Local Volume Estimate for Manifolds with $L^2$-bounded Curvature
Ye Li
我们得到了一个局部体积增长的完整的，非紧的黎曼流形与小的积分界和巴赫张量有有限的$L^2$范数在4维。
We obtain a local volume growth for complete, noncompact Riemannian manifolds with small integral bounds and with Bach tensor having finite $L^2$ norm in dimension 4.
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We are concerned with the study of different notions of curvature on graphs. We show that if a graph has stronger inner-outer curvature growth than a model graph, then it has faster volume growth too. We also study the relationhips of volume growth with other kind of curvatures, such as the Ollivier-Ricci curvature.
We give several Bishop-Gromov relative volume comparisons with integral Ricci curvature which improve the results in \cite{PW1}. Using one of these volume comparisons, we derive an estimate for the volume entropy in terms of integral Ricci curvature which substantially improves an earlier estimate in \cite{Au2} and give an application on the algebraic entropy of its fundamental group. We also extend the almost minimal volume rigidity of \cite{BBCG} to integral Ricci curvature.
In this note, we prove that for a complete noncompact three dimensional Riemannian manifold with bounded sectional curvature, if it has uniformly positive scalar curvature, then there is a uniform lower bound on the injectivity radius.
On Ricci curvature and volume growth in dimension three
证明了R^3上的完备度量减去一个具有非负里奇曲率和二次Ricci曲率衰减的球，具有立方体积增长。
We study the provenance of singularity formation under mean curvature flow and volume preserving mean curvature flow in an axially symmetric setting. We prove that if the mean curvature is uniformly bounded on any finite time interval, then no singularities can develop during that time under both mean curvature flow and volume preserving mean curvature flow.
Bounded curvature manifolds without Euclidean isometric immersions of bounded mean curvature
本文构造了任意维数下具有有界曲率和单射半径的完备黎曼度量，证明其不存在有界平均曲率的欧几里得等距浸入，从而否定了Yau问题52。
The prescribed scalar curvature problem for metrics with unit total volume
我们解决了修改后的卡兹丹-华纳问题，找到规定的数量曲率和单位总体积的度量。
Short-time persistence of bounded curvature under the Ricci flow
我们使用一阶能量量来证明里奇流的唯一性的加强声明。这个陈述的一个结果是，如果一个非紧流形上的完全解具有一致有界的里奇曲率，那么它的截面曲率将在一个短时间内保持有界，如果它最初是有界的。换句话说，如果里奇曲率保持有界，则里奇流的完全解的外尔曲率张量不可能瞬间变为无界。
Open manifolds with asymptotically nonnegative Ricci curvature and large volume growth
本文研究了具有渐近非负里奇曲率和大体积增长的完备非紧黎曼流形的拓扑。证明了在一定的曲率衰减和体积增长条件下，它们具有有限的拓扑型。利用Abresch-Gromoll的超额函数估计的推广，我们将其推广到具有$k$ -次渐近非负里奇曲率的流形上.
Bounds on the volume entropy and simplicial volume in Ricci curvature $L^p$ bounded from below
设$(M,g)$是一个具有里奇曲率几乎有界的紧致流形，$π:\bar{M}\to M$是一个正规的黎曼覆盖.我们表明，对于任何非负函数$f$上的$M$，平均值$føπ$上的测地球的$\bar{M}$是可比的平均值$f$上的$M$。结合对数体积估计，这意味着在里奇曲率$L^p$的几个拓扑不变量（体积熵，单纯体积，第一贝蒂数和介绍的基本组）的界限-有界从下面。
Remark on the Calabi flow with bounded curvature
本文证明了：若Calabi流的曲率张量沿Calabi流一致有界，且Mabuchi能量适当，则Calabi流收敛于一个常数的数量曲率度量.
Scalar curvature rigidity with a volume constraint
受Brendle-Marques-Neves对Min-Oo猜想的反例的启发，我们证明了一个适用于半球的体积约束数量曲率刚性定理.
