Alexander Mramor
In this article, we use the recently developed mean curvature flow with surgery for 2 convex hypersurfaces to prove several isotopy existence and finally extrinsic finiteness results (in the spirit of Cheeger's compactness theorem) for the space of 2 convex hypersurfaces in $\mathbb{R}^{n+1}$.
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K-area is an invariant for Riemannian manifolds introduced by Gromov as an obstruction to the existence of positive scalar curvature. However in general it is difficult to determine whether K-area is finite or not. though the definition of K-area is quite natural. In this paper, we study how the invariant changes under surgery.
In this article, we extend the mean curvature flow with surgery to mean convex hypersurfaces with entropy less than $Λ_{n-2}$. In particular, 2-convexity is not assumed. Next we show the surgery flow with just the initial convexity assumption $H - \frac{\langle x, ν\rangle}{2} > 0$ is possible and as an application we use the surgery flow to show that smooth $n$-dimensional closed self shrinkers with entropy less than $Λ_{n-2}$ are isotopic to the round $n$-sphere.
In this paper, we introduce a parameterized discrete curvature ($α$-curvature) for piecewise linear metrics on polyhedral surfaces, which is a generalization of the classical discrete curvature. A discrete uniformization theorem is established for the parameterized discrete curvature, which generalizes the discrete uniformization theorem obtained by Gu-Luo-Sun-Wu. We also prove the global rigidity of parameterized discrete curvature with respect to the discrete conformal factors, which confirms a generalized Luo conjecture on rigidity of discrete curvatures. We further introduce a parameterize…
Surgery and positive Bakry-Émery Ricci curvature
本文研究手术中保持正Bakry-Émery Ricci曲率加权黎曼度量的问题，建立连通和与高维球面手术的局部性定理，应用证明闭单连通自旋5-流形存在该度量，并提供正Ricci曲率流形新例子。
We prove a monotonicity formula for mean curvature flow with surgery. This formula differs from Huisken's monotonicity formula by an extra term involving the mean curvature. As a consequence, we show that a surgically modified flow which is sufficiently close to a smooth flow in the sense of geometric measure theory is, in fact, free of surgeries. This result has applications to the longtime behavior of mean curvature flow with surgery in Riemannian three-manifolds.
We define a notion of mean curvature flow with surgery for two-dimensional surfaces in $\mathbb{R}^3$ with positive mean curvature. Our construction relies on the earlier work of Huisken and Sinestrari in the higher dimensional case. One of the main ingredients in the proof is a new estimate for the inscribed radius established by the first author.
Huisken and Sinestrari have recently defined a surgery process for mean curvature flow when the initial data is a two-convex hypersurface. The process depends on a parameter H. Its role is to initiate a surgery when the maximum of the mean curvature of the evolving hypersurface becomes H, and to control the scale at which each surgery is performed. We prove that as H goes to infinity the surgery process converges to level set flow.
We give a simple criterion for a pointwise curvature condition to be stable under surgery. Namely, a curvature condition $C$, which is understood to be an open, convex, O(n)-invariant cone in the space of algebraic curvature operators, is stable under surgeries of codimension at least $c$ provided it contains the curvature operator corresponding to $S^{c-1} \times \reals^{n-c+1}$, $c \geq 3$. This is used to generalize the well-known classification result of positive scalar curvature in the simply-connected case in the following way: Any simply-connected manifold $M^n$, $n \geq 5$, which is ei…
Generalized Surgery on Riemannian Manifolds of Positive Ricci Curvature
本研究推广Wraith的正Ricci曲率手术定理，用带core metric的球面丛粘合替代球面与圆盘乘积的附加操作，在底空间容许core metric的2维球面丛上构造出core metric，得到新的6维正Ricci曲率流形例子。
We consider a regular smooth curve in $\mathbb{E}^n$ such that its coordinates' components are the fundamental solutions of the differential equation $ y^{(n)} (x) - y(x) = 0 ,$ $x \in \mathbb{R} $ of order $n$. We show that the total first curvature of this curve is infinite for odd $n$ and is finite for even $n$.
