Michelle Bucher-Karlsson
We compute the value of the simplicial volume for closed, oriented Riemannian manifolds covered by $\mathbb{H}^{2}\times\mathbb{H}^{2}$ explicitly, thus in particular for products of closed hyperbolic surfaces. This gives the first exact value of a nonvanishing simplicial volume for a manifold of nonconstant curvature.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
We estimate the upper bound for the $\ell^{\infty}$-norm of the volume form on $\mathbb{H}^2\times\mathbb{H}^2\times\mathbb{H}^2$ seen as a class in $H_{c}^{6}(\mathrm{PSL}_{2}\mathbb{R}\times\mathrm{PSL}_{2}\mathbb{R}\times\mathrm{PSL}_{2}\mathbb{R};\mathbb{R})$. This gives the lower bound for the simplicial volume of closed Riemennian manifolds covered by $\mathbb{H}^{2}\times\mathbb{H}^{2}\times\mathbb{H}^{2}$. The proof of these facts yields an algorithm to compute the lower bound of closed Riemannian manifolds covered by $\big(\mathbb{H}^2\big)^n$.
We define the ideal simplicial volume for compact manifolds with boundary. Roughly speaking, the ideal simplicial volume of a manifold $M$ measures the minimal size of possibly ideal triangulations of $M$ "with real coefficients", thus providing a variation of the ordinary simplicial volume defined by Gromov in 1982, the main difference being that ideal simplices are now allowed to appear in representatives of the fundamental class. We show that the ideal simplicial volume is bounded above by the ordinary simplicial volume, and that it vanishes if and only if the ordinary simplicial volume doe…
We provide sharp lower bounds for the simplicial volume of compact $3$-manifolds in terms of the simplicial volume of their boundaries. As an application, we compute the simplicial volume of several classes of $3$-manifolds, including handlebodies and products of surfaces with the interval. Our results provide the first exact computation of the simplicial volume of a compact manifold whose boundary has positive simplicial volume. We also compute the minimal number of tetrahedra in a (loose) triangulation of the product of a surface with the interval.
We show that the simplicial volume of a contractible 3-manifold not homeomorphic to $\mathbb{R}^3$ is infinite. As a consequence, the Euclidean space may be characterized as the unique contractible $3$-manifold with vanishing minimal volume, or as the unique contractible $3$-manifold supporting a complete finite-volume Riemannian metric with Ricci curvature uniformly bounded from below. On the contrary, we show that in every dimension $n\geq 4$ there exists a contractible $n$-manifold with vanishing simplicial volume not homeomorphic to $\mathbb{R}^n$. We also compute the spectrum of the simpl…
We show that closed manifolds supporting a nonpositively curved metric with negative $([\frac{n}{4}]+1)$-Ricci curvature, have positive simplicial volume. This answers a special case of a conjecture of Gromov.
We study the question when a manifold that fibers over a sphere can be rationally essential, or even have positive simplicial volume. More concretely, we show that mapping tori of manifolds (whose fundamental groups can be quite arbitrary) of odd dimension at least 7 with non-zero simplicial volume are very common. This contrasts the case of fiber bundles over a sphere of dimension d > 1: we prove that their total spaces are rationally inessential if d is at least 3, and always have simplicial volume 0. Using a result by Dranishnikov, we also deduce a surprising property of macroscopic dimensi…
The simplicial volume introduced by Gromov provides a topologically accessible lower bound for the minimal volume. Lafont and Schmidt proved that the simplicial volume of closed, locally symmetric spaces of non-compact type is positive. In this paper, we present a generalization of this result to certain non-compact locally symmetric spaces of finite volume, to so-called Hilbert modular varieties. The key idea is to reduce the problem to the compact case by first relating the simplicial volume of these manifolds to the Lipschitz simplicial volume and then taking advantage of a proportionality …
Let $(M,g)$ be a compact manifold with Ricci curvature almost bounded from below and $π:\bar{M}\to M$ be a normal, Riemannian cover. We show that, for any nonnegative function $f$ on $M$, the means of $føπ$ on the geodesic balls of $\bar{M}$ are comparable to the mean of $f$ on $M$. Combined with logarithmic volume estimates, this implies bounds on several topological invariants (volume entropy, simplicial volume, first Betti number and presentations of the fundamental group) in Ricci curvature $L^p$-bounded from below.
