Dong Pu, Jingjing Su, Hongwei Xu
In this paper, we investigate the mean curvature flow of compact surfaces in $4$-dimensional space forms. We prove the convergence theorems for the mean curvature flow under certain pinching conditions involving the normal curvature, which generalise Baker-Nguyen's convergence theorem.
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In this paper, we proved a compactness result about Riemannian manifolds with an arbitrary pointwisely pinched Ricci curvature tensor.
In this paper, we prove that expanding gradient Ricci solitons with (positively) pinched Ricci curvature are trivial ones. Namely, they are either compact or flat.
We make rigorous an old idea of using mean curvature flow to prove a theorem of Richard Hamilton on the compactness of proper hypersurfaces with pinched, bounded curvature.
There is a conjecture that a complete Riemannian 3-manifold with bounded sectional curvature, and pointwise pinched nonnegative Ricci curvature, must be flat or compact. We show that this is true when the negative part (if any) of the sectional curvature decays quadratically.
We show that a compact Riemannian manifold with weakly 1/4-pinched sectional curvatures is either locally symmetric or diffeomorphic to a space form.
We prove that codimension two surfaces satisfying a nonlinear curvature condition depending on normal curvature are smoothly deformed by mean curvature flow to round points.
Compact four-manifolds with pinched self-dual Weyl curvature
本文研究了具有调和自双曲Weyl曲率和挤压条件的紧致四流形的性质。
This paper has been withdrawn by the author due to a serious gap in the proof of the main theorem.
Pinched self-dual Weyl curvature on Einstein four-manifolds
研究Einstein四维流形中挤压自双曲Weyl曲率的性质，证明其在特定条件下具有反自双曲结构。
We prove that any $n$--dimensional complete gradient Ricci soliton with pinched Weyl curvature is a finite quotient of $\RR^{n}$, $\RR \times \SS^{n-1}$ or $\SS^{n}$. In particular, we do not need to assume the metric to be locally conformally flat.
In this paper, we obtain classification of four-dimensional Einstein manifolds with positive Ricci curvature and pinched sectional curvature. In particular, the first result concerns with an upper bound of sectional curvature, improving a theorem of E. Costa. The second is a generalization of D. Yang's result assuming an upper bound on the difference between sectional curvatures.
We say that a nonnegatively curved manifold $(M,g)$ has quarter pinched flag curvature if for any two planes which intersect in a line the ratio of their sectional curvature is bounded above by 4. We show that these manifolds have nonnegative complex sectional curvature. By combining with a theorem of Brendle and Schoen it follows that any positively curved manifold with strictly quarter pinched flag curvature must be a space form. This in turn generalizes a result of Andrews and Nguyen in dimension 4. For odd dimensional manifolds we obtain results for the case that the flag curvature is pinc…
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.
In a previous paper, we proved a number of optimal rigidity results for Riemannian manifolds of dimension greater than four whose curvature satisfy an integral pinching. In this article, we use the same integral Bochner technique to extend the results in dimension three. Then, by using the classification of closed three-manifolds with nonnegative scalar curvature and a few topological considerations, we deduce optimal sphere theorems for three-dimensional manifolds with integral pinched curvature.
We show that the blow-ups of compact solutions to the mean curvature flow in $\mathbb{R}^N$ initially satisfying the pinching condition $|A|^2 < c |H|^2$ for a suitable constant $c = c(n)$ must be codimension one.
For a convex domain $D$ bounded by the hypersurface $\partial D$ in a space of constant curvature we give sharp bounds on the width $R-r$ of a spherical shell with radii $R$ and $r$ that can enclose $\partial D$, provided that normal curvatures of $\partial D$ are pinched by two positive constants. Furthermore, in the Euclidean case we also present sharp estimates for the quotient $R/r$. From the obtained estimates we derive stability results for almost umbilical hypersurfaces in the constant curvature spaces.
Planarity and convexity for pinched ancient solutions of mean curvature flow
本文证明了高余维平均曲率流的面性估计的尺度不变变体，并证明了余维一受夹紧完备古代解的凸性，进而刻画了高余维中的某些古代解和收缩子。
In this paper, we prove that any $κ$-noncollapsed gradient steady Ricci soliton with nonnegative curvature operator and horizontally $ε$-pinched Ricci curvature must be rotationally symmetric. As an application, we show that any $κ$-noncollapsed gradient steady Ricci soliton $(M^n, g,f)$ with nonnegative curvature operator must be rotationally symmetric if it admits a unique equilibrium point and its scalar curvature $R(x)$ satisfies $\lim_{r(x)\rightarrow\infty}R(x)f(x)=C_0\sup_{x\in M}R(x)$ with $C_0>\frac{n-2}{2}$.
