Dorel Fetcu, Harold Rosenberg
We prove a Simons type equation for non-minimal surfaces with parallel mean curvature vector (pmc surfaces) in $M^n(c)\times\mathbb{R}$, where $M^n(c)$ is an $n$-dimensional space form. Then, we use this equation in order to characterize complete non-minimal pmc surfaces with non-negative Gaussian curvature.
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This paper continues the study of Alexandrov-Fenchel inequalities for quermassintegrals for $k$-convex domains. It focuses on the application to the Michael-Simon type inequalities for $k$-curvature operators. The proof uses optimal transport maps as a tool to relate curvature quantities defined on the boundary of a domain.
Mean curvature flow converging to an minimizing cone and its Hardt-Simon foliation
本文构造了一族平均曲率流，在 type I 和 type II 重标度下分别收敛到严格稳定的极小超锥及其 Hardt-Simon 叶状结构，并去除了 Velázquez 先前结果中的对称性条件。
A proof of the Simon conjecture and curvature rigidities
本文通过谱比较方法证明了单位球面中闭极小曲面的Simon猜想，并推广到平行平均曲率向量曲面及高维旋转对称情形。
In this paper, we first investigate a new locally constrained mean curvature flow (1.5) and prove that if the initial hypersurface is of smoothly compact starshaped, then the solution of the flow (1.5) exists for all time and converges to a sphere in smooth topology. Following this flow argument, not only do we achieve a new proof of the celebrated sharp Michael-Simon inequality for mean curvature in (n+1) dimensional Euclidean space, but we also get the necessary and sufficient condition for the establishment of the equality. In the second part of this paper, we study a mean curvature type fl…
Curvature bounds for the spectrum of closed Einstein spaces and Simon conjecture
本文证明了闭Einstein空间上关于特征值谱的Simon猜想，并进一步将其推广至下一个特征值间隙，给出了最佳曲率界。
Simons formulas in complex space forms and product spaces
该研究针对复空间形式与乘积空间中的平行平均曲率子流形推导Simons型方程，并用其刻画部分此类子流形，丰富了子流形几何的相关理论。
Optimal Transport Approach to Michael-Simon-Sobolev Inequalities in Manifolds with Intermediate Ricci Curvature Lower Bounds
该研究将最优输运的McCann定理推广至子流形场景，证明了满足中间Ricci曲率下界条件的流形内子流形的Michael-Simon-Sobolev不等式，涵盖了已有相关研究结论的变体情形。
A Simons' type formula for spacelike submanifolds in semi-Riemannian warped product
本文建立半黎曼扭曲乘积中类空子流形的 Simons 型公式，并据此推广平行与伪平行浸入结果，得出 de Sitter 与 Einstein-de Sitter 时空中的分类结论。
Locally constrained flows and sharp Michael-Simon inequalities in hyperbolic space
本文利用局部约束的平均曲率流和逆曲率流，在双曲空间中针对具有较弱凸性条件的超曲面，成功证明了关于平均曲率及高阶平均曲率的精确Michael-Simon型不等式。
Sobolev inequalities in manifolds with nonnegative intermediate Ricci curvature
本文利用Alexandrov-Bakelman-Pucci方法，将Michael-Simon型Sobolev不等式推广到具有非负中间Ricci曲率的Riemannian流形中的子流形上，并由此导出极小子流形的等周不等式。
We compute the Laplacian of the squared norm of the second fundamental form of a surface in Sol_3 and then use this Simons type formula to obtain some gap results for compact constant mean curvature surfaces of this space.
Equations of Simons type are presented. They are satisfied by a pair of special operators associated to the immersion $Σ^2\looparrowright M^2(c)\times\mathbb{R}$ with constant mean curvature. Some immersions are characterized.
We use a Simons type equation in order to characterize complete non-minimal pmc surfaces with non-negative Gaussian curvature.
A single induction proof of Simons' Riemannian holonomy theorem
本文通过单一强归纳法，利用平坦子空间、根中心化子及迹消失引理，给出了 Simons 黎曼和乐定理的完整代数证明，并推导了流形的局部对称性。
A few formulas and theorems for statistical structures are proved. They deal with various curvatures as well as with metric properties of the cubic form or its covariant derivative. Some of them generalize formulas and theorems known in the case of Lagrangian submanifolds or affine hypersurfaces.