We show that every closed, locally homogeneous Riemannian manifold with positive simplicial volume must be homeomorphic to a locally symmetric space of non-compact type.
We show that surface bundles over surfaces with base and fiber of genus at least 2 have non-vanishing simplicial volume.
We show that compact, locally symmetric spaces of non-compact type have positive simplicial volume. This gives a positive answer to a question that was first raised by Gromov in 1982. We provide a summary of results that are known to follow from positivity of the simplicial volume.
Positivity of simplicial volume for closed nonpositively curved four-manifolds with nonzero Euler characteristic
研究具有非零欧拉特征的闭非正曲率四维流形，运用高斯 - 博内定理证明其单纯体积为正，部分解决相关猜想，还证实了格罗莫夫关于负里奇曲率时单纯体积正性的猜想。
Simplicial volume and 0-strata of separating filtrations
本文用Papasoglu的面积最小化分离集方法，为Guth和Braun--Sauer关于闭定向流形在万有覆盖球体积有界时单纯形体积有上界的定理提供了替代证明，并给出了显式常数。
We prove that given a hyperbolic manifold endowed with an auxiliary Riemannian metric whose sectional curvature is negative and whose volume is sufficiently small in comparison to the hyperbolic one, we can always find for any radius at least $1$ a ball in its universal cover whose volume is bigger than the hyperbolic one. This result is deduced from a non-sharp macroscopic version of a conjecture by R. Schoen about scalar curvature, whose proof is a variation of an argument due to M. Gromov and based on a smoothing technique. We take the opportunity of this work to present a full account of t…
Simplicial volume and isolated, closed totally geodesic submanifolds of codimension one
本文证明含余维数一孤立闭全测地子流形的非正曲率闭流形具有正单纯体积，并推出非正弯曲解析流形在万有覆叠含余维数一平坦时的二分结论。
We introduce an axiomatic definition for the Kodaira dimension and classify Thurston geometries in dimensions $\leq 5$ in terms of this Kodaira dimension. We show that the Kodaira dimension is monotone with respect to the partial order defined by maps of non-zero degree between 5-manifolds. We study the compatibility of our definition with traditional notions of Kodaira dimension, especially the highest possible Kodaira dimension. To this end, we establish a connection between the simplicial volume and the holomorphic Kodaira dimension, which in particular implies that any smooth Kähler 3-fold…
We study the Lipschitz simplicial volume, which is a metric version of the simplicial volume. We introduce the piecewise straightening procedure for singular chains, which allows us to generalize the proportionality principle and the product inequality to the case of complete Riemannian manifolds of finite volume with sectional curvature bounded from above. We obtain also yet another proof of the proportionality principle in the compact case by a direct approximation of the smearing map.
Gromov's Simplicial Volume Vanishing Conjecture for Positive Scalar Curvature
本文在基本群满足一种自然弱化的快速衰减性质的条件下，证明了Gromov猜想：任何允许正标量曲率度量的闭流形具有消失的单纯体积。
We show that codimension one dimensional Jacobian of the barycentric straightening map is uniformly bounded for most of the higher rank symmetric spaces. As a consequence, we prove that the locally finite simplicial volume of most $\mathbb Q$-rank $1$ locally symmetric spaces is positive, which has been open for many years. Finally we improve the degree theorem for $\mathbb Q$-rank $1$ locally symmetric spaces of Connell and Farb. We also address the issue of surjectivity of the comparison map in real rank $2$ case.
For a non-vanishing gradient-like vector field on a compact manifold $X^{n+1}$ with boundary, a discrete set of trajectories may be tangent to the boundary with reduced multiplicity $n$, which is the maximum possible. (Among them are trajectories that are tangent to $\partial X$ exactly $n$ times.) We prove a lower bound on the number of such trajectories in terms of the simplicial volume of $X$ by adapting methods of Gromov, in particular his "amenable reduction lemma". We apply these bounds to vector fields on hyperbolic manifolds.
We consider the minimal entropy problem, namely the question of whether there exists a smooth metric of minimal entropy, for certain classes of 3-manifolds. Among other resulsts, we show that if M is a closed, orientable, geometrizable 3-manifold with zero simplicial volume, then the minimal entropy can be solved for M if and only if M admits a metric modelled on 4 of the 8 standard 3-dimensional geometries, namely $S^3$, $S^2\times R$, $E^3$, or Nil.
