Galina Guzhvina
In early 80's M.Gromov showed that there exists a constant $ε$ such that any compact Riemannian manifold $M^n$ with $|K|_{M^n} \cdot diam^2(M^n) \leq ε$ can be finitely covered by a nilmanifold. The present paper illustrates by an explicit example that the pinching constant $ε$ depends on the dimension $n$ of the manifold, in particular, it decreases with the dimension at least as $\frac{12}{n^2}.$
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 show that the 2-jet bundle of local Riemannian metrics on an arbitrary differentiable manifold admits a section which pointwise fulfills the curvature relation sec(g)=a for any real number a. It follows by Gromov's h-principle for open, invariant differential relations that every noncompact differentiable manifold carries arbitrarily pinched (incomplete) Riemannian metrics.
Let $M$ be a complete Riemannian metric of sectional curvature within $[-a^2,-1]$ whose fundamental group contains a $k$-step nilpotent subgroup of finite index. We prove that $a\ge k$ answering a question of M. Gromov. Furthermore, we show that for any $ε>0$, the manifold $M$ admits a complete Riemannian metric of sectional curvature within $[-(k+ε)^2,-1]$.
We study metrics on conic 2-spheres when no Einstein metrics exist. In particular, when the curvature of a conic metric is positive, we obtain the best curvature pinching constant. We also show that when this best pinching constant is approached, the conic 2-sphere has an explicit Gromov-Hausdorff limit. This is a generalization of the previous results of Chen-Lin and Bartolucci for 2-spheres with one or two conic points.
Sectional Curvature Pinching of Two-Step Nilmanifolds
本文研究二步幂零流形的截面曲率拼挤常数，证明其位于[-3,-3/2]，上界具有刚性，下界非刚性，并给出代数刻画。
The pinching constant for closed minimal submanifolds of high codimension in the sphere
本文针对单位球面中余维数 $q\geqslant 2$ 的闭极小子流形，推导了非全测地情形下其第二基本形式平方长度最大值的下界估计。
We generalize the second pinching theorem for minimal hypersurfaces in a sphere due to Peng-Terng, Wei-Xu, Zhang, and Ding-Xin to the case of hypersurfaces with small constant mean curvature. Let $M^n$ be a compact hypersurface with constant mean curvature $H$ in $\mathbb{S}^{n+1}$. Denote by $S$ the squared norm of the second fundamental form of $M$. We prove that there exist two positive constants $γ(n)$ and $δ(n)$ depending only on $n$ such that if $|H|\leqγ(n)$ and $β(n,H)\leq S\leqβ(n,H)+δ(n)$, then $S\equivβ(n,H)$ and $M$ is one of the following cases: (i) $\mathbb{S}^{k}(\sqrt{\frac{k}{…
A Pinching constant for harmonic manifolds
本文将证明调和流形的截面曲率在两边都有界。事实上，我们将给出所有调和流形的一个拼挤常数。我们将利用Z.I.Szabo证明的调和流形的嵌入定理和Hilbert空间中螺旋线的描述来证明这个结果。
We prove an estimate for solutions to the linearized Ricci flow system on closed 3-manifolds. This estimate is a generalization of Hamilton's pinching is preserved estimate for the Ricci curvatures of solutions to the Ricci flow on 3-manifolds with positive Ricci curvature. In our estimate we make no assumption on the curvature of the initial metric. We show that the norm of the solution of the Lichnerowicz Laplacian heat equation (coupled to the Ricci flow) is bounded by a constant (depending on time) times the scalar curvature plus a constant. This relies on a Bochner type formula and establ…
Motivated by a previous work of Zheng and the second named author, we study pinching constants of compact Kähler manifolds with positive holomorphic sectional curvature. In particular we prove a gap theorem following the work of Petersen and Tao on Riemannian manifolds with almost quarter-pinched sectional curvature.
