Georgi Ganchev, Ognian Kassabov
In dimension greater than four, we prove that if a Hermitian non-Kaehler manifold is of pointwise constant antiholomorphic sectional curvatures, then it is of constant sectional curvatures.
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A classification theorem for nearly Kähler manifolds of constant antiholomorphic sectional curvature is proved.
It is proved that if an AK2-manifold of dimension greater or equal to 6 is of pointwise constant antiholomorphic sectional curvature, then it is a 6-dimensional manifold of constant negative sectional curvature or a Kähler manifold of constant holomorphic sectional curvature.
The following theorem is proved: If an AH3-manifold M of dimension greather or equal to 6 is of pointwise constant antiholomorphic sectional curvature, then M is a real space form or a complex space form.
It is proved that if an almost Kähler manifold of dimension greater or equal to 8 is of pointwise constant antiholomorphic sectional curvature, then it is a complex space form.
We prove the following results: An almost Hermitian manifold of indefinite metric is of pointwise constant holomorphic sectional curvature if the holomorphic sectional curvature is bounded from above and from below. If the antiholomorphic sectional curvature is bounded either from above or from below, then the manifold is of pointwise constant antiholomorphic sectional curvature. Similar results are obtained for almost Hermitian manifolds of definite metric.
Criterions for constancy of the holomorphic sectional curvature and the antiholomorphic sectional curvature are proved for almost Hermitian manifolds. It is shown, that an almost Hermitian manifold satisfying the axiom of antiholomorphic planes or the axiom of antiholomorphic spheres is a real or a complex space form.
It is proved, that if a quasi-Kähler manifold $M$ of dimension greater or equal to 6 is of pointwise constant antiholomorphic sectional curvature $ν$, then $ν$, the scalar curvature and the $*$-scalar curvature of $M$ are constants.
Let M be an almost Hermitian manifold of dimension greater or equal to 6. The following theorems are proved: Theorem 1. If M is of pointwise constant θ-holomorphic sectional curvature for a number θ in (0,π/2) then M is of constant sectional curvature or a Kähler manifold of constant holomorphic sectional curvature. Theorem 2. If M is of pointwise constant antiholomorphic sectional curvature and M is an RK-manifold (or AH3-manifold), then M is of constant antiholomorphic sectional curvature.
The Schur's theorem of antiholomorphic type is proved for arbitrary almost Hermitian manifolds, namely: If a connected almost Hermitian manifold of dimension greater or equal to 6 is of pointwise constant antiholomorphic sectional curvature, then this curvature is a global constant.
We study a class of Riemannian manifolds with respect to the covariant derivative of their curvature tensors. We introduce geometrically the class of directed Riemannian manifolds of pointwise constant relative sectional curvature and give a tensor characterization for such manifolds. We prove that all rotational hypersurfaces are directed and find the rotational hypersurfaces of pointwise constant relative sectional curvature. For the class of directed Riemannian manifolds of pointwise constant relative sectional curvature having a totally umbilical scalar distribution we prove a structural t…
Motivated by a recent work of Chen-Zheng [8] on Strominger space forms, we prove that a compact Hermitian surface with pointwise constant holomorphic sectional curvature with respect to a Gauduchon connection $\nabla^t $ is either Kähler, or an isosceles Hopf surface with an admissible metric and $t=-1$ or $t=3$. In particular, a compact Hermitian surface with pointwise constant Lichnerowicz holomorphic sectional curvature is Kähler. We further generalize the result to the case for the two-parameter canonical connections introduced by Zhao-Zheng [30], which extends a previous result by Apostol…
We study the asymptotic Dirichlet problem for the minimal graph equation on a Cartan-Hadamard manifold $M$ whose radial sectional curvatures outside a compact set satisfy an upper bound $$K(P)\le - \frac{ϕ(ϕ-1)}{r(x)^2}$$ and a pointwise pinching condition $$|K(P)|\le C_K|K(P')|$$ for some constants $ϕ>1$ 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 for dimensions $n>4/ϕ+1$.
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.
An isometric immersion $f: M^{n} \rightarrow \tilde M^{m}$ from an $n$-dimensional Riemannian manifold $M^{n}$ into an almost Hermitian manifold $\tilde M^{m}$ of complex dimension $m$ is called pointwise slant if its Wirtinger angles define a function defined on $M$. In this paper we establish the existence and uniqueness theorems for pointwise slant immersions of Riemannian manifolds $M^{n}$ into a complex space form $\tilde M^{n}(c)$ of constant holomorphic sectional curvature $c$.
