Zhiqin Lu
In this paper, we proved the Normal Scalar Curvature Conjecture and the Bottcher-Wenzel Conjecture. We also established some new pinching theorems for minimal submanifolds in spheres.
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In this paper, we proved the normal scalar curvature conjecture and the Bottcher-Wenzel conjecture.
Constant holomorphic sectional curvature conjecture and Fino-Vezzoni conjecture
本文探讨了非Kaehler几何中两个重要猜想的历史和最新进展，包括常holomorphic截面曲率猜想和Fino-Vezzoni猜想。
We prove a criterion of existence of solutions conjectured by C. C. Chen and C. S. Lin [20] for the prescribed scalar curvature problem on the standard n-dimensional sphere.
We prove a uniqueness theorem for immersed spheres of prescribed (non-constant) mean curvature in homogeneous three-manifolds. In particular, this uniqueness theorem proves a conjecture by A.D. Alexandrov about immersed spheres of prescribed Weingarten curvature in R3 for the special but important case of prescribed mean curvature. As a consequence, we extend the classical Hopf uniqueness theorem for constant mean curvature spheres to the case of immersed spheres of prescribed antipodally symmetric mean curvature in R3.
In this paper, we show that if the mean curvature of a closed smooth embedded mean curvature flow in R^3 is of type-I, then the rescaled flow at the first finite singular time converges smoothly to a self-shrinker flow with multiplicity one. This result confirms Ilmanen's multiplicity-one conjecture under the assumption that the mean curvature is of type-I. As a corollary, we show that the mean curvature at the first singular time of a closed smooth embedded mean curvature flow in R^3 is at least of type-I.
Decomposition of the Curvature Operator and Applications to the Hopf Conjecture
本文通过将曲率算子分解为 Hermitian 分量，建立截面曲率特征值判据与 Betti 数零化定理，证实小 Weyl 曲率情形的 Hopf 猜想并获得新刚性结果。
In this paper angular curvature measures are investigated. Our first result is a complete classification of translation-invariant angular smooth curvature measures on $\mathbb{R}^n$. Subsequently, we use this result to show that the class of angular curvature measures on a Riemannian manifold is preserved by both the pullback by isometric immersions and the action of the Lipschitz-Killing algebra. The latter confirms the angularity conjecture formulated by A. Bernig, J.H.G. Fu, and G. Solanes.
In this short note we review some known results on the structure and regularity of spaces with lower Ricci curvature bounds. We present some known and new open questions about next steps.
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
The critical point equation arises as a critical point of the total scalar curvature functional defined on the space of constant scalar curvature metrics of a unit volume on a compact manifold. In this equation, there exists a function $f$ on the manifold that satisfies the following $$ (1+f){\rm Ric} = Ddf + \frac{nf +n-1}{n(n-1)}sg. $$ It has been conjectured that if $(g, f)$ is a solution of the critical point equation, then $g$ is Einstein and so $(M, g)$ is isometric to a standard sphere. In this paper, we show that this conjecture is true if the given Riemannian metric has positive isotr…
Gaussian curvature conjecture for minimal graphs
本文通过将单位圆盘上极小图的Gaussian曲率猜想归结为内接双心四边形上Scherk型极小曲面的曲率估计，并利用复分析方法给出尖锐估计，最终证明了该长期未决的猜想。
The Willmore conjecture states that any immersion F:T^2 -> R^n of a 2-torus into flat euclidean space satisfies $\int_{T^2} H^2\geq 2π^2$. We prove it under the condition that the L^p-norm of the Gaussian curvature is sufficiently small.
We formulate several conjectures on mean convex domains in the Euclidean spaces, as well as in more general spaces with lower bonds on their scalar curvatures, and prove a few theorems motivating these conjectures.
The Besse's conjecture was posted on the well-known book Einstein manifolds by Arthur L. Besse, which describes the critical point of Hilbert-Einstein functional with constraint of unit volume and constant scalar curvature. In this article, we show that there is an interesting connection between Besse's conjecture and positive mass theorem for Brown-York mass. With the aid of positive mass theorem, we investigate the geometric structure of the so-called CPE manifolds, which provides us further understanding of Besse's conjecture. As a related topic, we also have a discussion of corresponding r…
Expanders for Mean Curvature Flow and Counterexamples to Ilmanen's Genus-Reduction Conjecture
本文构造了新的均曲率流扩张器，这些扩张器光滑地趋于某些收缩器产生的锥体。对于每个锥体，证明了存在任意高亏格的扩张器，反驳了伊尔曼恩的亏格减少猜想。
The Hopf sign conjecture states that a compact Riemannian 2d-manifold M of positive curvature has Euler characteristic X(M)>0 and that in the case of negative curvature X(M) (-1)^d >0. The Hopf product conjecture asks whether a positive curvature metric can exist on product manifolds like S^2 x S^2. By formulating curvature integral geometrically, these questions can be explored for finite simple graphs, where it leads to linear programming problems. In this more expository document we aim to explore also a bit of the history of the Hopf conjecture and mention some strategies of attacks which …
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.