Volume of minimal hypersurfaces in manifolds with nonnegative Ricci curvature
建立了具有非负里奇曲率的闭黎曼流形的体积宽度的极小极大估计。更精确地说，我们证明了每一个封闭的黎曼流形的非负里奇曲率承认PL莫尔斯函数的水平集体积有界的体积的流形。作为这种扫出估计的结果，存在一个嵌入的，封闭的（可能是奇异的）极小超曲面，其体积是有界的流形的体积。
Lipschitz-Volume rigidity on limit spaces with Ricci curvature bounded from below
本文证明了具有里奇曲率的流形的非塌陷Gromov-Hausdorff极限的Lipschitz-体积刚性定理。这是亚历山德罗夫几何中Lipschitz体积刚性的对应。
Volume growth and positive scalar curvature
本文研究三维完备非紧黎曼流形在非负里奇曲率和一致正数量曲率条件下的体积增长，证明了尖锐线性增长比率并给出刚性刻画。
On Almost Isometry Theorem in Alexandrov Spaces with Curvature Bounded Below
本文给出了曲率有界的Alexandrov空间中一个几乎等距定理的新证明。
Curvature estimates for surfaces with bounded mean curvature
第第二基本型$|A|$的范数估计在曲面几何的研究中起着至关重要的作用。事实上，当$|A|$有界时，曲面不能弯曲得太厉害。本文证明了对于$|A|$的$L^2$范数有界的嵌入测地圆盘，如果其中曲率的$W^{1,p}$范数足够小，则$|A|$内点有界，即$p>2$ .在这样做，我们推广了一些著名的估计$|A|$的极小曲面。
Curvature estimates for properly immersed $ϕ_{h}$-bounded submanifolds
Jorge-Koutrofiotis和Pigola-Rigoli-Setti证明了外有界子流形的锐截面曲率估计。Bessa和Montenegro证明了这些估计在适当浸入的圆柱有界子流形上成立。另一方面，Bessa和Dajczer证明了适当浸入圆柱有界子流形的精确中曲率估计.在本文中，我们证明了这些截面和中曲率估计的一个更大的类的子流形，适当浸入$ϕ$ -有界子流形。
Integral Ricci curvature bounds for possibly collapsed spaces with Ricci curvature bounded from below
假设一个完整的黎曼流形的里奇曲率的下界，$p< 1/2$，我们显示存在的局部$L^p$规范的里奇曲率，只依赖于尺寸和提高体积塌陷的界限。
A volume stability theorem on toric manifolds with positive Ricci curvature
本文证明了一个体积稳定性定理：如果n维复曲面流形$M$有一个不变量Kähler度量$\mathbb{T}^n$且其里奇曲率不小于1，且其体积接近于$\mathbb{CP}^n$的体积，则$M$是$\mathbb{CP}^n$的双全纯.
Manifolds with Quadratic Curvature Decay and Slow Volume Growth
我们证明了对于非紧流形，存在拓扑障碍，使得它可以接受一个曲率二次衰减的黎曼度量和一个比同维数的欧几里得空间慢的体积增长.
Weakly convex closed subsets of spaces with bounded nonpositive curvature
在具有有界非正曲率的黎曼度量的R^n上，弱凸闭子集是拓扑平凡的。这种子集在交下的稳定性刻画了欧几里得空间。
Positive Scalar Curvature and Volume Growth
针对非负里奇曲率的完备黎曼流形，通过证明两个精确体积增长阶估计，解决了格罗莫夫1986年提出的余维二体积增长猜想。
Manifolds with quadratic curvature decay and fast volume growth
给出了球面空间型上曲率二次衰减的非紧黎曼流形具有端点为锥的有限拓扑型的充分条件.
We give a new proof of a recent result of Munteanu--Wang relating scalar curvature to volume growth on a $3$-manifold with non-negative Ricci curvature. Our proof relies on the theory of $μ$-bubbles introduced by Gromov as well as the almost splitting theorem due to Cheeger--Colding.
Entire bounded constant mean curvature Killing graphs
本文证明了在一定曲率条件下，位于板内的具有常中曲率的整Killing图必是全测地切片。
On compactness of Kähler metrics with bounded entropy and bounded $L^{2n+1}$ scalar curvature
本文针对Chen和Cheng关于紧Kähler流形上Kähler度量势的估计结果，研究放宽标量曲率有界性条件的问题，核心是将标量曲率的有界假设弱化为$L^{2n+1}$有界，推进了Kähler度量紧性相关的先验估计研究。
Volume preserving curvature flows in Lorentzian manifolds
设N是具有紧Cauchy超曲面的（n+1）维全局双曲Lorentz流形.考虑N中具有不同曲率函数F（包括中曲率、高斯曲率和第二初等对称多项式）和一个保体积项的曲率流.在适当的假设下，我们证明了流的长时间存在性和相应的图在$C^\infty$ -拓扑中指数收敛到一个常F-曲率超曲面.此外，我们还研究了常F-曲率超曲面的稳定性和叶理。
Biharmonic submanifolds in manifolds with bounded curvature
考虑截面曲率由非负常数$c$上界的黎曼流形中的完备双调和子流形$ϕ:(M,g)\rightarrow (N,h)$.假设中曲率由$\sqrt c$从下到上限定。如果（i）对于某些$0<p<\infty$，$\int_M (|{\bf H}|^2-c)^{p}dv_g<\infty$，或（ii）$M$的里奇曲率从下有界，则中曲率是$\sqrt c$。此外，如果$M$是紧的，那么我们得到相同的结果没有假设（i）或（ii）。
We prove that Berwald spaces whose flag curvature is nowhere vanishing are in fact Riemannian spaces. This means that any Berwald space with flag curvature bounded below by a positive number must be also Riemannian. This rigidity result shows the importance of non-Riemannian examples when imposing flag curvature bounds on Finsler spaces.
