Isoperimetry and volume preserving stability in real projective spaces

不可定向曲面上保测度微分同胚群的曲率

Curvature of Measure-Preserving Diffeomorphism Groups of Non-Orientable Surfaces

该研究聚焦不可定向紧曲面的保测度微分同胚群曲率，算出克莱因瓶、实射影平面多方向的曲率等指标，拓展经典方法给出两类曲面上信风流模型的天气不可预测性估计。

Positivity preserving along a flow over projective bundle

In this paper, we introduce a flow over the projective bundle $p:P(E^*)\to M$, which is a natural generalization of both Hermitian-Yang-Mills flow and Kähler-Ricci flow. We prove that the semipositivity of curvature of the hyperplane line bundle $\mathcal{O}_{P(E^*)}(1)$ is preserved along this flow under the null eigenvector assumption. As applications, we prove that the semipositivity is preserved along the this flow if the base manifold $M$ is a curve, which implies that the Griffiths semipositivity is preserved along the Hermitian-Yang-Mills flow over a curve. And we also reprove that the …

Quasiconformal distortion of projective transformations and discrete conformal maps

We consider the quasiconformal dilatation of projective transformations of the real projective plane. For non-affine transformations, the contour lines of dilatation form a hyperbolic pencil of circles, and these are the only circles that are mapped to circles. We apply this result to analyze the dilatation of the circumcircle preserving piecewise projective interpolation between discretely conformally equivalent triangulations. We show that another interpolation scheme, angle bisector preserving piecewise projective interpolation, is in a sense optimal with respect to dilatation. These two in…

某些模空间的辛结构和Poisson结构。II。上紧离散平面群的射影表示

Symplectic and Poisson structures of certain moduli spaces. II. Projective representations of cocompact discrete planar groups

设$G$是具有双不变度量的李群，不一定正定。本文证明了在早期文献中对闭曲面的基本群所作的某种构造可以推广到欧氏或非欧氏运动的任意无限保向余紧平面离散群$π$上，并得到（i）包含同态空间$\roman{Hom}(π,G)$的光滑流形$\Cal M$上的辛结构，此外，（ii）一个哈密顿$G$ -作用在$\Cal M$上，保持辛结构以及动量映射，使得约化空间等于表示空间$\roman{Rep}(π,G)$。更一般地说，这种构造也适用于某些射影表示空间。对于$G$紧的，所得到的表示空间继承了{ \it分层辛空间\/}的结构，使得分层具有有限的辛体积。例如，具有有理权的半稳定全纯抛物丛的{ \smc Mehta-Seshadri}模空间或与它们密切相关的空间通过{ \it在有限维中的辛约化以这种方式产生。

Projective superflows. II. $O(3)$ and the icosahedral group

Let $X\in\mathbb{R}^{n}$. For $ϕ:\mathbb{R}^{n}\mapsto\mathbb{R}^{n}$ and $t\in\mathbb{R}$, we put $ϕ^{t}=t^{-1}ϕ(Xt)$. A projective flow is a solution to the projective translation equation $ϕ^{t+s}=ϕ^{t}\circϕ^{s}$, $t,s\in\mathbb{R}$. The projective superflow is a projective flow with a rational vector field which, among projective flows with a given symmetry, is in a sense unique and optimal. In this second part we classify $3$-dimensional real superflows. Apart from the superflow $ϕ_{\hat{\mathbb{T}}}$ (with a group of symmetries being all symmetries of a tetrahedron) and the superflow $ϕ…

Projective and polynomial superflows. I

Let $x\in\mathbb{R}^{n}$. For $ϕ:\mathbb{R}^{n}\mapsto\mathbb{R}^{n}$ and $t\in\mathbb{R}$, we put $ϕ^{t}=t^{-1}ϕ(xt)$. A projective flow is a solution to the projective translation equation $ϕ^{t+s}=ϕ^{t}\circϕ^{s}$, $t,s\in\mathbb{R}$. Previously we have developed an arithmetic, topologic and analytic theory of $2$-dimensional projective flows: rational, algebraic, unramified, abelian flows, commuting flows. The current paper is devoted to highly symmetric flows - superflows. Within flows with a given symmetry, superflows are unique and optimal. Our first result classifies all $2$-dimensiona…