Combinatorial curvature flows with surgery for inversive distance circle packings on surfaces
本文为欧几里得反演距离圆填充引入了组合Calabi流等曲率流，通过加权Delaunay条件下的边翻转手术处理奇点，并利用离散共形理论证明了其长时间存在性与全局收敛性，为寻找具有预定组合曲率的分段欧几里得度量提供了有效算法。
In this article we use the mean curvature flow with surgery to derive regularity estimates for the level set flow going past Brakke regularity in certain special conditions allowing for 2-convex regions of high density. We also show a stability result for the plane under the level set flow.
We give a new proof for the existence of mean curvature flow with surgery of 2-convex hypersurfaces in $R^N$, as announced in arXiv:1304.0926. Our proof works for all $N \geq 3$, including mean convex surfaces in $R^3$. We also derive a priori estimates for a more general class of flows in a local and flexible setting.
In this paper, we give a finiteness result on the diffeomorphism types of curvature-adapted equifocal hypersurfaces in a simply connected compact symmetric space. Furthermore, the condition curvature-adapted can be dropped if the symmetric space is of rank one.
We construct smooth mean curvature flows with surgery that approximate weak mean curvature flows with only spherical and neck-pinch singularities. This is achieved by combining the recent work of Choi-Haslhofer-Hershkovits, and Choi-Haslhofer-Hershkovits-White, establishing canonical neighbourhoods of such singularities, with suitable barriers to flows with surgery. A limiting argument is then used to control these approximating flows. We conclude by improving the entropy bound on the low-entropy Schoenflies conjecture.
A finiteness principle for distance functions on Riemannian surfaces with Hölder continuous curvature
本文研究 Hölder 连续 Gauss 曲率 Riemann 曲面上测地线到点的距离函数，证明类似 Whitney 延拓理论的有限性原理，并将其用于带曲率半范数控制的等距嵌入。
We utilize a condition for algebraic curvature operators called surgery stability as suggested by the work of S. Hoelzel to investigate the space of riemannian metrics over closed manifolds satisfying these conditions. Our main result is a parametrized Gromov-Lawson construction with not necessarily trivial normal bundles and shows that the homotopy type of this space of metrics is invariant under surgeries of a suitable codimension. This is a generalization of a well-known theorem by V. Chernysh and M. Walsh for metrics of positive scalar curvature. As an application of our method, we show th…
Positive $τ$-bi-Ricci curvature and Mean curvature flow with surgery in hyperbolic space
该研究证明双曲空间中满足正$τ$-双Ricci曲率的光滑闭连通浸入超曲面存在带有限次手术且可终止的平均曲率流，并推导了相应流形的微分同胚类型与紧区域的拓扑性质。
Ricci Flow with Surgery on Four-manifolds with Positive Isotropic Curvature
本文研究了具有正迷向曲率且不具有本质不可压缩空间型的紧致四流形上的里奇流。我们的目的是双重的。一是给出了具有正迷向曲率且不具有本质不可压缩空间型的四维流形上的Hamilton分类定理的完整证明;二是将Perelman关于三维里奇流的一些最新结果推广到四维流形上。在证明我们实际上已经提供了，到轻微的修改，所有必要的细节部分从第1节至第5节佩雷尔曼的第二篇论文的里奇流。
Finiteness of totally geodesic hypersurfaces
本文研究闭负曲率解析黎曼流形中全测地超曲面的数量问题，证明其有限性，除非流形为常曲率（此时等距于算术双曲流形），揭示了全测地超曲面数量与流形曲率性质的深刻关联。
Surgery and total mean curvature
本文通过手术方法证明了Gromov关于填充物总平均曲率的猜想，结合正质量定理和新方法，解决了旋量和非旋量填充物的总平均曲率问题。