We studied the axiom of anti-invariant 2-spheres and the axiom of co-holomorphic $(2n+1)$-spheres. We proved that a nearly Kählerian manifold satisfying the axiom of anti-invariant 2-spheres is a space of constant holomorphic sectional curvature. We also showed that an almost Hermitian manifold $M$ of dimension $2m\geq6$ satisfying the axiom of co-holomorphic $(2n+1)$-spheres for some $n,$ where $(1\leq n\leq m-1)$, the manifold $M$ has pointwise constant type $α$ if and only if $M$ has pointwise constant anti-holomorphic sectional curvature $α$.
The paper provides a different proof of the result of Brendle-Schoen on the differential sphere theorem. It is shown directly that the invariant cone of curvature operators with positive (or non-negative) complex sectional curvature is preserved by the Ricci flow. This implies, by a result of Böhm-Wilking, that the normalized Ricci flow deforms such a metric to a metric of constant positive curvature. Using earlier work of Yau and Zheng it can be shown that a metric with strictly (pointwise) 1/4-pinched sectional curvature has positive complex sectional curvature. This gives a direct proof of …
We present some formulae related to the Chern-Ricci curvatures and scalar curvatures of special Hermitian metrics. We prove that a compact locally conformal Kähler manifold with constant nonpositive holomorphic sectional curvature is Kähler. We also give examples of complete non-Kähler metrics with pointwise negative constant but not globally constant holomorphic sectional curvature, and complete non-Kähler metric with zero holomorphic sectional curvature and nonvanishing curvature tensor.
Hermitian connections with parallel torsion and constant holomorphic sectional curvature
本文研究具有平行挠率和常全纯截面曲率的Hermitian联络，证明非零常数迫使挠率为零，从而度量是Kähler的且局部为复空间形式，并应用于Chern和Bismut联络。
A rigidity theorem for Einstein $4$-manifolds with sectional curvature of a fixed sign, and its consequences
研究具有半定截面曲率的4维爱因斯坦度量，通过刻画满足特定逐点不等式的此类度量，得出一系列推论，包括在非正曲率闭爱因斯坦及凯勒 - 爱因斯坦4维流形研究中的应用，还得到了凯勒 - 爱因斯坦情形下的尖锐不等式。
Curvature Sign Rigidity and Sharp Pointwise Pinching Thresholds
本文研究黎曼流形上截面曲率与Ricci曲率符号在逐点夹逼条件下能否共存，通过支撑刚性定理证明尖锐阈值，并给出显式构造与紧化障碍。
Given a closed 3-manifold with an initial Riemannian metric of negative sec- tional curvature, we consider the cross curvature flow an evolution equation of metric on M3. We prove long-time existence of a solution to the cross curvature flow via the maximum principle theorem. Besides, we demonstrate the solution exists for all time and converges pointwisely to a hyperbolic metric
Rigidity of $k$-extremal submanifolds in a sphere
本文研究球面中$k(\ge 1)$-极值子流形的刚性，并在逐点和积分意义上的截面与Ricci曲率等不同条件下证明了多种夹挤定理。
On Kahler manifolds with Ricci curvature lower bound, assuming the real analyticity of the metric, we establish a sharp relative volume comparison theorem for small balls. The model spaces being compared to are complex space forms, i.e, Kahler manifolds with constant holomorphic sectional curvature. Moreover, we give an example showing that on Kahler manifolds, the pointwise Laplacian comparison theorem does not hold when the Ricci curvature is bounded from below.
Sharp Weitzenböck and PIC2 Estimates from Sectional-Scalar Curvature Pinching
研究\(n\)维欧几里得向量空间上代数曲率张量的尖锐估计，通过应用该估计到流形曲率张量分解，在特定挤压条件下得出\(H^2(M;\mathbb{R})\)消失等结论，还给出截面 - 标量挤压准则及相关流形性质。
Some new differentiable sphere theorems are obtained via the Ricci flow and stable currents. We prove that if $M^n$ is a compact manifold whose normalized scalar curvature and sectional curvature satisfy the pointwise pinching condition $R_0>σ_{n}K_{\max}$, where $σ_n\in (\frac{1}{4},1)$ is an explicit positive constant, then $M$ is diffeomorphic to a spherical space form. This gives a partial answer to Yau's conjecture on pinching theorem. Moreover, we prove that if $M^n(n\geq3)$ is a compact manifold whose $(n-2)$-th Ricci curvature and normalized scalar curvature satisfy the pointwise condi…
We derive some integral inequalities for holomorphic maps between complex manifolds. As applications, some rigidity and degeneracy theorems for holomorphic maps without assuming any pointwise curvature signs for both the domain and target manifolds are proved, in which key roles are played by total integration of the function of the first eigenvalue of second Ricci curvature and an almost nonpositivity notion for holomorphic sectional curvature introduced in our previous work. We also apply these integral inequalities to discuss the infinite-time singularity type of the Kähler-Ricci flow. The …
Torsion algebras of Hermitian manifolds and rigidity
该研究将埃尔米特流形的陈挠率视为全纯切丛上的斜对称复双线性乘积，证明类陈-凯勒紧致连通埃尔米特流形的挠率乘积满足雅可比恒等式，强拟正陈全纯截面曲率时挠率李代数幂零且为零，进而推出度量为凯勒度量，还构造了非单连通非有理连通Hopf曲面的正实双截曲率度量。
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.
