We prove the Gromov--Ros conjecture for every rank-one symmetric space of noncompact type: finite-perimeter isoperimetric regions are precisely the geodesic balls, up to ambient isometry and null sets. The proof is organized uniformly in the root multiplicities $(p,q)$. We introduce the volume radius \[ R(r)^n=n\int_0^r\sinh^{n-1}s\,\cosh^q s\,ds, \] which converts the polar volume form into $R^{n-1}dR\,d\sigma$. Hence radial--angular stretches $R\mapsto e^{t\varphi(\theta)}R$ have the exact Jacobian $e^{nt\varphi}$. Starting from a volume geometric median, a log-partition correction produces an exactly volume-preserving family of global bi-Lipschitz stretches. The reduced-boundary area formula and the ambient cofactor yield a universal pointwise trace identity depending only on $(p,q)$ and the horizontal and vertical components of the measure-theoretic normal. Its specializations for $q=1,3,7$ admit explicit strict sign certificates, forcing the normal of an isoperimetric region to be radial almost everywhere. Isotropy invariance in BV and a one-dimensional weighted endpoint comparison then force a single ball. For complex hyperbolic space we additionally prove that every smooth bounded fixed-volume stable critical domain is a geodesic ball. The resulting complex-hyperbolic isoperimetric theorem removes the geometric hypothesis in several sharp weighted-Bergman contraction, Faber--Krahn, and Lieb--Wehrl inequalities, and their high-weight scaling limit recovers holomorphic Gaussian hypercontractivity.
Let $S^n$ be the unit round sphere with its intrinsic angular metric, normalized so that $\operatorname{diam}S^n=\pi$. For finite homogeneous metric spaces $X$, put \[ \delta_n=\inf_X d_{GH}(X,S^n). \] The main open problem is whether $\inf_{n\ge2}\delta_n>0$. Gelander's theorem gives $\delta_n>0$ in each fixed dimension, but not uniformly. An abstract cross-polytope construction gives the universal upper bound $\delta_n\le\pi/4$. In the opposite direction, ChatGPT combines the passage from small Gromov--Hausdorff error to an approximate finite action on the sphere, logarithmic stability of approximate inner-product-preserving maps due to Cuesta, operator-norm stability of almost representations, and Green's width theorem for finite transitive sets. This gives the quantitative bound \[ \delta_n\ge \frac{c}{(1+\log(n+1))^2} \] for all sufficiently large $n$.
Let $X=G/K$ be a symmetric space of noncompact type, of dimension $n$ and rank $r$, and let $Y=\Gamma\backslash X$. Sarnak's local bound for an $L^2$-normalized spherical joint eigenfunction with regular tempered parameter of size $T$ is $\|\phi\|_\infty\ll T^{(n-r)/2}$. We prove $o(T^{(n-r)/2})$ locally uniformly on every quotient. On finite-volume real hyperbolic manifolds this is uniform in the expanding cusp range $y\leq T^\beta$, $\beta<1/2$. If the injectivity radius is bounded below, we prove the global estimate $T^{(n-r)/2}(\log T)^{-r/2}$.
In this paper, we address the following question: if a flat torus $\mathbb{T}^n$ is isometrically and minimally embedded into a sphere $\mathbb{S}^N$, must its translation group extend to the isometry group of the ambient sphere? As shown by Robert Bryant, for $n=2$ the answer is positive. Furthermore, while Ying Lu, Peng Wang, and Zhenxiao Xie recently demonstrated that the answer is negative for immersions when $n \geq 3$, the question for embeddings remained open. This problem is deeply tied to the work of Mikhail Gromov and Anton Petrunin concerning optimal curvature bounds. Petrunin proved that any immersion of a torus into a unit ball must have a maximum normal curvature of at least $\sqrt{\frac{3n}{n+2}}$. This bound is attained, for example, by families of tori constructed by Gromov. We call the tori that attain this optimal bound"Gromov tori". In this work, we first demonstrate that any Gromov torus is intrinsically flat, lies within a sphere, and is minimal inside it. We then establish the necessary and sufficient conditions for defining these tori. Finally, we present our main result: for dimensions $n \ge 3$, there exists a non-equivariant embedded Gromov torus, which provides a definitive negative answer to the question above.