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.
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.
On Hopf's conjecture and positive second intermediate Ricci curvature
针对正第二中间 Ricci 曲率下 Hopf 猜想结论可能失效的问题，本文证明维数被四整除且具有秩至少十的环面对称性时，Euler 示性数为正，并推广了 Four Periodicity Theorem。
We show that any normal metric on a closed biquotient with finite fundamental group has positive Ricci curvature.
For biharmonic maps, there is a famous conjecture named Chen's conjecture. In later paper, Wang and Ou gave an affirmative partial answer to submersion version of Chen's conjecture. In this paper, we give an affirmative partial answer to submersion version of generalized Chen's conjecture, that is, triharmonic Riemannian submersions from a 3-dimensional space form into a surface is harmonic.
A Counterexample to Yau's Conjectured Asymptotic Scalar-Curvature Integral Bound
通过结合环面透镜、角裤和自适应混合块，构造了正Ricci曲率完备单端三维流形，其数量曲率积分发散，给出Yau猜想反例。
On a compact $n$-dimensional manifold, it is well known that a critical metric of the total scalar curvature, restricted to the space of metrics with unit volume is Einstein. It has been conjectured that a critical metric of the total scalar curvature, restricted to the space of metrics with constant scalar curvature of unit volume, will be Einstein. This conjecture was proposed in 1987 by Besse, but has yet to be proved. In this paper, we prove the Besse conjecture with a weaker condition than harmonic curvature for $n\geq 3$.
We classify the connected pseudo-Riemannian manifolds of signature $(p,q)$ with $q\ge5$ so that at each point of $M$ the skew-symmetric curvature operator has constant rank 2 and constant Jordan normal form on the set of spacelike 2 planes and so that the skew-symmetric curvature operator is not nilpotent for at least one point of $M$.
Chern Conjecture on Minimal Willmore Hypersurfaces with Constant Scalar Curvature
本文证明了具有常标量曲率的极小Willmore超曲面的Chern猜想，通过建立关于二阶基本形式平方范数的不等式条件，验证了第二间隙的猜想。
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。
Let M be a compact pseudo-umbilical submanifold of the unit sphere S. In the present note, it is shown that if the normal curvature, scalar curvature S and square of the length of second fundamental form satisfy certain conditions, then M is totally geodesic.
In this paper, we study normal complex contact metric manifolds and we get some general results on them. Moreover, we obtained the general expression of the curvature tensor field for arbitrary vector fields. Furthermore, we show that the necessary and succient conditions to be normal a complex contact metric manifold. Also, we give some new equailities for Ricci curvature of normal complex contact metric manifolds.
In this paper, we give a proof of the DDVV conjecture which is a pointwise inequality involving the scalar curvature, the normal scalar curvature and the mean curvature on a submanifold of a real space form. Furthermore we solved the problem of its equality case.
We propose a new approach to the study of compact Riemannian manifolds with nonnegative Ricci curvature and strictly convex boundary or positive Ricci curvature and convex boundary. Several conjectures are formulated. Some partial results that support these conjectures are established.
In this paper, we study the following conjecture of Hamilton: Any compact gradient shrinking Ricci soliton with positive curvature operator must be Einstein. We first derive several identities. Then we show that the conjecture is true under an additional condition. Furthermore, such a soliton must be of constant curvature.
On a compact $n$-dimensional manifold $M$, it is well known that a critical metric of the total scalar curvature, restricted to the space of metrics with unit volume, is Einstein. It has been conjectured that a critical metric of the total scalar curvature, restricted to the space of metrics with constant scalar curvature of unit volume, will also be Einstein. It was shown that this conjecture is true when $M$ together with a critical metric has harmonic curvature or the metric is Bach flat. In this paper, we tried to prove this conjecture with a divergence-free Bach tensor.
