Hessian of Hausdorff dimension on purely imaginary directions
Hessian of Hausdorff dimension on purely imaginary directions
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DOI：
10.1112/blms.12612
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发表时间：
2020-10
影响因子：
0.9
通讯作者：
M. Bridgeman;Béatrice Pozzetti;Andr'es Sambarino;Anna Wienhard
中科院分区：
文献类型：
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作者：
M. Bridgeman;Béatrice Pozzetti;Andr'es Sambarino;Anna Wienhard
Bridgeman-Taylor (Math. Ann. 341 (2008), 927–943) 和 McMullen (Invent. Math. 173 (2008), 365-425) 表明，Teichmüller 空间上的 Weil-Petersson 度量可以通过观察某些准 Fuchsian 变形的 Hausdorff 维数的无穷小变化来实现。在本文中，我们对 Bridgeman–Canary–Labourie–Sambarino (Geom. Dedicata 192 (2018), 57–86) 在 PSLd(R)${{\mathsf {PSL}}}_d(\mathbb {R})$ 的希钦分量上引入的谱间隙压力度量给出了类似的几何解释。更一般地，我们研究 Hausdorff 维数的 Hessian 矩阵作为 (1,1,2)-超凸表示空间上的函数，该类在 (J. reine angew.Math.774 (2021), 1-51) 中引入，其中包括 Hitchin 表示和 θ$\Theta$ 正表示的小复杂变形。作为另一个应用，我们证明当 Γ${{\Gamma }}$ 在PO(n,1)${{\mathsf {PO}}}(n,1)$ （除非 n=2$n=2$ 并且变形与 X(Γ,PO(2,1))$\mathfrak {X}({{\Gamma }}, {{\mathsf {PO}}}(2,1))$ 相切）。
Bridgeman–Taylor (Math. Ann. 341 (2008), 927–943) and McMullen (Invent. Math. 173 (2008), 365–425) showed that the Weil–Petersson metric on Teichmüller space can be realized by looking at the infinitesimal change of the Hausdorff dimension of certain quasi‐Fuchsian deformations. In this article, we give a similar geometric interpretation of the spectral gap pressure metric introduced by Bridgeman–Canary–Labourie–Sambarino (Geom. Dedicata 192 (2018), 57–86) on the Hitchin component for PSLd(R)${{\mathsf {PSL}}}_d(\mathbb {R})$ . More generally, we investigate the Hessian of the Hausdorff dimension as a function on the space of (1,1,2)‐hyperconvex representations, a class introduced in (J. reine angew. Math. 774 (2021), 1–51) which includes small complex deformations of Hitchin representations and of Θ$\Theta$ ‐positive representations. As another application, we prove that the Hessian of the Hausdorff dimension of the limit set at the inclusion Γ→PO(n,1)→PU(n,1)${{\Gamma }}\rightarrow {{\mathsf {PO}}}(n,1)\rightarrow {{\mathsf {PU}}}(n,1)$ is positive definite when Γ${{\Gamma }}$ is co‐compact in PO(n,1)${{\mathsf {PO}}}(n,1)$ (unless n=2$n=2$ and the deformation is tangent to X(Γ,PO(2,1))$\mathfrak {X}({{\Gamma }}, {{\mathsf {PO}}}(2,1))$ ).