Bray's football theorem (\cite{bray2009penrose}) is a weakening of Bishop theorem in dimension 3. It gives a sharp volume upper bound for a three dimensional manifold with scalar curvature larger than $n(n-1)$ and Ricci curvature larger than $\varepsilon$. This paper extends Bray's football theorem in high dimensions, assuming the manifold is axis symmetric or the Ricci curvature has an upper bound.
Remark about the spectrum of the p-form Laplacian under a collapse with curvature bounded below
给出了曲率有界的p-型Laplacian在某类坍缩中小的正特征值个数的一个下界.
Volume Growth and Curvature Decay of Positively Curved Kähler manifolds
本文得到了关于无穷远处完备非紧正曲Kähler流形几何的三个结果。第一个定理指出，具有正对分曲率的完备非紧Kähler流形的体积增长阶至少是实维数的一半（即，复杂的维度）。第二个定理指出具有正对分曲率的完备非紧Kähler流形的曲率至少在平均意义下线性衰减。第三个结果是关于体积增长和曲率衰减之间的关系。证明了具有非负曲率算子的完备非紧Kähler流形的曲率衰减在一定平均意义下是二次的.
Singularities of mean curvature flow with bounded mean curvature and Morse index
研究具有有界平均曲率和莫尔斯指数的平均曲率流奇点多重性，针对3≤n≤6时\(\mathbb{R}^{n + 1}\)中闭光滑嵌入流，证明首个奇异时间平均曲率或莫尔斯指数会爆炸。
In this note we provide several lower bounds for the volume of a geodesic ball within the injectivity radius in a $3$-dimensional Riemannian manifold assuming only upper bounds for the Ricci curvature.
On the boundedness of the sectional curvature of almost Hermitian manifolds
证明了如下结果：不定度量的几乎Hermitian流形具有逐点常全纯截面曲率，如果全纯截面曲率上下有界.如果反全纯截面曲率从上或从下有界，则流形具有逐点常反全纯截面曲率。对于有定度量的几乎厄米流形也得到了类似的结果。
In this paper, we prove a classification for complete embedded constant weighted mean curvature hypersurfaces $Σ\subset\mathbb{R}^{n+1}$. We characterize the hyperplanes and generalized round cylinders by using an intrinsic property on the norm of the second fundamental form. Furthermore, we prove an equivalence of properness, finite weighted volume and exponential volume growth for submanifolds with weighted mean curvature of at most linear growth.
This paper introduces and investigates a generalization of the notion of a pointed Riemannian manifold having its radial curvature bounded from above by that of a model surface of revolution.
Optimal volume bound and volume growth for Ricci-nonnegative manifolds with positive Bi-Ricci curvature
本文在维数小于八的条件下，证明了具有非负Ricci曲率且紧集外双Ricci曲率下界为n-2的完备黎曼流形的最优体积增长，回答了Antonelli-Xu的问题，并给出正双Ricci曲率下Gromov体积界猜想的类比。
Higher order mean curvature estimates for bounded complete hypersurfaces
得到了浸入完备黎曼流形中的完备有界超曲面的高阶中曲率的精确估计。对于Lorentz环境空间中的完备类空超曲面也给出了类似的结果。
Smoothing Riemannian Metrics with Bounded Ricci Curvatures in Dimension Four, II
本文是作者论文\cite{Li}的续篇。证明了如果4-流形的度量$g$具有有界的里奇曲率且曲率处处无局部集中，则它可以被光滑化为具有有界截面曲率的度量.这里我们不假设$g$的局部Sobolev常数的界，因此该平滑结果可以应用于塌陷情况。
We prove a parametrized compactness theorem on manifolds of bounded Ricci curvature, upper bounded diameter and lower bounded injectivity radius.
Complete quasi-Yamabe gradient solitons with bounded scalar curvature
该研究依据标量曲率相对于孤立子常数的上下界条件，分别分类了完备非平凡的收缩、稳定及膨胀拟 Yamabe 梯度孤立子。
A mean curvature estimate for cylindrically bounded submanifolds
本文将文献[1]中关于乘积流形中以欧几里得空间为因子的圆柱有界真子流形的中曲率的估计推广到具有翘曲乘积结构的一般乘积环境空间.