Immersions with small normal curvature
本文研究了单位球体内最小法向曲率浸入问题，确定了$S^n\times S^1$的法向曲率最小值，并证明了可微球体定理和极小解的存在性。
In this paper we consider Lorentzian surfaces in the 4-dimensional pseudo-Riemannian sphere $\mathbb S^4_2(1)$ with index 2 of curvature one. We obtain the complete classification of minimal Lorentzian surfaces $\mathbb S^4_2(1)$ whose Gaussian and normal curvatures are constants. We conclude that such surfaces have the Gaussian curvature $1/3$ and the absolute value of normal curvature $2/3$. We also give some explicit examples.
On a compact n-dimensional manifold, it has been conjectured that a critical point metric of the total scalar curvature, restricted to the space of metrics with constant scalar curvature of unit volume, will be Einstein. This conjecture was proposed in 1984 by Besse, but has yet to be proved. In this paper, we prove that if the manifold with the critical point metric has harmonic curvature, then it is isometric to a standard sphere.
A well known conjecture in complex geometry states that a compact Hermitian manifold with constant holomorphic sectional curvature must be Kähler if the constant is non-zero and must be Chern flat if the constant is zero. The conjecture is confirmed in complex dimension $2$, by the work of Balas-Gauduchon in 1985 (when the constant is zero or negative) and by Apostolov-Davidov-Muskarov in 1996 (when the constant is positive). For higher dimensions, the conjecture is still largely unknown. In this article, we restrict ourselves to the class of complex nilmanifolds and confirm the conjecture in …
We study a pointwise inequality for submanifolds in real space forms involving the scalar curvature, the normal scalar curvature and the mean curvature. We translate it into an algebraic problem, allowing us to prove a slightly weaker version of it. We also prove the conjecture for certain types of submanifolds of $\mathbb C^n$.
Counterexamples to preservation of flat normal bundles under mean curvature flow
本文构造反例，证明平均曲率流下平坦法丛性质不保持，并给出平行主法向条件不保持的例子。
In this paper we form relations for the determination of the elements of the Eötvös matrix of the Earth's normal gravity field. In addition a relation between the Gauss curvature of the normal equipotential surface and the Gauss curvature of the actual equipotential surface both passing through the point P is presented. For this purpose we use a global Cartesian system (X, Y, Z) and use the variables X, and Y to form a local parameterization a normal equipotential surface to describe its fundamental forms and the plumbline curvature. The first and second order partial derivatives of the normal…
We collect a few guesses on possible implications of a lower bound on the scalar curvature of a Riemannian manifold on the size and shape of this manifold.
On the Scalar Curvature Compactness Conjecture in the Conformal Case
本文研究共形于n维标准球面的正数量曲率黎曼流形序列的紧性，证明共形因子在解析意义下紧致并在小体积奇异集外C^0收敛，且在总数量曲率有界时极限具有弱正数量曲率。
A proof of the Cartan-Hadamard conjecture for small volumes under a Ricci curvature lower bound
本文在 Ricci 曲率下界条件下证明了小体积情形的广义 Cartan-Hadamard 猜想，并给出任意体积情形向等式刚性归约的关键简化。
The Han-Li conjecture states that: Let $(M,g_0)$ be an $n$-dimensional $(n\geq 3)$ smooth compact Riemannian manifold with boundary having positive (generalized) Yamabe constant and $c$ be any real number, then there exists a conformal metric of $g_0$ with scalar curvature $1$ and boundary mean curvature $c$. Combining with Z. C. Han and Y. Y. Li's results, we answer this conjecture affirmatively except for the case that $n\geq 8$, the boundary is umbilic, the Weyl tensor of $M$ vanishes on the boundary and has a non-zero interior point.
Veronese minimizes normal curvatures
本文研究高维欧氏球中闭子流形的几何刚性，通过建立法曲率的最优界证明其必为球面，并刻画四个射影平面的 Veronese 嵌入为边界情形。
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。
We construct new examples of algebraic curvature tensors so that the Jordan normal form of the higher order Jacobi operator is constant on the Grassmannian of subspaces of type $(r,s)$ in a vector space of signature $(p,q)$. We then use these examples to establish some results concerning higher order Osserman and higher order Jordan Osserman algebraic curvature tensors.
In the first part of this paper, we prove the extensibility of an arbitrary boundary metric to a positive scalar curvature (PSC) metric inside for a compact manifold with boundary, which completely solves an open problem due to Gromov (see Question \ref{extension1}). Then we introduce a fill-in invariant (see Definition \ref{fillininvariant}) and discuss its relationship with the positive mass theorems for asymptotically flat (AF) and asymptotically hyperbolic (AH) manifolds. Moreover, we prove that the positive mass theorem for AH manifolds implies that for AF manifolds. In the end, we give s…
On Hermitian manifolds with constant mixed curvature
本文验证了Kai Tang关于非零常数混合曲率紧致厄米流形必为凯勒流形的猜想，针对复幂零流形、可解流形等特殊类型及平衡三维流形，提供了部分证据。
The main result of this paper was already obtained in the paper `Some Remarks on the Geometry of Austere Manifolds', by Robert L. Bryant.