We construct random open covers of higher-rank locally symmetric spaces using a construction we call scaffolded Poisson processes. Let $X=G/K$ be a symmetric space of noncompact type and real rank at least $2$. We prove a general vanishing theorem for the normalized torsion in first homology along sequences of torsion-free lattices in $G$. In particular, if $G$ is simple, we get \[ \dfrac{\log |H_1(M_n;\mathbb{Z})_{\operatorname{tors}}|}{\mathrm{vol}(M_n)} \longrightarrow 0 \] for any sequence of distinct manifolds $M_n = \Gamma_n \backslash X$. This answers a question of Ab\'ert, Gelander, and Nikolov, and confirms the degree-one vanishing with trivial integral coefficients predicted by a conjecture of Bergeron and Venkatesh in the higher-rank setting. In addition, we get quantitative bounds with respect to the minimal injectivity radius for both the torsion in first homology and the minimal number of generators of $\Gamma$. Finally, we prove the analogous statements for affine buildings.
Let $S$ be a closed oriented surface of genus $g\geq 2$. We consider an $n$-pleated representation $\rho: \pi_1(S) \to \mathrm{PSL}_n(\mathbb{C})$ obtained by bending a Hitchin representation $\rho_0:\pi_1(S) \to \mathrm{PSL}_n(\mathbb{R})$ along a maximal geodesic lamination. The space of such $n$-pleated representations was recently introduced by Maloni-Martone-Mazzoli-Zhang who provided a parametrization via shear-bend cocycles. Our first result is that $\rho_0$ dominates $\rho$ in the Hilbert length spectrum and the translation-length spectrum; this generalizes our earlier result for finite laminations on punctured surfaces. Using this, we prove entropy rigidity results: namely, the Hilbert entropy of a quasi-Hitchin representation in the bending fiber is strictly greater than that of $\rho_0$, and the same for the translation-length entropy in the case that $\rho_0$ is $n$-Fuchsian. The proof involves analyzing the weighted planar networks for finite approximants of the monodromy matrix, and establishing a strict domination for \emph{most} curves using the equidistribution of closed geodesics in the unit tangent bundle of $S$.
We study the topology and low-degree Hodge theory of moduli spaces of semistable sheaves on bielliptic surfaces. For primitive rank-zero Mukai vectors $\mathbf{v}=(0,c_1(L),\chi)$ satisfying the positivity condition $\operatorname{nt}(L)\geq3$, the distinguished fixed-determinant component $M_{H,S}(\mathbf{v},L)$ admits a support morphism to $|L|$ and is interpreted as a relative compactified Jacobian. Using this fibration, Lefschetz-type properties of positive linear systems, monodromy, and mixed Hodge structures, we construct a surjective homomorphism $ \pi_1(S)\twoheadrightarrow\pi_1\bigl(M_{H,S}(\mathbf{v},L)\bigr) $ and compute the first two Betti numbers of an Albanese fiber $F$, $M_{H,S}(\mathbf{v},L)$, and the distinguished component $M^\circ_{H,S}(\mathbf{v})$. Fourier-Mukai transforms and Bridgeland wall crossing extend these computations to primitive admissible Mukai vectors of positive rank. We further prove that $H^2(F,\mathbb{C})$ is of pure Hodge type $(1,1)$. If $\lambda_S=\ell(\mathbf{v})=1,$ then $F$ is a strict irreducible Calabi-Yau variety up to a finite quasi-\'etale cover. Under the additional genericity assumption on the pullback polarization, $M^\circ_{H,S}(\mathbf{v})$ is smooth and is noncanonically birational to $\operatorname{Pic}^0(S)\times\operatorname{Hilb}^{\mathbf{v}^2/2}(S)$.
A. Piskunov· 0 citations
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