We give a direct calculation of the curvature of the Hitchin connection, in geometric quantization on a symplectic manifold, using only differential geometric techniques. In particular, we establish that the curvature acts as a first-order operator on the quantum spaces. Projective flatness follows if the Kähler structures do not admit holomorphic vector fields. Following Witten, we define a complex variant of the Hitchin connection on the bundle of prequantum spaces. The curvature is essentially unchanged, so projective flatness holds in the same cases. Finally, the results are applied to qua…
A note on parallel mean curvature surfaces and Codazzi operators
本文提出用Codazzi算子刻画的曲面内蕴Klotz-Osserman型定理，并利用Simons公式研究乘积空间中具非正Gauss曲率的平行平均曲率曲面。
We prove a sharp Sobolev inequality on manifolds with nonnegative Ricci curvature. Moreover, we prove a Michael-Simon inequality for submanifolds in manifolds with nonnegative sectional curvature. Both inequalities depend on the asymptotic volume ratio of the ambient manifold.
The Michael-Simon-Sobolev inequality on manifolds for positive symmetric tensor fields
本研究利用ABP方法，在非负截面曲率黎曼流形的紧子流形上，建立了光滑对称一致正定(0,2)张量场的Michael-Simon-Sobolev不等式，推广了S. Brendle的已有结果。
We prove a Simons type formula for submanifolds with parallel mean curvature vector field in product spaces of type $M^n(c)\times\mathbb{R}$, where $M^n(c)$ is a space form with constant sectional curvature $c$, and then we use it to characterize some of these submanifolds.
The focal submanifolds of isoparametric hypersurfaces in spheres are all minimal Willmore submanifolds, mostly being $\mathcal{A}$-manifolds in the sense of A.Gray but rarely Ricci-parallel (\cite{QTY},\cite{LY},\cite{TY3}). In this paper we study the geometry of the focal submanifolds via Simons formula. We show that all the focal submanifolds with $g\geq3$ are not normally flat by estimating the normal scalar curvatures. Moreover, we give a complete classification of the semiparallel submanifolds among the focal submanifolds.
Łojasiewicz-Simon inequalities for minimal networks: stability and convergence
本文证明平面三重点网络的Łojasiewicz-Simon梯度不等式，并由此建立极小网络在曲率运动下的稳定性与平滑收敛性，同时构造了收敛到退化网络的实例。
J.J.L. Vel$\acute{a}$zquez in 1994 used the degree theory to show that there is a perturbation of Simons' cone, starting from which the mean curvature flow develops a type $\mathrm{II}$ singularity at the origin. He also showed that under a proper time-dependent rescaling of the solution around the origin, the rescaled flow converges in the $C^{0}$ sense to a minimal hypersurface which is tangent to Simons' cone at infinity. In this paper, we prove that the rescaled flow actually converges locally smoothly to the minimal hypersurface, which appears to be the singularity model of the type $\mat…
We derive curvature estimates for minimal submanifolds in Euclidean space for arbitrary dimension and codimension via Gauss map. Thus, Schoen-Simon-Yau's results and Ecker-Huisken's results are generalized to higher codimension. In this way we improve Hildebrandt-Jost-Widman's result for the Bernstein type theorem.
Isoperimetric and Michael-Simon inequalities on manifolds with asymptotically nonnegative curvature
该研究在渐近非负曲率流形上证明等周不等式成立，在紧集外非负曲率流形上推导Michael-Simon不等式，采用ABP方法并结合几何与谱性质完成证明，拓展了相关不等式的适用条件。
In this note we prove uniqueness of nondegenerate compact blowups for the motion by curvature of planar networks. The proof follows ideas introduced in "Lojasiewicz-Simon inequalities for minimal networks: stability and convergence" for the study of stability properties of critical points of the length functional and it is based on the application of a Lojasiewicz-Simon gradient inequality.
Sobolev inequalities involving 2-tensor fields in manifolds with nonnegative sectional curvature
本文应用ABP方法，在非负截面曲率流形中为光滑对称一致正定(0,2)张量场建立了Log Sobolev型及Michael Simon Sobolev不等式。
Application of Chern-Simons gauge theory to the enclosed volume of constant mean curvature surfaces in the 3-sphere
基于Hitchin的工作，本文推导出三维球面中紧致CMC曲面所围体积的公式，该公式由Chern-Simons丛上的和乐与Willmore泛函表示，并通过多例验证了其有效性。
We present a reduction of codimension theorem for surfaces with parallel mean curvature in symmetric spaces.
Michael-Simon type inequalities in hyperbolic space $\mathbb{H}^{n+1}$ via Brendle-Guan-Li's flows
本文利用 Brendle-Guan-Li 流，在 $h$-凸条件下建立了双曲空间中平均曲率及第 $k$ 个平均曲率的锐化 Michael-Simon 不等式，并推广了 Minkowski 型及加权 Alexandrov-Fenchel 不等式。
In this paper, we obtain an Ecker-Huisken type result for entire graphs with parallel mean curvature.
