For every $N\ge 2$, we prove that the Bergman metric on the regular locus of a finite ball quotient $\mathbb{B}^N/\Gamma$, where $\Gamma\subset \mathrm{U}(N)$ is finite and fixed-point-free, is K\"ahler-Einstein if and only if $\Gamma$ is trivial. Consequently, if $\Omega$ is an $N$-dimensional normal Stein space with isolated singularities and compact, smooth, strongly pseudoconvex boundary admitting a real-algebraic CR realization, then the Bergman metric on $\Omega_{\mathrm{reg}}$ is K\"ahler-Einstein if and only if $\Omega$ is biholomorphic to $\mathbb{B}^N$. This proves an algebraic version of the Cheng-Huang-Xiao conjecture in every complex dimension $N\ge 2$.
Let $n\geq2$ and let $M^n$ be a closed connected smooth manifold. Let $R^\gamma(M)$ be the space of smooth Riemannian metrics $g$ on $M$ for which the generalized conformal Laplace operator $-\gamma\Delta_g+\mathrm{R}_g$ is strictly positive. We prove that if $n=2$ and $\gamma>0$, or if $n\ge3$ and $0<\gamma \leq 4(n-1...
G. Antonelli, Georg Frenck, Bernhard Hanke· 0 citations
Let $(L,e^{-\phi})$ be a positive Hermitian holomorphic line bundle over a compact Riemann surface $X$, and let $\omega=i\partial\overline{\partial}\phi$. We obtain explicit pointwise estimates for the Bergman form of the tensor power $mL$. If $\mathrm{Ric}\,\omega\leq\omega$ and the shortest nonconstant closed geodesi...
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 $M_t$ 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...
We prove that for every integer $n\geq 2$ and $8\pi(n-1)<\rho<8\pi n$, the singular Liouville equation $\Delta u+\e^u=\rho\delta_0$ on a rectangular torus $E_{\mathrm{i}b}=\mathbb{C}/(\mathbb Z+\mathrm{i} b\mathbb Z)$ has exactly $n$ solutions, which are all axisymmetric. Together with previous results by Chen-Lin and...
Let $D\subset\mathbb C^n$, $n\geq 2$, be a bounded pseudoconvex domain with smooth boundary, and let $H^p_\omega(D)$ be the Hardy space defined using a weighted boundary measure $\omega\,d\sigma$, where $\omega$ is bounded above and bounded away from zero. For every $0<p<\infty$, $p\neq2$, we prove that each surjective...
Ren-Yu Chen, Song-Ying Li, Su-Juan Long et al.· 0 citations
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