Let $\|M\|_{\Delta}$ denote the simplicial volume of $M$, $V_r(X,h)=\sup_{x\in X}\operatorname{Vol}_h\big(B_h(x,r)\big)$, and $\mathbb{H}^n$ denotes hyperbolic $n$-space. We prove that, if a closed oriented $n$-manifold $M$ admits a hyperbolic metric, then there is a dimensional constant $\delta_n>0$ such that every Riemannian metric $g$ on $M$ with \[ \frac{\operatorname{Vol}_g(M)}{\|M\|_{\Delta}}<\delta_n \] satisfies \[ V_r(\widetilde M,\widetilde g)\ge V_r(\mathbb{H}^n) \quad\text{for every }r\ge 1. \]
Let $M$ be a connected smooth $n$-manifold without boundary, where $n\geq3$, and let $\kappa\in\mathbb{R}$, with $\kappa\leq0$ if $M$ is open. We prove that every smooth Riemannian metric $g_0$ with $\mathrm{Scal}_{g_0}\geq\kappa$ is a locally uniform limit of smooth Riemannian metrics $g_i$ with $\mathrm{Scal}_{g_i}=\...
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...
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 \ma...
Let $\mathcal{X}$ be a complex manifold and $X$ a compact complex manifold with boundary in $\mathcal{X}$. For a complex Lie group $G$ and a regularity class $$\mathfrak{r}\in \big\{\mathcal{C}^k|\ k\in\mathbb{N}\cup\{\infty\}\big\}\cup\big\{\Lambda^r_{\rm loc}|\ r\in (0,\infty)\big\} $$ we define the sheaf of groups $...
Let $M_t$ denote the normalized average over the lattice points in the Euclidean ball of radius $t$ in $\mathbb{Z}^d$. We prove that the full maximal operator $f\mapsto\sup_{t\geq0}\lvert M_t f\rvert$ is bounded on $\ell^p(\mathbb{Z}^d)$, for every $1<p\leq\infty$, with a constant independent of the dimension. In parti...
Let $(M^3, g_{\mathrm{hyp}})$ be a closed oriented hyperbolic three-manifold normalized so that $\operatorname{sec}_{g_{\mathrm{hyp}}} \equiv -1$. We prove a sharp lower bound for the volume of hypersurfaces in $M^3 \times \mathbb{S}^1$ representing the slice class $[M^3 \times \{ \mathrm{pt} \}] \in H_3(M^3 \times \ma...
Li-Zhi Chen, Kun-Tao Jin· 0 citations
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