This paper investigates the combinatorial $α$-curvature for vertex scaling of piecewise hyperbolic metrics on polyhedral surfaces, which is a parameterized generalization of the classical combinatorial curvature. A discrete uniformization theorem for combinatorial $α$-curvature is established, which generalizes Gu-Guo-Luo-Sun-Wu's discrete uniformization theorem for classical combinatorial curvature. We further introduce combinatorial $α$-Yamabe flow and combinatorial $α$-Calabi flow for vertex scaling to find piecewise hyperbolic metrics with prescribed combinatorial $α$-curvatures. To handle…
High Codimension Mean Curvature Flow with Surgery
本文提出了一种高余维数平均曲率流的手术方法，通过二次2-凸性条件实现对奇点的拓扑控制，并获得了子流形的精确分类。
Surgery and equivariant Yamabe invariant
本文考虑了等变Yamabe问题，即紧李群G的G-不变度量空间上的Yamabe问题。G-Yamabe不变量被类似地定义为单位体积G-不变度量的常数数量曲率的上确界，该上确界使它们的G-不变共形子类中的总数量曲率泛函最小化。我们证明了一个关于G-Yamabe不变量在余维为3或更大的情况下如何变化的公式，并计算了一些G-Yamabe不变量。
We study mean curvature flow in $\mathbb S_K^{n+1}$, the round sphere of sectional curvature $K>0$, under the quadratic curvature pinching condition $|A|^{2} < \frac{1}{n-2} H^{2} + 4 K$ when $n\ge 4$ and $|A|^{2} < \frac{3}{5}H^{2}+\frac{8}{3}K$ when $n=3$. This condition is related to a famous theorem of Simons, which states that the only minimal hypersurfaces satisfying $\vert A\vert^2<nK$ are the totally geodesic hyperspheres. It is related to but distinct from two-convexity. Notably, in contrast to two-convexity, it allows the mean curvature to change sign. We show that the pinching condi…
Mean convex flows with surgery
本文构造了从任意紧致平均凸超曲面出发的带手术平均曲率流，通过非退化柱状奇点由流自身执行拓扑手术，并需在有限个光滑时刻微调。
In this article, we give a complete and self--contained account of Chernysh's strengthening of the Gromov--Lawson surgery theorem for metrics of positive scalar curvature. No claim of originality is made.
Singular sets on spaces with integral curvature bounds and diffeomorphism finiteness for manifolds
本文推广并改进了 Cheeger 及 Cheeger-Naber 对具有积分曲率界的非塌缩黎曼流形奇异集的维数估计，并证明了临界情形下流形的微分同胚有限性定理。
We present a new curvature condition which is preserved by the Ricci flow in higher dimensions. For initial metrics satisfying this condition, we establish a higher dimensional version of Hamilton's neck-like curvature pinching estimate. Using this estimate, we are able to prove a version of Perelman's Canonical Neighborhood Theorem in higher dimensions. This makes it possible to extend the flow beyond singularities by a surgery procedure in the spirit of Hamilton and Perelman. As a corollary, we obtain a classification of all diffeomorphism types of such manifolds in terms of a connected sum …
In this paper, we establish compactness for various geometric curvature energies including integral Menger curvature, and tangent-point repulsive potentials, defined a priori on the class of compact, embedded $m$-dimensional Lipschitz submanifolds in ${\mathbb{R}}^n$. It turns out that due to a smoothing effect any sequence of submanifolds with uniformly bounded energy contains a subsequence converging in $C^1$ to a limit submanifold. This result has two applications. The first one is an isotopy finiteness theorem: there are only finitely many isotopy types of such submanifolds below a given e…
We point out that Ecker's local integral quantity agrees with Huisken's global integral quantity at infinity for ancient mean curvature flows if Huisken's one is finite on each time-slice. In particular, this means that the finiteness of Ecker's integral quantity at infinity implies the finiteness of the entropy at infinity.