In this paper, we give pinching Theorems for the first nonzero eigenvalue $λ$ of the Laplacian on the compact hypersurfaces of the Euclidean space. Indeed, we prove that if the volume of $M$ is 1 then, for any $ε>0$, there exists a constant $C\_ε$ depending on the dimension $n$ of $M$ and the $L\_{\infty}$-norm of the mean curvature $H$, so that if the $L\_{2p}$-norm $\|H\|\_{2p}$ ($p\geq 2$) of $H$ satisfies $n\|H\|\_{2p}-C\_ε<λ$, then the Hausdorff-distance between $M$ and a round sphere of radius $(n/λ)^{1/2}$ is smaller than $ε$. Furthermore, we prove that if $C$ is a small enough constant…
A variant of Gromov's H{ö}lder-equivalence problem, motivated by a pinching problem in Riemannian geometry, is discussed. A partial result is given. The main tool is a general coarea inequality satisfied by packing energies of maps.
In this note we characterize compact hypersurfaces of dimension $n\geq 2$ with constant mean curvature $H$ immersed in space forms of constant curvature and satisfying an optimal integral pinching condition: they are either totally umbilical or, when $n\geq 3$ and $H\neq 0$, they are locally contained in a rotational hypersurface. In dimension two, the integral pinching condition reduces to a topological assumption and we recover the classical Hopf-Chern result.
In this paper, by using monotonicity formulas for vector bundle-valued $p$-forms satisfying the conservation law, we first obtain general $L^2$ global rigidity theorems for locally conformally flat (LCF) manifolds with constant scalar curvature, under curvature pinching conditions. Secondly, we prove vanishing results for $L^2$ and some non-$L^2$ harmonic $p$-forms on LCF manifolds, by assuming that the underlying manifolds satisfy pointwise or integral curvature conditions. Moreover, by a Theorem of Li-Tam for harmonic functions, we show that the underlying manifold must have only one end. Fi…
In this paper we study collapsing sequences M_{i}-> X of Riemannian manifolds with curvature bounded or bounded away from a controlled subset. We introduce a structure over X which in an appropriate sense is dual to the N-structure of Cheeger, Fukaya and Gromov. As opposed to the N-structure, which live over the M_{i} themselves, this structure lives over X and allows for a convenient notion of global convergence as well as the appropriate background structure for doing analysis on X. This structure is new even in the case of uniformly bounded curvature and as an application we give a generali…
An improved second pinching theorem for minimal hypersurfaces with constant scalar curvature in spheres
本文改进了球面中常数量曲率闭极小子流形的第二拼挤定理，将下界从 $10n/7$ 提升至 $85n/59$，并给出归一化二次缺陷平均值的正下界，无需对三次迹作常值假设。
Let $C$ be a strictly convex domain in a $3$-dimensional Riemannian manifold with sectional curvature bounded above by a constant and let $Σ$ be a constant mean curvature surface with free boundary in $C$. We provide a pinching condition on the length of the traceless second fundamental form on $Σ$ which guarantees that the surface is homeomorphic to either a disk or an annulus. Furthermore, under the same pinching condition, we prove that if $C$ is a geodesic ball of $3$-dimensional space forms, then $Σ$ is either a spherical cap or a Delaunay surface.