Total curvature and isoperimetric inequalities in pinched Cartan-Hadamard manifolds
本文针对具有挤压负曲率的Cartan-Hadamard流形，建立凸超曲面总Gauss-Kronecker曲率的尖锐下界，结合Chern-Gauss-Bonnet定理等完成证明，还应用该结果得到5维中Cartan-Hadamard猜想的等周不等式。
Three-manifolds with non-negatively pinched Ricci curvature
研究非负曲率pinching的三维流形的Ricci流解，证明其具有标度不变曲率衰减和pinching保持，结合Lott和Deruelle-Schulze-Simon的工作，证明Hamilton的pinching猜想。
Manifolds with PIC1 pinched curvature
本文通过构造从任意 PIC1 夹挤的完备非紧流形出发、永恒存在的 Ricci 流，将三维非负夹挤 Ricci 曲率流形的平坦或紧性结论推广到高维。
We establish a nice orthonormal frame field on a closed surface minimally immersed in a unit sphere $S^{n}$, under which the shape operators take very simple forms. Using this frame field, we obtain an interesting property $K+K^{N}=1$ for the Gauss curvature $K$ and the normal curvature $K^{N}$ if the Gauss curvature is positive. Moreover, using this property we obtain the pinching on the intrinsic curvature and normal curvature, the pinching on the normal curvature, respectively.
We consider a complete noncompact Riemannian manifold M and give conditions on a compact submanifold K of M so that the outward normal exponential map off of the boundary of K is a diffeomorphism onto M\K. We use this to compactify M and show that pinched negative sectional curvature outside K implies M has a compactification with a well defined Hölder structure independent of K. The Hölder constant depends on the ratio of the curvature pinching. This extends and generalizes a 1985 result of Anderson and Schoen.
We give an estimate of the first eigenvalue of the Laplace operator on a complete noncompact stable minimal hypersurface $M$ in a complete simply connected Riemannian manifold with pinched negative sectional curvature. In the same ambient space, we prove that if a complete minimal hypersurface $M$ has sufficiently small total scalar curvature then $M$ has only one end. We also obtain a vanishing theorem for $L^2$ harmonic 1-forms on minimal hypersurfaces in a Riemannian manifold with sectional curvature bounded below by a negative constant. Moreover we provide sufficient conditions for a minim…
Four-dimensional shrinkers with pinched curvatures
本文针对四维收缩子，在不同曲率挤压条件下完成分类，并建立Hitchin-Thorpe型不等式，丰富了收缩子的几何研究成果。
We show that every one-harmonic map, in the sense of Trapani and Valli, from a Hadamard surface of pinched negative curvature to $\mathbb{H}^2$ has image the interior of the convex hull of a subset of $\partial_\infty\mathbb{H}^2$. The proof relies on Minkowski geometry, by interpreting one-harmonic maps as the Gauss maps of convex surfaces.
Bach-pinched metrics on closed manifolds
本研究借助Aubin提出的形变方法，证明了所有四维闭流形上均存在满足Bach张量被标量曲率夹挤条件的度量，推进了微分几何中相关度量存在性问题的研究。
Pairs of Embedded Spheres with Pinched Prescribed Mean Curvature
针对单位三维球面上满足挤压条件的正函数或变号函数h，证明存在至少两个具有预设平均曲率h的嵌入二维球面，拓展了预设平均曲率嵌入球面的存在性结果。
A Riemannian manifold $M$ has higher hyperbolic rank if every geodesic has a perpendicular Jacobi field making sectional curvature -1 with the geodesic. If in addition, the sectional curvatures of $M$ lie in the interval $[-1,-\frac14]$, and $M$ is closed, we show that $M$ is a locally symmetric space of rank one. This partially extends work by Constantine using completely different methods. It is also a partial converse to Hamenstädt's hyperbolic rank rigidity result for sectional curvatures $\leq -1$, and complements well-known results on Euclidean and spherical rank rigidity.
We study solutions of high codimension mean curvature flow defined for all negative times, usually referred to as ancient solutions. We show that any compact ancient solution whose second fundamental form satisfies a certain natural pinching condition must be a family of shrinking spheres. Andrews and Baker have shown that initial submanifolds satisfying this pinching condition, which generalises the notion of convexity, converge to round points under the flow. As an application, we use our result to simplify their proof.
We consider the evolution by mean curvature flow of Lagrangian submanifolds of the complex projective space CP^n. We prove that, if the initial value satisfies a suitable pinching condition, then the flow exists for all times and the manifold converges to a totally geodesic submanifold. As a corollary, we obtain that a Lagrangian submanifold satisfying our pinching condition is diffeomorphic to a real projective space.
We consider the evolution by mean curvature flow of a closed submanifold of the complex projective space. We show that, if the submanifold has small codimension and satisfies a suitable pinching condition on the second fundamental form, then the evolution has two possible behaviors: either the submanifold shrinks to a round point in finite time, or it converges smoothly to a totally geodesic limit in infinite time. The latter behavior is only possible if the dimension is even. These results generalize previous works by Huisken and Baker on the mean curvature flow of submanifolds of the sphere.
