In this paper, we investigate divergent orbits for cone actions on products of certain homogeneous spaces. We introduce a notion of essential singularity for such actions, and estimate the Hausdorff dimension of the corresponding singular set. In particular, let $G/\Gamma=\mathrm{SL}(2,\mathbb{R})^s/\mathrm{SL}(2,\mathbb{Z})^s$, and let $C$ be a cone in the positive Weyl chamber with angular aperture $\epsilon>0$. Then the Hausdorff dimension of the set of points with essential divergent orbits under $C$ satisfies that when $\epsilon\in (0,\frac{1}{64})$, $$ 3s-\frac{1}{2}-4(s-1)\epsilon \leq \dim D^e(C, G/\Gamma)\leq 3s-\frac{1}{2}-\frac{1}{3}\epsilon. $$ This extends the previous result of An--Guan--Marnat--Shi \cite{AGMS} to higher-dimensional cone actions.
Let $X$ be a compact Riemann surface of genus $g\geq2$, and let $\M$ be the moduli space of rank-two trace-free Higgs bundles with fixed determinant of odd degree. For the normalization of the Hitchin metric used in this paper, we prove that the volume of geodesic ball in the Hitchin moduli space is given by $$ \Vol_{g...
We consider scale-invariant curvature energies for immersions of closed manifolds of even dimension $n=2h$ into $\mathbb R^m$, with principal term $\int_{\Sigma} \big|\nabla^{(h-1)} \vec{\mathrm{I\!I}}\big|_g^2\,d\text{vol}_g$ and arbitrary lower-order polynomial extrinsic invariants of the same scaling. Following the...
For $r>2$, we study the moduli space parameterising fixed degree morphisms $\mathbb P^r\to X$ where $X$ is a smooth hypersurface of low degree. More precisely, we prove the following result: let $n\geq 2, e\geq 1$ and $X\subset \mathbb P^{n-1}$ a smooth degree $d\geq 2$ hypersurface over an algebraically closed field o...
For a locally symmetric space $\Gamma\backslash G /K$ we study the spectrum of the invariant differential operators of $G/K$ on $L^2(\Gamma\backslash G/K)$. This gives a purely analytical proof of the recent theorem of arXiv:2402.02530 that the quasiregular representation $L^2(\Gamma \backslash G)$ is tempered if the l...
For every $n \geq 1$ there is a distinguished $\mathfrak{sl}_2(\mathbb{R})$ action on the Poincar\'e disk, arising as the infinitesimal isometries of a hyperbolic metric with conical singularity of order $n-1$ at the origin. For $n=1$ this is the standard infinitesimal M\"obius action, and for $n>1$ these vector fields...
We study proper actions on homogeneous spaces of the group $\mathrm{B}(n)$ of invertible upper triangular matrices from a coarse-geometric viewpoint. We obtain sufficient conditions for the properness of the natural $L$-action on $\mathrm{B}(n)/H$, where $L,H$ are closed subgroups of $\mathrm{B}(n)$. We also give an ex...
Hiroaki Nagaya· 0 citations
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