Let $M^n$ be a compact hypersurface with constant mean curvature $H$ in $\mathbb{S}^{n+1}$. Denote by $S$ the squared norm of the second fundamental form of $M$. We prove that there exists a positive constant $γ(n)$ depending only on $n$ such that if $|H|\leqγ(n)$ and $β(n,H)\leq S\leqβ(n,H)+\frac{n}{23}$, then $S\equivβ(n,H)$ and $M$ is one of the following cases: (i) $\mathbb{S}^{k}(\sqrt{\frac{k}{n}})\times \mathbb{S}^{n-k}(\sqrt{\frac{n-k}{n}})$, $\,1\le k\le n-1$; (ii) $\mathbb{S}^{1}(\frac{1}{\sqrt{1+μ^2}})\times \mathbb{S}^{n-1}(\fracμ{\sqrt{1+μ^2}})$. Here $β(n,H)=n+\frac{n^3}{2(n-1)}H…
We derive some elliptic differential inequalities from the Weitzenböck formulas for the traceless Ricci tensor of a Kähler manifold with constant scalar curvature and the Bochner tensor of a Kähler-Einstein manifold respectively. Using elliptic estimates and maximum principle, some $L^p$ and $L^\infty $ pinching results are established to characterize Kähler-Einstein manifolds among Kähler manifolds with constant scalar curvature, and others are given to characterize complex space forms among Kähler-Einstein manifolds. Finally, these pinching results may be combined to characterize complex spa…
We prove that some Riemannian manifolds with boundary under an explicit integral pinching are spherical space forms. Precisely, we show that 3-dimensional Riemannian manifolds with totally geodesic boundary, positive scalar curvature and an explicit integral pinching between the $L^2$-norm of their scalar curvature and the $L^2$-norm of their Ricci tensor are spherical space forms with totally geodesic boundary. Moreover, we prove also that 4-dimensional Riemannian manifolds with umbilic boundary, positive Yamabe invariant and an explicit integral pinching between the total integral of their $…
In this paper we consider the uniqueness problem of the constant mean curvature spheres in asymptotically flat 3-manifolds. We require the metric have the form g_{ij}=δ_{ij}+h_{ij} with h_{ij}=O_{4}(r^{-1}) and R=O(r^{-3-τ}),τ>0. We do not require the metric to be close to Schwarzschild metric in any sense or to satisfy RT conditions. We prove that, when the mass is not 0, stable CMC spheres that separate a certain compact part from infinity satisfy the radius pinching estimate r_{1}\leq Cr_{0} , which in many cases is critical to prove the uniqueness of the CMC spheres. As applications of thi…
We prove a curvature pinching result for the Ricci flow on asymptotically flat manifolds: if an asymptotically flat manifold of dimension $n\geq 3$ has scale-invariant integral norm of curvature sufficiently pinched relative to the inverse of its Sobolev constant, then the Ricci flow starting from this manifold exists for all positive times and converges to flat Euclidean space. In particular our result implies that the initial manifold must have been diffeomorphic to $\mathbb{R}^n$.
Let $M^n$ be a closed convex hypersurface lying in a convex ball $B(p,R)$ of the ambient $(n+1)$-manifold $N^{n+1}$. We prove that, by pinching Heintze-Reilly's inequality via sectional curvature upper bound of $B(p,R)$, 1st eigenvalue and mean curvature of $M$, not only $M$ is Hausdorff close and almost isometric to a geodesic sphere $S(p_0,R_0)$ in $N$, but also its enclosed domain is $C^{1,α}$-close to a geodesic ball of constant curvature.
We prove that an n($\geq$ 4)-dimensional compact Bach-flat manifold with positive constant $σ_2$ is an Einstein manifold, provided that its Weyl curvature satisfies a suitable pinching condition.
Ricci pinching on complete Kahler manifolds
本文研究完备凯勒流形的里奇捏缩问题，证明了两类流形的平坦性/里奇平坦性、复维数≥2的特定凯勒流形的紧性，引入里奇形式积分估计等关键要素，完善了相关拓扑里奇质量定理的应用条件。
Some new differentiable sphere theorems are obtained via the Ricci flow and stable currents. We prove that if $M^n$ is a compact manifold whose normalized scalar curvature and sectional curvature satisfy the pointwise pinching condition $R_0>σ_{n}K_{\max}$, where $σ_n\in (\frac{1}{4},1)$ is an explicit positive constant, then $M$ is diffeomorphic to a spherical space form. This gives a partial answer to Yau's conjecture on pinching theorem. Moreover, we prove that if $M^n(n\geq3)$ is a compact manifold whose $(n-2)$-th Ricci curvature and normalized scalar curvature satisfy the pointwise condi…
In this article we exhibit the largest constant in a quadratic isoperimetric inequality which ensures that a geodesic metric space is Gromov hyperbolic. As a particular consequence we obtain that Euclidean space is a borderline case for Gromov hyperbolicity in terms of the isoperimetric function. We prove similar results for the linear filling radius inequality. Our results strengthen and generalize theorems of Gromov, Papasoglu and others.
