2n + 1)$-dimensional standard lens space $L^n(p) \operatorname{mod} p$, where $p$ is a prime. Furthermore, we study stable extendibility of the bundle $\pi^*_n (\tau(\mathbf{C}P^n))$ induced by the natural projection $\pi_n : L^n(p)\to \mathbf{C}P^n$ from the complex tangent bundle $\tau(\mathbf{C}P^n)$ of the complex projective $n$-space $\mathbf{C}P^n$. As an application, we have a result on stable extendibility of $\tau(\mathbf{C}P^n)$ which gives another proof of Schwarzenberger’s theorem.

Hensel-Przytycki-Webb proved that the curve graphs of all orientable surfaces are 17-hyperbolic. In this paper, we show that the curve graphs of nonorientable surfaces are 17-hyperbolic by applying Hensel-Przytycki-Webb’s argument. We also show that the arc graphs of non-orientable surfaces are 7-hyperbolic, and arccurve graphs of (non-)orientable surfaces are 9-hyperbolic.

The moduli space of left-invariant pseudo-Riemannian metrics on a given Lie group is defined as the orbit space of a certain isometric action on some pseudo- Riemannian symmetric space. In terms of the moduli space, we formulate a procedure to obtain a generalization of Milnor frames for left-invariant pseudo-Riemannian metrics on a given Lie group. This procedure is an analogue of the recent studies on left-invariant Riemannian metrics. In this paper, we describe the orbit space of the action of a particular parabolic subgroup, and then apply it to obtain a generalization of Milnor frames for so-called the Lie groups of real hyperbolic spaces, and also for the three-dimensional Heisenberg group. As a corollary we show that all left-invariant pseudo-Riemannian metrics of arbitrary signature on the Lie groups of real hyperbolic spaces have constant sectional curvatures.