In this paper, we will show vanishing theorem of $p$ harmonic $1$ form on submanifold $M$ in $ \bar{M} $ whose BiRic curvature satisfying $ \overline{\mathrm{BiRic}}^a \geq Φ_a(H,S) $. As an corollary, we can get the corresponding theorem for $ p $ harmonic function and $ p $ harmonic map. We also investigate the finiteness problem of $p$ harmonic $1$ form on submanifold $M$ in $ \bar{M} $ whose BiRic curvature satisfying $ \overline{\mathrm{BiRic}}^a \geq -k^2 $.
A discrete uniformization theorem for decorated piecewise hyperbolic metrics on surfaces
本文研究了曲面上高斯曲率的自然离散化，建立了离散单值化定理，并引入带手术的组合$\alpha$-Ricci流和组合$\alpha$-Calabi流，证明了其长期存在性与收敛性，从而解决了预定组合$\alpha$-曲率的带装饰分片双曲度量的存在性问题。
We study mean curvature flow of $n$-dimensional submanifolds of $S_K^{n+\ell}$, the round $(n+\ell)$-sphere of sectional curvature $K>0$, under the quadratic curvature pinching condition $|A|^{2} < \frac{1}{n-2}|H|^{2} + 4K$ when $n\geq 8$, $|A|^{2} < \frac{4}{3n}|H|^{2}+\frac{n}{2}K$ when $n=7$, and $|A|^2<\frac{3(n+1)}{2n(n+2)}|H|^2+\frac{2n(n-1)}{3(n+1)}K$ when $n=5$ or $6$. This condition is related to a theorem of Li and Li [Arch. Math., 58:582--594, 1992] which states that the only $n$-dimensional minimal submanifolds of $S_K^{n+\ell}$ satisfying $|A|^2<\frac{2n}{3}K$ are the totally geo…
We study the Ricci flow for initial metrics with positive isotropic curvature (strictly PIC for short). In the first part of this paper, we prove new curvature pinching estimates which ensure that blow-up limits are uniformly PIC in all dimensions. Moreover, in dimension $n \geq 12$, we show that blow-up limits are weakly PIC2. This can be viewed as a higher-dimensional version of the fundamental Hamilton-Ivey pinching estimate in dimension $3$. In the second part, we develop a theory of ancient solutions which have bounded curvature; are $κ$-noncollapsed; are weakly PIC2; and are uniformly PI…
Some applications of Ricci flow to 3-manifolds
我们描述并公布了一些结果（与G.贝松湖贝西埃湾Boileau和J.Porti）关于3-流形的几何和拓扑。大部分的文章主要是作为一个非专家介绍几何化的3-流形，里奇流与手术，和单纯体积的方法崩溃定理。在最后一节中，我们讨论了开3-流形上的带正数量曲率的障碍的带外科手术的里奇流。
The class of Riemannian orbifolds of dimension n defined by a lower bound on the sectional curvature and the volume and an upper bound on the diameter has only finitely many members up to orbifold homeomorphism. Furthermore, any class of isospectral Riemannian orbifolds with a lower bound on the sectional curvature is finite up to orbifold homeomorphism.
The $π_2$-diffeomorphism finiteness result (\cite{FR1,2}, \cite{PT}) asserts that the diffeomorphic types of compact $n$-manifolds $M$ with vanishing first and second homotopy groups can be bounded above in terms of $n$, and upper bounds on the absolute value of sectional curvature and diameter of $M$. In this paper, we will generalize this $π_2$-diffeomorphism finiteness by removing the condition that $π_1(M)=0$ and asserting the diffeomorphism finiteness on the Riemannian universal cover of $M$.