In this paper, via a new Hardy type inequality, we establish some cohomology vanishing theorems for free boundary compact submanifolds $M^n$ with $n\geq2$ immersed in the Euclidean unit ball $\mathbb{B}^{n+k}$ under one of the pinching conditions $|Φ|^2\leq C$, $|A|^2\leq \widetilde{C}$, or $|Φ|\leq R(p,|H|)$, where $A$ $(Φ)$ is the (traceless) second fundamental form, $H$ is the mean curvature, $C,\widetilde{C}$ are positive constants and $R(p,|H|)$ is a positive function. In particular, we remove the condition on the flatness of the normal bundle, solving the first question, and partially an…
We classify compact conformally flat $n$-dimensional manifolds with constant positive scalar curvature and satisfying an optimal integral pinching condition: they are covered isometrically by either $\mathbb{S}^{n}$ with the round metric, $\mathbb{S}^{1}\times \mathbb{S}^{n-1}$ with the product metric or $\mathbb{S}^{1}\times \mathbb{S}^{n-1}$ with a rotationally symmetric Derdziński metric.
In this paper we study the relationship of hyperbolicity and (Cheeger) isoperimetric inequality in the context of Riemannian manifolds and graphs. We characterize the hyperbolic manifolds and graphs (with bounded local geometry) verifying this isoperimetric inequality, in terms of their Gromov boundary. Furthermore, we characterize the trees with isoperimetric inequality (without any hypothesis). As an application of our results, we obtain the solvability of the Dirichlet problem at infinity for these Riemannian manifolds and graphs, and that the Martin boundary is homeomorphic to the Gromov b…
Inspired by the study of $V$-static manifold about classification, in this article, we apply the recent results obtained by Freitas and Gomes (Compact gradient Einstein-type manifolds with boundary, 2022) to prove the rigidity results for compact gradient Einstein-type manifolds with nonempty boundary and constant scalar curvature under some suitable pinching conditions. As a special case of gradient Einstein-type manifold, we also give a rigidity result of $(m,ρ)$-quasi-Einstein manifold with boundary.
In an appendix to an earlier paper (cf. arXiv:1703.00984) we showed we showed how to construct tunnels of positive scalar curvature and of arbitrarily small length and volume connecting points in a \emph{three dimensional} manifold of \emph{constant sectional curvature}. Here we generalize the construction to arbitrary dimensions and require only positivity of the scalar curvature. This version contains an added section on existence of arbitrarily narrow tunnels of prescribed length.
We study the $C^4$ smooth convex bodies $\mathbb{K}\subset\mathbb{R}^{n+1}$ satisfying $K(x)=u(x)^{1-p}$, where $x\in\mathbb{S}^n$, $K$ is the Gauss curvature of $\partial\mathbb{K}$, $u$ is the support function of $\mathbb{K}$, and $p$ is a constant. In the case of $n=2$, either when $p\in[-1,0]$ or when $p\in(0,1)$ in addition to a pinching condition, we show that $\mathbb{K}$ must be the unit ball. This partially answers a conjecture of Lutwak, Yang, and Zhang about the uniqueness of the $L_p$-Minkowski problem in $\mathbb{R}^3$. Moreover, we give an explicit pinching constant depending onl…
A proof of Gromov's cube inequality on scalar curvature
针对 Gromov 关于高维正标量曲率立方体相对面距离不等式的猜想，本文采用 Dirac 算子方法证明其在所有维数下以最优常数成立，并给出极小曲面方法难以获得的强化版本。
We discuss the behavior of $(λ_{1. p}(M))^{1/p}$ with respect to the Gromov-Hausdorff topology and the variable $p$, where $λ_{1, p}(M)$ is the first positive eigenvalue of the $p$-Laplacian on a compact Riemannian manifold $M$. Applications include new estimates for the first eigenvalues of the $p$-Laplacian on Riemannian manifolds with lower Ricci curvature bounds, and isoperimetric inequalities on Gromov-Hausdorff limit spaces. We also establish a new Lichnerowicz-Obata type theorem.