Simplicial Volume and Scalar Curvature on Closed Kähler Surfaces
该研究针对闭凯勒曲面证实了格罗莫夫定量数量曲率-单纯体积猜想的核心估计，还构造了无穷多满足相同估计的一般类型非凯勒辛4流形，推进了相关几何猜想的验证与拓展。
Growth in the universal cover under large simplicial volume
本文研究闭流形上度量 $g$ 的万有覆盖中球的体积何时超过双曲体积，证明当 $(\mathrm{Vol}(M,g))^2$ 小于双曲体积的某个小常数倍时，对所有 $r\geq1$ 成立，加强了 Sabourau 定理并推进 Guth 猜想。
Theorem A. Let $M^n$ denote a closed Riemannian manifold with nonpositive sectional curvature and let $\tilde M^n$ be the universal cover of $M^n$ with the lifted metric. Suppose that the universal cover $\tilde M^n$ contains no totally geodesic embedded Euclidean plane $\mathbb{R}^2$ (i.e., $M^n$ is a visibility manifold). Then Gromov's simplicial volume $\| M^n \|$ is non-zero. Consequently, $M^n$ is non-collapsible while keeping Ricci curvature bounded from below. More precisely, if $Ric_g \ge -(n-1)$, then $vol(M^n, g) \ge \frac{1}{(n-1)^n n!} \| M^n \| > 0. Theorem B. (Perelman) Let $M^3$…
We introduce the volume entropy semi-norm in real homology and show that it satisfies functorial properties similar to the ones of the simplicial volume. Answering a question of M. Gromov, we prove that the volume entropy semi-norm is equivalent to the simplicial volume semi-norm in every dimension. We also establish a roughly optimal upper bound on the systolic volume of the multiples of any homology class.
Let n>2 and let M be an orientable complete finite volume hyperbolic n-manifold with (possibly empty) geodesic boundary having Riemannian volume vol(M) and simplicial volume ||M||. A celebrated result by Gromov and Thurston states that if M has empty boundary then the ratio between vol(M) and ||M|| is equal to v_n, where v_n is the volume of the regular ideal geodesic n-simplex in hyperbolic n-space. On the contrary, Jungreis and Kuessner proved that if the boundary of M is non-empty, then such a ratio is strictly less than v_n. We prove here that for every a>0 there exists k>0 (only depending…
Topological volumes of certain complete affine manifolds
本文通过估计基本群含无限顺从正规子群的完备仿射流形的顺从范畴，证明其单纯体积为零，并进一步得到稳定整系数单纯体积与最小体积熵消失，从而回答 Lück 的问题并验证积分逼近。
Let the complexity of a closed manifold M be the minimal number of simplices in a triangulation of M. Such a quantity is clearly submultiplicative with respect to finite coverings, and by taking the infimum on all finite coverings of M normalized by the covering degree we can promote it to a multiplicative invariant, a characteristic number already considered by Milnor and Thurston, which call the "stable complexity" of M. We study here the relation between the stable complexity of M and Gromov's simplicial volume ||M||. It is immediate to show that ||M|| is smaller or equal than the stable co…
A fundamental result by Gromov and Thurston asserts that, if M is a closed hyperbolic n-manifold, then the simplicial volume |M| of M is equal to vol(M)/v_n, where v_n is a constant depending only on the dimension of M. The same result also holds for complete finite-volume hyperbolic manifolds without boundary, while Jungreis proved that the ratio vol(M)/|M| is strictly smaller than v_n if M is compact with non-empty geodesic boundary. We prove here a quantitative version of Jungreis' result for n>3, which bounds from below the ratio |M|/vol(M) in terms of the ratio between the volume of the b…
We give some applications of the Chern Simons gauge theory to the study of the set ${\rm vol}(N,G)$ of volumes of all representations $ρ\coπ_1N\to G$, where $N$ is a closed oriented three-manifold and $G$ is either ${\rm Iso}_e\t{\rm SL_2(\R)}$, the isometry group of the Seifert geometry, or ${\rm Iso}_+{\Hi}^3$, the orientation preserving isometry group of the hyperbolic 3-space. We focus on three natural questions: (1) How to find non-zero values in ${\rm vol}(N, G)$? or weakly how to find non-zero elements in ${\rm vol}(\t N, G)$ for some finite cover $\t N$ of $N$? (2) Do these volumes sat…
We provide a proof for an inequality between volume and L2-Betti numbers of aspherical manifolds for which Gromov outlined a strategy based on general ideas of Connes. The implementation of that strategy involves measured equivalence relations, Gaboriau's theory of L2-Betti numbers of R-simplicial complexes, and other themes of measurable group theory. Further, we prove new vanishing theorems for L2-Betti numbers that generalize a classical result of Cheeger and Gromov. As one of the corollaries, we obtain a gap theorem which implies vanishing of L2-Betti numbers of an aspherical manifold when…
We study the asymptotic behavior of volume forms on a degenerating family of compact complex manifolds. Under rather general conditions, we prove that the volume forms converge in a natural sense to a Lebesgue-type measure on a certain simplicial complex. In particular, this provides a measure-theoretic version of a conjecture by Kontsevich--Soibelman and Gross--Wilson, bearing on maximal degenerations of Calabi--Yau manifolds.