We present some results on the boundedness of the mean curvature of proper biharmonic submanifolds in spheres. A partial classification result for proper biharmonic submanifolds with parallel mean curvature vector field in spheres is obtained. Then, we completely classify the proper biharmonic submanifolds in spheres with parallel mean curvature vector field and parallel Weingarten operator associated to the mean curvature vector field.
A submanifold of a pseudo-Riemannian manifold is said to have parallel mean curvature vector if the mean curvature vector field H is parallel as a section of the normal bundle. Submanifolds with parallel mean curvature vector are important since they are critical points of some natural functionals. In this paper, we survey some classical and recent results on submanifolds with parallel mean curvature vector. Special attention is paid to the classification of space-like and Lorentz surfaces with parallel mean curvature vector in Riemannian and indefinite space forms.
We classify complete biharmonic surfaces with parallel mean curvature vector field and non-negative Gaussian curvature in complex space forms.
We prove a Simons type equation for non-minimal surfaces with parallel mean curvature vector (pmc surfaces) in $M^3(c)\times\mathbb{R}$, where $M^3(c)$ is a 3-dimensional space form. Then, we use this equation in order to characterize certain complete non-minimal pmc surfaces.
On Simon's third gap conjecture for minimal surfaces in spheres
本文基于前人工作，研究Simon猜想中最小曲面在单位球中的第三间隙问题，通过发展改进的第三阶Simons型积分恒等式和建立更高阶曲率项的新下界，获得平方法向量场范数在整个区间[5/3,9/5]上的正间隙结果，包括端点情况，并应用此结果证明闭合自收缩曲面的刚性结果。
We prove a sharp logarithmic Sobolev inequality which holds for compact submanifolds without boundary in Riemannian manifold with nonnegative sectional curvature of arbitrary dimension and codimension, while the ambient manifold needs to have a specific Euclid-like property. Like the Michael-Simon Sobolev inequality, this inequality includes a term involving the mean curvature. This extends a recent result of S. Brendle with Euclidean setting.
We prove that the Euclidean plane is the only Riemannian plane with total curvature and free of conjugate points that satisfies Playfair's version of the parallel postulate.
We provide a classification of Einstein submanifolds in space forms with flat normal bundle and parallel mean curvature. This extends a previous result due to Dajczer and Tojeiro for isometric immersions of Riemannian manifolds with constant sectional curvature.
We explicitly determine tori that have a parallel mean curvature vector, both in the complex projective plane and the complex hyperbolic plane
Spheres with parallel mean curvature in $\mathbb{S}^2 \times \mathbb{H}^2$
本文研究球面与双曲平面乘积空间中平行平均曲率曲面，分类了因拓扑原因使全纯二次微分消失的球面。
We determine all helix surfaces with parallel mean curvature vector field, which are not minimal or pseudo-umbilical, in spaces of type $M^n(c)\times\mathbb{R}$, where $M^n(c)$ is a simply-connected $n$-dimensional manifold with constant sectional curvature $c$.
Scale-Separated Curvature Estimates for Strongly Stable Hypersurfaces
本文通过将Young不等式替换为Hölder不等式，简化了曲率估计的论证过程，得到了更小的常数，并自然扩展到常均曲率（CMC）设定中。研究还证明了在强稳定的CMC超曲面中，Hölder机制产生了一个具有两个竞争项的积分曲率估计，并在特定条件下将CMC估计转化为最小曲面形式。
We find a Simons type formula for submanifolds with parallel mean curvature vector (pmc submanifolds) in product spaces $M^n(c)\times\mathbb{R}$, where $M^n(c)$ is a space form with constant sectional curvature $c$, and then we use it to prove a gap theorem for the mean curvature of certain complete proper-biharmonic pmc submanifolds, and classify proper-biharmonic pmc surfaces in $\mathbb{S}^n(c)\times\mathbb{R}$.
Let (M,g) be a complete noncompact riemannian manifold with bounded geometry and parallel Ricci curvature. We show that some operators, "affine" relatively to the Ricci curvature, are locally invertible, in some classical Sobolev spaces, near the metric g.
We show that any left invariant metric with harmonic curvature on a solvable Lie group is Ricci-parallel. We show the same result for any Lie group of dimension $\leq$ 6.