Positive sectional curvature on a family of exotic fifteen-sphere
本文在至少508个两两非微分同胚的奇异十五维球面上构造严格正截面曲率度量，通过四元数线丛和手术方法，并利用Eells--Kuiper不变量区分光滑型，满足有界表示传递定理假设。
We show that any collection of n-dimensional orbifolds with sectional curvature and volume uniformly bounded below, diameter bounded above, and with only isolated singular points contains orbifolds of only finitely many orbifold homeomorphism types. This is a generalization to the orbifold category of a similar result for manifolds proven by Grove, Petersen, and Wu. It follows that any Laplace isospectral collection of orbifolds with sectional curvature uniformly bounded below and having only isolated singular points also contains only finitely many orbifold homeomorphism types. The main steps…
Let a compact Lie group act isometrically on a non-collapsing sequence of compact Alexandrov spaces with fixed dimension and uniform lower curvature and upper diameter bounds. If the sequence of actions is equicontinuous and converges in the equivariant Gromov--Hausdorff topology, then the limit space is equivariantly homeomorphic to spaces in the tail of the sequence. As a consequence, the class of Riemannian orbifolds of dimension $n$ defined by a lower bound on the sectional curvature and the volume and an upper bound on the diameter has only finitely many members up to orbifold homeomorphi…
In this article, we prove gradient estimates under Bakry-Emery curvature bounds for unbounded graph Laplacians which satisfy an ellipticity assumption. As applications, we study completeness and finiteness of stochastically complete graphs under Bakry-Emery curvature bounds.
Positive bisectional curvature, Ricci mass, and the fundamental group
本文证明了正双截曲率完备非紧 Kähler 流形上 Ricci 形式最高次幂积分有限，解决 Yau-Yang 猜想特例，并推论基本群有限，方法为构造有限 Monge-Ampère 质量的多次调和权函数。
On a conformal gap and finiteness theorem for a class of four manifolds
在本文中，我们发展了一类退化的黎曼度量满足某些整体共形界的4维紧致流形上的气泡树结构。利用冒泡树结构，我们建立了该类的间隙定理、复同态型的有限性定理以及满足适当曲率方程的共形度量族的直径界.
Curvature Estimates for the Ricci Flow II
本文给出了里奇流解的几个曲率估计和收敛性结果。曲率估计依赖于黎曼曲率张量的范数的某些局部时空积分的小性，而收敛结果需要黎曼曲率张量的范数的时空积分的有限性。它们也是爆破奇点的特征。
This is the third in a series of papers on the geometry and analysis of singular area minimizing hypersurfaces. We show how to derive obstruction and structure theories for scalar curvature constraints without imposing dimensional or topological restrictions on the underlying manifold. To this end, we use skin structures to merge minimal hypersurface methods with surgery arguments.
We study global aspects of the mean curvature flow of non-separating hypersurfaces $S$ in closed manifolds. For instance, if $S$ has non-vanishing mean curvature, we show its level set flow converges smoothly towards an embedded minimal hypersurface $Γ$. We prove a similar result for the flow with surgery in dimension 2. As an application we show the existence of monotone incompressible isotopies in manifolds with negative curvature. Combining this result with min-max theory, we show that quasi-Fuchsian and hyperbolic $3$-manifolds fibered over $\mathrm{S}^1$ admit smooth entire foliations who…
We demonstrate that the property of being Alexandrov immersed is preserved along mean curvature flow. Furthermore, we demonstrate that mean curvature flow techniques for mean convex embedded flows such as noncollapsing and gradient estimates also hold in this setting. We also indicate the necessary modifications to the work of Brendle--Huisken to allow for mean curvature flow with surgery for the Alexandrov immersed, $2$-dimensional setting.