Small scale index theory, scalar curvature, and Gromov's simplicial norms
本文研究具有数量曲率下界流形的拓扑复杂度，提出小尺度指标定理，利用泛覆盖的数量曲率、体积和单射半径界建立 Gromov 单纯范数上界，推广了 Lichnerowicz 消没定理并类比 Cheeger 有限性定理。
We study the set ${\rm vol}\left(M,G\right)$ of volumes of all representations $ρ\coπ_1M\to G$, where $M$ is a closed oriented $3$-manifold and $G$ is either ${\rm Iso}_+{\Hi}^3$ or ${\rm Iso}_e\t{\rm SL_2(\R)}$. By various methods, including relations between the volume of representations and the Chern--Simons invariants of flat connections, and recent results of surfaces in 3-manifolds, we prove that any 3-manifold $M$ with positive Gromov simplicial volume has a finite cover $\t M$ with ${\rm vol}(\t M,{\rm Iso}_+{\Hi}^3)\ne \{0\}$, and that any non-geometric 3-manifold $M$ containing at le…
This paper shows that the Seifert volume of each closed non-trivial graph manifold is virtually positive. As a consequence, for each closed orientable prime 3-manifold $N$, the set of mapping degrees $cD(M,N)$ is finite for any 3-manifold $M$, unless $N$ is finitely covered by either a torus bundle, or a trivial circle bundle, or the 3-sphere.
The circumcenter of mass of a simplicial polytope $P$ is defined as follows: triangulate $P$, assign to each simplex its circumcenter taken with weight equal to the volume of the simplex, and then find the center of mass of the resulting system of point masses. The so obtained point is independent of the triangulation. The aim of the present note is to give a definition of the circumcenter of mass that does not rely on a triangulation. To do so we investigate how volumes of polytopes change under Möbius transformations.
We study noncompact, complete, finite volume, negatively curved manifolds $M$. We construct $M$ with infinitely generated fundamental groups in all dimensions $n \geq 2$. We construct $M$ whose cusp cross sections are compact hyperbolic manifolds in all dimension $n\geq 3$. In contrast we show that if sectional curvature $-1<K(M)<0$, then cusp cross sections have zero simplicial volume. We construct negatively curved lattices that do not contain any parabolic isometries. We show that there are $M$ such that $\widetilde{M}$ does not satisfy the visibility axiom. We give a condition on the curva…
We generalize an inequality of Besson-Courtois-Gallot about volume and simplicial volume of closed manifolds to the $\ell_1$-norm of all the homology classes of complete manifolds. The inequality involves the critical exponent of the fundamental group of the manifold and the mass of the homology classes.
We prove the macroscopic cousins of three conjectures: 1) a conjectural bound of the simplicial volume of a Riemannian manifold in the presence of a lower scalar curvature bound, 2) the conjecture that rationally essential manifolds do not admit metrics of positive scalar curvature, 3) a conjectural bound of $L^2$-Betti numbers of aspherical Riemannian manifolds in the presence of a lower scalar curvature bound. The macroscopic cousin is the statement one obtains by replacing a lower scalar curvature bound by an upper bound on the volumes of $1$-balls in the universal cover.
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。