Stability of the Euclidean 3-ball under L2-curvature pinching
本文证明具有边界的紧致黎曼3-流形在曲率L2范数和边界基本形式差H1/2范数均很小的条件下微分同胚于欧氏球，并给出度量接近的定量最优界。
Complete hypersurfaces in $R^{n+1}$ with constant mean and scalar curvature
本文研究了在R^{n+1}中具有常均曲率和常标量曲率的完备超曲面的刚性问题，提出了在不同维度下的刚性定理和特征化结果。
An Abresch-Rosenberg type tensor and a gap theorem for CMC hypersurfaces in $\mathbb{S}^n\times\mathbb{R}$ and $\mathbb{H}^n\times\mathbb{R}$
该研究构造了S^n×R和H^n×R中CMC超曲面的Abresch-Rosenberg型对称无迹张量，结合积分Simons恒等式得到显式间隙定理，还确定了阈值的二阶大h渐近行为。
Gromov-Hausdorff Convergence of Constant Scalar Curvature Kähler Metrics near the Canonical Class
针对具有半丰富典范丛的紧凯勒流形，本文证明其典范类附近的常数量曲率凯勒度量在格罗莫夫-豪斯多夫拓扑下收敛到对应扭曲凯勒-爱因斯坦度量的完备化，还给出了相关收敛性定理的新证明。
The goal of this paper is to investigate the rigidity of 4-dimensional manifolds involving some pinching curvature conditions. To this end, we make use of the approach of biorthogonal curvature which is weaker than the sectional curvature. Here, we prove a rigidity result for 4-dimensional compact manifolds under a suitable lower bound condition on the minimum of the biorthogonal curvature. From this, we improve the pinching constants considered by some preceding works on a rigidity result for 4-dimensional manifolds.
We consider Gromov-Thurston examples of negatively curved n-manifolds which do not admit metrics of constant sectional curvature. We show that for each n some of the Gromov-Thurston manifolds admit strictly convex real-projective structures.
Spectral almost rigidity on Kähler manifolds with positive Ricci lower bound
本文证明了紧 Kähler 流形上谱几乎刚性的尖锐定理，通过前 n^2+3 个特征值夹逼确定双全纯类型，并给出临界特征值重数的上界与达到条件。
We provide an upper bound for the Gromov width of compact homogeneous Hodge manifolds $(M, ω)$ with $b_2(M)=1$. As an application we obtain an upper bound on the Seshadri constant $ε(L)$ where $L$ is the ample line bundle on $M$ such that $c_1(L)=[\fracωπ]$.
The lower mean curvature bound in Gromov's mean-of-the-mean-curvature conjecture
本文探讨格罗莫夫平均平均曲率猜想中是否需额外假设边界平均曲率下界，证实二维情形无需该假设但三维及以上维数需该假设
Upper bound for the total mean curvature of spin fill-ins
研究旋量填充的总均曲率上界，证明Gromov猜想在特定条件下成立，给出不同边界情况下的常数表达式。
The quantum cohomology algebra of the (full) flag manifold is a fundamental example in quantum cohomology theory, with connections to combinatorics, algebraic geometry, and integrable systems. Using a differential geometric approach, we give an algorithm for computing the multiplicative structure constants of this algebra (the 3-point genus zero Gromov-Witten invariants). The algorithm involves a Grobner basis calculation and the solution by quadrature of a system of differential equations. In particular we obtain quantum Schubert polynomials in a natural fashion.