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.
For any closed smooth Riemannian manifold H. Weyl has defined a sequence of numbers called today intrinsic volumes. They include volume, Euler characteristic, and integral of the scalar curvature. We conjecture that absolute values of all intrinsic volumes are bounded by a constant depending only on the dimension of the manifold, upper bound on its diameter, and lower bound on the sectional curvature. Furthermore we conjecture that intrinsic volumes can be defined for some (so called smoothable) Alexandrov spaces with curvature bounded below and state few of the expected properties of them, pa…
The well known Chen's conjecture on biharmonic submanifolds states that a biharmonic submanifold in a Euclidean space is a minimal one ([10-13, 16, 18-21, 8]). For the case of hypersurfaces, we know that Chen's conjecture is true for biharmonic surfaces in $\mathbb E^3$ ([10], [24]), biharmonic hypersurfaces in $\mathbb E^4$ ([23]), and biharmonic hypersurfaces in $\mathbb E^m$ with at most two distinct principal curvatures ([21]). The most recent work of Chen-Munteanu [18] shows that Chen's conjecture is true for $δ(2)$-ideal hypersurfaces in $\mathbb E^m$, where a $δ(2)$-ideal hypersurface i…
Rigidity of Critical Point Metrics under some Ricci curvature constraints
本文在特定里奇曲率约束下证明临界点度量的刚性性，证明在traceless里奇算子范数为常数或满足特定条件时，临界点度量为爱因斯坦度量。
Biharmonic Wintgen ideal submanifolds in Riemannian manifolds of constant sectional curvature
本文证明非正常截面曲率黎曼流形中的双调和Wintgen理想子流形是极小的，正常截面曲率情形下每个连通分量具有常平均曲率，部分肯定了Chen猜想、双曲空间中的广义Chen猜想以及球面中的Balmuş-Montaldo-Oniciuc猜想。
Cao's splitting theorem says that for any complete Kähler-Ricci flow $(M,g(t))$ with $t\in [0,T)$, $M$ simply connected and nonnegative bounded holomorphic bisectional curvature, $(M,g(t))$ is holomorphically isometric to $\C^k\times (N,h(t))$ where $(N,h(t))$ is a Kahler-Ricci flow with positive Ricci curvature for $t>0$. In this article, we show that $k=n-r$ where $r$ is the Ricci rank of the initial metric. As a corollary, we also confirm a splitting conjecture of Wu-Zheng when curvature is assumed to be bounded.
We prove that for the mean curvature flow of closed embedded hypersurfaces, the intrinsic diameter stays uniformly bounded as the flow approaches the first singular time, provided all singularities are of neck or conical type. In particular, assuming Ilmanen's multiplicity one conjecture and no cylinder conjecture, we conclude that in the two-dimensional case, the diameter always stays bounded. We also obtain sharp $L^{n-1}$ bound for the curvature. The key ingredients for our proof are the Lojasiewicz inequalities by Colding-Minicozzi and Chodosh-Schulze, and the solution of the mean-convex n…
We prove some rigidity results for compact manifolds with boundary. In particular for a compact Riemannian manifold with nonnegative Ricci curvature and simply connected mean convex boundary, it is shown that if the sectional curvature vanishes on the boundary, then the metric must be flat.
This note corrects a mistake in the original book in the evolution equations of total curvature for the curve-shrinking flow in an ambient Ricci Flow. The resulting upper bound for the evolution of total curvature is an exponential bound in time. The change involves the multiplicative constant. Here we show that it depends on the initial total curvature and the initial length, rather than just on the initial total curvature as was asserted before. This change does not affect the application of these results to prove finite-time extinction when the third homotopy group of the manifold is non-tr…
A long-standing conjecture in complex geometry says that a compact Hermitian manifold with constant holomorphic sectional curvature must be Kähler when the constant is non-zero and must be Chern flat when the constant is zero. The conjecture is known in complex dimension $2$ by the work of Balas-Gauduchon in 1985 (when the constant is zero or negative) and by Apostolov-Davidov-Muskarov in 1996 (when the constant is positive). For higher dimensions, the conjecture is still largely unknown. In this article, we restrict ourselves to pluriclosed manifolds, and confirm the conjecture for the specia…
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