Rigidity Results for Compact Submanifolds with Pinched Ricci Curvature in Euclidean and Spherical Space Forms
本文研究了在欧几里得和球面空间形式中具有受限曲率的紧致子流形，证明其要么是爱因斯坦克利福德 тор斯，要么是拓扑球体，或具有部分同调群消失，扩展了多项经典刚性结果。
G. Pipoli and C. Sinestrari considered the mean curvature flow starting from a closed submanifold in the complex projective space. They proved that if the submanifold is of small codimension and satisfies a suitable pinching condition for the second fundamental form, then the flow has two possible behaviors: either the submanifold collapses to a round point in finite time, or it converges smoothly to a totally geodesic submanifold in infinite time. In this paper, we prove the similar results for the mean curvature flow starting from pinched closed submanifolds in (general) rank one symmetric s…
In this paper, we study closed four-dimensional manifolds. In particular, we show that under various new pinching curvature conditions (for example, the sectional curvature is no more than 5/6 of the smallest Ricci eigenvalue) then the manifold is definite. If restricting to a metric with harmonic Weyl tensor, then it must be self-dual or anti-self-dual under the same conditions. Similarly, if restricting to an Einstein metric, then it must be either the complex projective space with its Fubini-Study metric, the round sphere or their quotients. Furthermore, we also classify Einstein manifolds …
Is a complete Riemannian manifold with positively pinched Ricci curvature compact
本文针对Hamilton关于欧氏空间中带夹挤第二基本形式的凸超曲面必为紧集的结果，利用作者1998年的拟共形映射结论给出无需额外假设的完整替代证明且可推广，以此纪念Ricci流创立者Hamilton。
We investigate the evolution of closed strictly convex hypersurfaces in $\mathbb{R}^{n+1}$, n=3, for contracting normal velocities, including powers of the mean curvature, of the norm of the second fundamental form, and of the Gauss curvature. We prove convergence to a round point for 2-pinched initial hypersurfaces. In $\mathbb{R}^{n+1}$, n=2, natural quantities exist for proving convergence to a round point for many normal velocities. Here we present their counterparts for arbitrary dimensions $n\in\mathbb{N}$.
We show that any ancient solution to the Ricci flow which satisfies a suitable curvature pinching condition must have constant sectional curvature.
Long-time Ricci flow existence and topological rigidity from manifolds with pinched scale-invariant integral curvature
证明了在曲率有界且尺度不变积分曲率相对于Sobolev常数足够夹挤的完备流形上，里奇流长时间存在；若曲率次临界L^p可积，则流形等距于标准平坦R^n，从而得到拓扑刚性。
Ricci pinched compact submanifolds in spheres
本文研究圆球紧子流形在仅依赖平均曲率向量长度的 Ricci 曲率下界条件下的拓扑，并指出该条件仅在特殊情形可推出外在几何的强结论。
Ricci pinched compact submanifolds in space forms
本文研究非负曲率空间形式中 Ricci 曲率受平均曲率向量长度控制的紧子流形，证明无需平均曲率向量平行即可肯定 Xu–Gu 猜想。
Let (M,g_0) be a compact Riemannian manifold with pointwise 1/4-pinched sectional curvatures. We show that the Ricci flow deforms g_0 to a constant curvature metric. The proof uses the fact, also established in this paper, that positive isotropic curvature is preserved by the Ricci flow in all dimensions. We also rely on earlier work of Hamilton and of Bohm and Wilking.
On locally conformally flat manifolds with positive pinched Ricci curvature
本文利用 Yamabe 流证明，满足正夹断 Ricci 曲率条件的完备局部共形平坦流形必为紧流形，从而在该附加几何条件下将 Hamilton 夹断猜想推广到高维。
We consider ancient solutions to the mean curvature flow in $\mathbb{R}^{n+1}$ ($n \geq 3$) that are weakly convex, uniformly two-convex, and satisfy derivative estimates $|\nabla A| \leq γ_1 |H|^2, |\nabla^2 A| \leq γ_2 |H|^3$. We show that such solutions are noncollapsed. As an application, in arbitrary codimension, we consider compact $n$-dimensional ($n \geq 5$) solutions to the mean curvature flow in $\mathbb{R}^N$ that satisfy the pinching condition $|H| > 0$ and $|A|^2 < c(n) |H|^2$, $c(n) = \min\{\frac{1}{n-2}, \frac{3(n+1)}{2n(n+2)}\}$. We conclude that any blow-up model at the first …
Ricci pinched compact hypersurfaces in spheres
本文研究球面中 Ricci 曲率受平均曲率约束的紧致超曲面的拓扑，利用 Bochner 技巧获得比任意余维情形更强的结果。
We study the asymptotic Dirichlet problem for $\mathcal{A}$-harmonic functions on a Cartan-Hadamard manifold whose radial sectional curvatures outside a compact set satisfy an upper bound $$ K(P)\le - \frac{1+\varepsilon}{r(x)^2 \log r(x)} $$ and a pointwise pinching condition $$ |K(P)|\le C_K |K(P')| $$ for some constants $\varepsilon>0$ and $C_K\ge 1$, where $P$ and $P'$ are any 2-dimensional subspaces of $T_xM$ containing the (radial) vector $\nabla r(x)$ and $r(x)=d(o,x)$ is the distance to a fixed point $o\in M$. We solve the asymptotic Dirichlet problem with any continuous boundary data …
We prove the central limit theorem of random variables induced by distances to Brownian paths and Green functions on the universal cover of Riemannian manifolds of finite volume with pinched negative curvature. We further provide some ergodic properties of Brownian motions and an application of the central limit theorem to the dynamics of geodesic flows in pinched negative curvature.