A remark on positively curved manifolds of dimensions 7 and 13
构造了无穷多个闭7-流形到13维正曲闭黎曼流形的全测地线嵌入。讨论了Pinching常数的计算问题和其它全测地线嵌入的存在性问题。
Effective stability of negatively curved Einstein metrics in dimensions at least $4$
该研究证明了至少四维闭合流形在满足一定条件时存在真正的负截面曲率爱因斯坦度量，并且 pinching 常数不受直径、体积或注入半径的限制。
In this paper we prove the existence of isoperimetric regions of any volume in Riemannian manifolds with Ricci bounded below assuming Gromov--Hausdorff asymptoticity to the suitable simply connected model of constant sectional curvature. The previous result is a consequence of a general structure theorem for perimeter-minimizing sequences of sets of fixed volume on noncollapsed Riemannian manifolds with a lower bound on the Ricci curvature. We show that, without assuming any further hypotheses on the asymptotic geometry, all the mass and the perimeter lost at infinity, if any, are recovered by…
Let $M^n$ be a closed Riemannian manifold. Larry Guth proved that there exists $c(n)$ with the following property: if for some $r>0$ the volume of each metric ball of radius $r$ is less than $({r\over c(n)})^n$, then there exists a continuous map from $M^n$ to a $(n-1)$-dimensional simplicial complex such that the inverse image of each point can be covered by a metric ball of radius $r$ in $M^n$. It was previously proven by Gromov that this result implies two by now famous Gromov's inequalities: $Fill Rad(M^n)\leq c(n)vol(M^n)^{1\over n}$ and, if $M^n$ is essential, then also $sys_1(M^n)\leq 6…
The measure concentration property of an mm-space $X$ is roughly described as that any 1-Lipschitz map on $X$ to a metric space $Y$ is almost close to a constant map. The target space $Y$ is called the screen. The case of $Y=\mathbb{R}$ is widely studied in many literature (see \cite{gromov}, \cite{ledoux}, \cite{mil2}, \cite{milsch}, \cite{sch}, \cite{tal}, \cite{tal2} and its reference). M. Gromov developed the theory of measure concentration in the case where the screen $Y$ is not necessarily $\mathbb{R}$ (cf. \cite{gromovcat}, {gromov2}, \cite{gromov}). In this paper, we consider the case …
We introduce the concept of pseudo symplectic capacities which is a mild generalization of that of symplectic capacities. As a generalization of the Hofer-Zehnder capacity we construct a Hofer-Zehnder type pseudo symplectic capacity and estimate it in terms of Gromov-Witten invariants. The (pseudo) symplectic capacities of Grassmannians and some product symplectic manifolds are computed. As applications we first derive some general nonsqueezing theorems that generalize and unite many previous versions, then prove the Weinstein conjecture for cotangent bundles over a large class of symplectic u…
We apply Gromov's ham sandwich method to get (1) domain monotonicity (up to a multiplicative constant factor); (2) reverse domain monotonicity (up to a multiplicative constant factor); and (3) universal inequalities for Neumann eigenvalues of the Laplacian on bounded convex domains in a Euclidean space.
We show that the Cheeger constant of compact surfaces is bounded by a function of the area. We apply this to isoperimetric profiles of bounded genus non-compact surfaces, to show that if their isoperimetric profile grows faster than $\sqrt t$, then it grows at least as fast as a linear function. This generalizes a result of Gromov for simply connected surfaces. We study the isoperimetric problem in dimension 3. We show that if the filling volume function in dimension 2 is Euclidean, while in dimension 3 is sub-Euclidean and there is a $g$ such that minimizers in dimension 3 have genus at most …
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。