Let $\Sigma$ be a closed oriented surface of genus $>1$ and $M$ a complete hyperbolic 3-manifold with a marking $i:\Sigma\longrightarrow M$. We consider the case that $M$ has no parabolic cusps and at least one of the two ends is simply degenerate. For $\varGamma=\pi_1(\Sigma)$, let $\rho_M:\varGamma\longrightarrow \mathrm{PSL}_2(\mathbb{C})$ be the holonomy of $M$ and $\rho:\varGamma\longrightarrow \mathrm{PSL}_2(\mathbb{C})$ any representation in $\mathrm{PSL}_2(\mathbb{C})$. We will show that, if $\rho$ is discrete and non-faithful, then \[ \|[\mathrm{Vol}(\rho)]-[\mathrm{Vol}(\rho_M)]\|_\infty\geq \boldsymbol{v}_3 \] holds, where $[\mathrm{Vol}(\rho)]$ denotes the bounded fundamental class of $\rho$ in the bounded cohomology $H_b^3(\varGamma,\mathbb{R})$ of $\varGamma$ and $\boldsymbol{v}_3$ is the volume of a regular ideal 3-simplex in $\mathbb{H}^3$. As an application, we present a rigidity theorem for $\rho_M$ in the set of representations $\rho$ of $\varGamma$ in $\mathrm{PSL}_2(\mathbb{C})$ in terms of $[\mathrm{Vol}(\rho)]$. The rigidity theorem implies that $\boldsymbol{v}_3$ is the optimal separating constant.
Let $X\subset \mathbb C^n$ be a closed pure $d$-dimensional complex analytic set. We associate to $X$ its set of projective asymptotic directions $$ \Sigma_\infty(X) :=\{\ell \in \mathbb P^{n-1}(\mathbb C):\ell\cap C_\infty(X)\ne\{0\}\}, $$ where $ C_\infty(X)$ is the total tangent cone at infinity. We prove the metric...
We prove zero density estimates for $L$-functions of cuspidal automorphic representations $\pi$ of $\mathrm{GL}_2(\mathbb{A}_{\mathbb{Q}})$. We show that $N_\pi(\sigma, T) \ll T^{\frac{5}{2}(1 - \sigma) + o(1)}$, where $N_\pi(\sigma, T)$ denotes the number of zeros $\rho = \beta + i\gamma$ of $L(s,\pi)$ with $\beta \ge...
Let $\bar{V}$ be a connected smooth complex projective variety. Let $D \subset \bar{V}$ be a normal crossing divisor. Let $\mathbb{V}$ be a complex polarizable variation of Hodge structure on $V := \bar{V} \setminus D$. Suppose that the period map of $\mathbb{V}$ is immersive at a point of $V$. We prove that for every...
Let $(M^{n+1},g)$ be a closed smooth Riemannian manifold, and let \[ 0=\lambda_0\leq\lambda_1\leq\ldots \] be the spectrum of the Laplace-Beltrami operator, with corresponding real eigenfunctions $\{\phi_j\}_{j\geq0}$. Let $\{\omega_p(M,g)\}_{p\geq1}$ denote Gromov's volume spectrum. In 2014, Marques-Neves conjectured...
Let $G$ be a compact, connected, simply connected semisimple Lie group with Weyl group $W$ and simple reflections $S$. For a simplicial complex $\mathcal K$ on $S$, form the homotopy colimit $X_{\mathcal K}(G)=\operatorname*{hocolim}_{I\in\mathcal K}G/G_I$ of standard partial flag manifolds. We compute its integral hom...
For $m=2,3$, we prove that every smooth immersion $F:\mathbb{R}\mathbb{P}^m\looparrowright\overline{\mathbb B}^{,N}(1)$ satisfies $\kappa(F)^2\ge 2m/(m+1)$, with equality only for the Veronese embedding, up to congruence. We also prove that a closed, connected, orientable three-manifold admitting an immersion into a Eu...
Tsz-Kiu Aaron Chow, Jingbo Wan· 0 citations
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