Generalizing results due to Brady and Farb we prove the existence of a bilipschitz embedded manifold of pinched negative curvature and dimension m_1+m_2-1 in the product X:=X_1^{m_1} times X_2^{m_2} of two Hadamard manifolds X_i^{m_i} of dimension m_i with pinched negative curvature. Combining this result with a Theorem by Buyalo and Schroeder we prove the additivity of the hyperbolic rank for products of manifolds with pinched negative curvature.
PIC1 pinched manifolds are flat or compact
本文证明了所有维度下，缩陷里奇曲率的流形要么平坦要么紧致，发展了处理无限远坍缩流形的提升技术，并改进了里奇流的曲率估计和间隙定理。
Inverse mean curvature flow and Ricci-pinched three-manifolds
本文利用逆平均曲率流重新证明了满足 Ricci-pinched 条件的非紧三维流形要么平坦、要么具有非欧体积增长，从而为 Hamilton 猜想提供了不依赖 Ricci 流的新证明。
We prove that a $n$-dimensional, $4 \leq n \leq 6$, compact gradient shrinking Ricci soliton satisfying a $L^{n/2}$-pinching condition is isometric to a quotient of the round $\mathbb{S}^{n}$. The proof relies mainly on sharp algebraic curvature estimates, the Yamabe-Sobolev inequality and an improved rigidity result for integral pinched Einstein metrics.
Geometric and topological rigidity of pinched submanifolds
针对空间形式中满足第二基本形式长度与平均曲率挤压条件的任意余维数紧致子流形，证明其要么使中间维度同调消失，要么确定子流形至全等，结果精确且无额外平均曲率假设地推广了前人成果。
Let (X,g) be a metrically complete, simply connected Riemannian manifold with bounded geometry and pinched negative curvature, i.e. there are constants a>b>0 such that -a^2<K<-b^2 for all sectional curvatures K. Here bounded geometry is used in the sense that all covariant derivatives of the Riemannian curvature tensor are bounded and the injectivity radius is uniformly bounded below by a positive constant. We show that there are no superexponentially decaying eigenfunctions of the Laplacian of g. We also show the analogous conclusion for other geometric operators, and prove a theorem with the…
In this paper we prove that, under an explicit integral pinching assumption between the $L^2$-norm of the Ricci curvature and the $L^2$-norm of the scalar curvature, a closed 3-manifold with positive scalar curvature admits an Einstein metric with positive curvature. In particular this implies that the manifold is diffeomorphic to a quotient of ${\Bbb S}^3$.
Geometric and topological rigidity of pinched submanifolds II
本文研究空间形式中满足夹条件的四维紧致子流形的几何与拓扑刚性，利用四维几何、非负各向同性曲率及Bochner技巧，得到精确结果并推广先前工作，无需额外假设。
We investigate the geometry and topology of submanifolds under a sharp pinching condition involving extrinsic invariants like the mean curvature and the length of the second fundamental form. Several homology vanishing results are given. Moreover, an integral bound is provided for the Bochner operator of compact Euclidean submanifolds in terms of the Betti numbers.
We prove that compact Kähler manifolds whose sectional curvatures are close to 1/4-pinched have ratios of Chern numbers close to the corresponding ratios of a complex hyperbolic space form. We deduce that the Mostow-Siu surfaces (and their three-dimensional analogues constructed by the first author) do not admit Kähler metrics with pinching close to 1/4.
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…
Non-existence of complete Kähler metric of negatively pinched holomorphic sectional curvature
本文证明了在有界弱伪凸域上，若Kähler度量满足特定曲率与几何条件，则其完备性决定了负向夹紧全纯截面曲率完备Kähler度量及完备Kähler-Einstein度量的存在性二分法，揭示了此类度量的不存在性充分条件。
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…
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