Ricci flow with surgery on three-manifolds
这是一篇技术论文，是数学DG/0211159的延续。在这里，我们构建了里奇流与手术和验证大多数的断言，在第13节的电子打印;例外情况是（1）可以在截面曲率上以局部下界坍缩的流形是图流形的陈述-这被推迟到单独的论文中，因为证明与里奇流无关，（2）关于极大角体积的下界和解的光滑性的要求，从某个时间点开始，证明是不合理的，另一方面，与其他结论无关。
The doubling conjecture for positive scalar curvature
本文研究正标量曲率 manifold 的倍增猜想，证明在满足特定基本群分裂条件时，该猜想成立，结合手术技术和面积最小超曲面理论，探讨如何调整 psc-metric 使给定超曲面成为极小、稳定极小或全测地。
On Mostow rigidity for variable negative curvature
证明了一类具有收缩负截面曲率、基本群固定、维数为$>2$的完备有限体积黎曼流形的有限性定理。其中一个关键的成分是，这样一个流形的基本群不允许一个小的非平凡的作用在$R$树。
We prove that the moduli space of 2-convex embedded n-spheres in R^{n+1} is path-connected for every n. Our proof uses mean curvature flow with surgery and can be seen as an extrinsic analog to Marques' influential proof of the path-connectedness of the moduli space of positive scalar curvature metics on three-manifolds.
Free boundary flow with surgery
本文证明了具有自由边界的平均凸曲面在具有手术的平均曲率流下的存在性，并将应用于解决凸球体中3个自由边界极小盘的存在性问题。
In this paper we study sectional curvature bounds for Riemannian manifolds with density from the perspective of a weighted torsion free connection introduced recently by the last two authors. We develop two new tools for studying weighted sectional curvature bounds: a new weighted Rauch comparison theorem and a modified notion of convexity for distance functions. As applications we prove generalizations of theorems of Preissman and Byers for negative curvature, the (homeomorphic) quarter-pinched sphere theorem, and Cheeger's finiteness theorem. We also improve results of the first two authors …
A discrete conformality for polyhedral metrics on surfaces is introduced in this paper which generalizes earlier work on the subject. It is shown that each polyhedral metric on a surface is discrete conformal to a constant curvature polyhedral metric which is unique up to scaling. Furthermore, the constant curvature metric can be found using a discrete Yamabe flow with surgery.
Nondegenerate neck pinches along the mean curvature flow
本文证明对于一般光滑紧致初始曲面，三维欧氏空间中的平均曲率流在首次奇异时刻会出现球形或非退化颈缩奇点，且这些奇点在时空中孤立，并据此给出了一种构造带手术的平均曲率流的新方法。
Hamilton's Ricci flow (RF) equations were recently expressed in terms of a sparsely-coupled system of autonomous first-order nonlinear differential equations for the edge lengths of a d-dimensional piecewise linear (PL) simplicial geometry. More recently, this system of discrete Ricci flow (DRF) equations was further simplified by explicitly constructing the Forman-Ricci tensor associated to each edge, thereby diagonalizing the first-order differential operator and avoiding the need to invert large sparse matrices at each time step. We recently showed analytically and numerically that these eq…
We show that an orientable 3-dimensional manifold M admits a complete riemannian metric of bounded geometry and uniformly pos- itive scalar curvature if and only if there exists a finite collection F of spherical space-forms such that M is a (possibly infinite) connected sum where each summand is diffeomorphic to S2xS1 or to some mem- ber of F. This result generalises G. Perelman's classification theorem for compact 3-manifolds of positive scalar curvature. The main tool is a variant of Perelman's surgery construction for Ricci flow.
A new framework of zero mean curvature surfaces in the isotropic 3-space
本文研究了isotropic 3-space中的零平均曲率表面，提出了ZMC-faces概念，并建立了三个Osserman型不等式，探讨了其等号条件与渐近行为的关系，并给出了达到等号的实例。
On dimension reduction in the Kähler-Ricci flow
本文考虑具有非负二分曲率的Kähler-里奇流解的降维问题。当复维数为$n=2$时，证明了具有非负二分曲率的完全平移Kähler-Ricci孤子的最优降维定理.在有限性假设下，证明了非紧复流形上具有非负对分曲率的里奇流的完备古解的一般降维定理.
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