We construct a special class of Lorentz surfaces in the pseudo-Euclidean 4-space with neutral metric which are one-parameter systems of meridians of rotational hypersurfaces with timelike or spacelike axis and call them meridian surfaces. We give the complete classification of the meridian surfaces with parallel mean curvature vector field. We also classify the meridian surfaces with parallel normalized mean curvature vector. We show that in the family of the meridian surfaces there exist Lorentz surfaces which have parallel normalized mean curvature vector field but not parallel mean curvatur…
We consider a quadratic form defined on the surfaces with parallel mean curvature vector of an any dimensional complex space form and prove that its $(2,0)$-part is holomorphic. When the complex dimension of the ambient space is equal to $2$ we define a second quadratic form with the same property and then determine those surfaces with parallel mean curvature vector on which the $(2,0)$-parts of both of them vanish. We also provide a reduction of codimension theorem and prove a non-existence result for $2$-spheres with parallel mean curvature vector.
Two holomorphic Hopf differentials for surfaces of non-null parallel mean curvature vector in S^2xS^2 and H^2xH^2 are constructed. A 1:1 correspondence between these surfaces and pairs of constant mean curvature surfaces of S^2xR and H^2xR is established. Using that, surfaces with vanishing Hopf differentials (in particular spheres with parallel mean curvature vector) are classified and a rigidity result for constant mean curvature surfaces of S^2xR and H^2xR is proved.
We compute a Simons' type formula for the stress-energy tensor of biharmonic maps from surfaces. Specializing to Riemannian immersions, we prove several rigidity results for biharmonic CMC surfaces, putting in evidence the influence of the Gaussian curvature on pseudo-umbilicity. Finally, the condition of biharmonicity is shown to enable an extension of the classical Hopf theorem to CMC surfaces in any ambient Riemannian manifold.
We establish variational formulas for Ricci upper and lower bounds, as well as a derivative formula for the Ricci curvature. As applications, constant curvature manifolds, Einstein manifolds and Ricci parallel manifolds are identified, respectively, with different integral-differential formulas and semigroup inequalities. Moreover, by using derivative and Hessian formulas for the heat semigroup $P_t$ developed from stochastic analysis, explicit Hessian estimates are derived on Einstein and Ricci parallel manifolds.
Einstein submanifolds with parallel mean curvature into $\mathbb{S}^n\times\mathbb{R}$
本文证明 $\mathbb{S}^n\times\mathbb{R}$ 中满足平坦法丛与平行平均曲率条件的 Einstein 子流形可表示为等距浸入的扭曲积。
Elementary proof of a Lemma for parallel mean curvature surfaces in complex space forms
本文为二维复空间形式中平行平均曲率曲面分类的重要引理提供了一种初等证明方法。
Pinching rigidity of surfaces with parallel mean curvature vector in spheres
研究浸入单位球中具有平行平均曲率向量和正高斯曲率的闭曲面，建立三个西蒙斯型积分恒等式，得到前两个尖锐端点间隙等结果，并结合分类定理刻画端点情形。
We prove a suite of asymptotically sharp quadratic curvature pinching estimates for mean curvature flow in the sphere which generalize Simons' rigidity theorem for minimal hypersurfaces. We then obtain derivative estimates for the second fundamental form which we utilize, via a compactness argument, to obtain a convexity estimate. Together, the convexity and cylindrical estimates yield a partial classification of singularity models. We also obtain new rigidity results for ancient solutions.
We give criteria for which a principal curvature becomes a bounded $C^\infty$-function at non-degenerate singular points of wave fronts by using geometric invariants. As applications, we study singularities of parallel surfaces and extended distance squared functions of wave fronts. Moreover, we relate these singularities to some geometric invariants of fronts.
Curvature criteria of A-simple singularities R,0-> R^2,0 and their parallel curves
本文提出了一种确定A-简单奇点R,0→R²,0曲率准则的方法，并探讨了其平行曲线的奇点类型。
We study the global geometry of surfaces in Sasakian space forms whose mean curvature vector is parallel in the normal bundle (these include the Riemannian Heisenberg space of dimension $2n+1$). We prove a codimension reduction theorem. We introduce two holomorphic quadratic differentials on anti-invariant such surfaces and use them to obtain classification theorems.
Diameter, Area, and Mean curvature
本研究将已有直径界推广至带边界子流形与弱曲率限制的非紧外围流形，得到依赖平均曲率积分和流形面积的直径界，并将其应用于极小、常平均曲率及极小极大构造的指定平均曲率曲面。
The purpose of this article is to determine explicitly the complete surfaces with parallel mean curvature vector, both in the complex projective plane and the complex hyperbolic plane. The main results are as follows: When the curvature of the ambient space is positive, there exists a unique such surface up to rigid motions of the target space. On the other hand, when the curvature of the ambient space is negative, there are `non-trivial' complete parallel mean curvature surfaces generated by Jacobi elliptic functions and they exhaust such surfaces.
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