Volume-preserving mean curvature flow for tubes in symmetric spaces
本文研究了对称空间中反射子流形的紧致闭域上的（非定半径）管道的保体积中曲率流。我们证明了在一定的条件下，沿着流动的tubeness被保存。
A counterexample to Yau's bounded-mean-curvature embedding question
针对Yau关于有界Ricci曲率与正单射半径流形是否存在有界平均曲率等距嵌入的问题，构造了四维及以上维度的反例，证明此类嵌入不存在。
In the paper two important theorems about complete affine spheres are generalized to the case of statistical structures on abstract manifolds. The assumption about constant sectional curvature is replaced by the assumption that the curvature satisfies some inequalities.
It is still an open problem that a complete open Kahler manifold with positive bisectional curvature is Stein. This paper partially resolve the problem by putting a restriction to volume growth condition. The partial solution here improves the observation in ([8], page 341). The improvement is based on assuming a weaker volume growth condition that is not sufficiently maximal.
We study a volume preserving curvature flow of convex hypersurfaces, driven by a power of the $k$-th elementary symmetric polynomial in the principal curvatures. Unlike most of the previous works on related problems, we do not require assumptions on the curvature pinching of the initial datum. We prove that the solution exists for all times and that the speed remains bounded and converges to a constant in an integral norm. In the case of the volume preserving scalar curvature flow, we can prove that the hypersurfaces converge smoothly and exponentially fast to a round sphere.
In this paper, we prove that a critical metric of the volume functional with pinched Weyl curvature is isometric to a geodesic ball in $\mathbb{S}^{n}.$ Moreover, we provide a necessary and sufficient condition on the norm of the gradient of the potential function in order to classify such critical metrics.
For Riemannian manifolds with a smooth measure $(M, g, e^{-f}dv_{g})$, we prove a generalized Myers compactness theorem when Bakry--Emery Ricci tensor is bounded from below and $f$ is bounded.
We consider the graphical mean curvature flow of strictly area decreasing maps $f:M\to N$, where $M$ is a compact Riemannian manifold of dimension $m>1$ and $N$ a complete Riemannian surface of bounded geometry. We prove long-time existence of the flow and that the strictly area decreasing property is preserved, when the bi-Ricci curvature $BRic_M$ of $M$ is bounded from below by the sectional curvature $σ_N$ of $N$. In addition, we obtain smooth convergence to a minimal map if $Ric_M\ge\sup\{0,{\sup}_Nσ_N\}$. These results significantly improve known results on the graphical mean curvature fl…
We prove that complete Riemannian manifolds with polynomial growth and Ricci curvature bounded from below, admit uniform Poincaré inequalities. A global, uniform Poincaré inequality for horospheres in the universal cover of a closed, $n$-dimensional Riemannian manifold with pinched negative sectional curvature follows as a corollary.
We obtain estimates on nonlocal quantities appearing in the Volume Preserving Mean Curvature Flow (VPMCF) in the closed, Euclidean setting. As a result we demonstrate that blowups of finite time singularities of VPMCF are ancient solutions to Mean Curvature Flow (MCF), prove that monotonicity methods may always be applied at finite times and obtain information on the asymptotics of the flow.
Inversion of some curvature operators near a parallel Ricci metric II: Non-compact manifold with bounded geometry
设（M，g）是具有有界几何和平行里奇曲率的完备非紧黎曼流形.我们证明了在某些经典的索伯列夫空间中，在度量g附近，某些相对于里奇曲率“仿射”的算子是局部可逆的.
Biharmonic hypersurfaces with bounded mean curvature
本文考虑球面上中曲率处处为零的向量场$ϕ:(M^m,g)\rightarrow (S^{m+1},h)$的完备双调和超曲面。如果第第二基本型$B$的平方范数由m和$\int_M H^{- p }dv_g<\infty$从上而下有界，对于某个$0<p<\infty$，则中曲率是常数。
Mixed type surfaces with bounded mean curvature in 3-dimensional space-times
本文证明了实解析洛伦兹三维流形中具有有界中曲率函数的类空曲面只有当其中曲率函数趋于零时，才能将其因果关系转化为类时曲面.此外，我们还将证明具有非零中曲率的曲面的存在性，并研究它们的性质。
Volume estimates for the singular sets of mean curvature flows
本文针对从$\boldsymbol{\rm R}^{n+1}$中光滑闭平均凸超曲面出发的平均曲率流，建立了其奇异集及定量奇异层的一致且精确的体积估计。
Curvature-free effects from volume growth and ends-counting and their applications
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Volume comparison of conformally compact manifolds with scalar curvature $R\geq -n\left(n-1\right)$
本文利用规范化Ricci-DeTurk流证明了严格稳定共形紧Einstein流形的一个稳定性结果。作为应用，我们给出了具有数量曲率$R\geq -n\left(n-1\right)$的共形紧致流形的局部体积比较，以及当重整化体积为零时的刚性结果。
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