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 …
A note on Almost Riemann Soliton and gradient almost Riemann soliton
本文研究三维非余辛法向近切触度量流形中的近Riemann孤立子与梯度近Riemann孤立子，推导了相关曲率与流形类型的等价条件，并构造了对应实例。
We prove that a compact Riemannian manifold of dimension $m \geq 3$ with harmonic curvature and $\lfloor\frac{m-1}{2}\rfloor$-positive curvature operator has constant sectional curvature, extending the classical Tachibana theorem for manifolds with positive curvature operator. The condition of $\lfloor\frac{m-1}{2}\rfloor$-positivity originates from recent work of Petersen and Wink, who proved a similar Tachibana-type theorem under the stronger condition that the manifold be Einstein. We show that the same rigidity property holds for complete manifolds assuming either parabolicity, an integral…
Biharmonic hypersurfaces in space forms
本文证明了非正截面曲率空间形式中任意维数的双调和超曲面均为极小，解决了双曲空间中广义陈省身猜想的超曲面情形；对单位球面中的双调和超曲面，推导出定量限制、逐点准则、数量曲率界及局部分类结果，为BMO猜想提供了证据。
This article aims to investigate the curvature operator of the second kind on Kähler manifolds. The first result states that an $m$-dimensional Kähler manifold with $\frac{3}{2}(m^2-1)$-nonnegative (respectively, $\frac{3}{2}(m^2-1)$-nonpositive) curvature operator of the second kind must have constant nonnegative (respectively, nonpositive) holomorphic sectional curvature. The second result asserts that a closed $m$-dimensional Kähler manifold with $\left(\frac{3m^3-m+2}{2m}\right)$-positive curvature operator of the second kind has positive orthogonal bisectional curvature, thus being biholo…
A central theme in Riemannian geometry is understanding the relationships between the curvature and the topology of a Riemannian manifold. Positive isotropic curvature (PIC) is a natural and much studied curvature condition which includes manifolds with pointwisequarter-pinched sectional curvatures and manifolds with positive curvature operator. By results of Micallef and Moore there is only one topological type of compact simply connected manifold with PIC; namely any such manifold must be homeomorphic to the sphere. On the other hand, there is a large class of nonsimply connected manifolds w…
In this paper, we obtain several new intrinsic and extrinsic differential sphere theorems via Ricci flow. For intrinsic case, we show that a closed simply connected $n(\ge 4)$-dimensional Riemannian manifold $M$ is diffeomorphic to $S^n$ if one of the following conditions holds pointwisely: $$ (i)\ R_0>\left(1-\frac{24(\sqrt{10}-3)}{n(n-1)}\right)K_{max};\quad \ (ii)\ \frac{Ric^{[4]}}{4(n-1)}>\left(1-\frac{6(\sqrt{10}-3)}{n-1}\right)K_{max}.$$ Here $K_{max}$, $Ric^{[k]}$ and $R_0$ stand for the maximal sectional curvature, the $k$-th weak Ricci curvature and the normalized scalar curvature. Fo…
Intrinsic and Normal Mean Ricci Curvatures: A Bochner--Weitzenboeck Identity for Simple d-Vectors
本文定义内蕴与正规平均里奇曲率，推导简单 d-向量的 Bochner--Weitzenboeck 恒等式，并应用于调和向量消失准则与 Hodge Laplacian 特征值下界。
A Compactness Theorem for Riemannian Manifolds with Boundary and Applications
本文证明带边黎曼流形在曲率、平均曲率、直径和边界体积一致有界下的$W^{1,p}$和$C^{0,\alpha}$预紧性，并应用于两个几何稳定性定理。
We show existence and uniqueness of very weak solutions of the Cauchy problem for the porous medium equation on Cartan-Hadamard manifolds satisfying suitable lower bounds on Ricci curvature, with initial data that can grow at infinity at a prescribed rate, that depends crucially on the curvature bounds. The curvature conditions we require are sharp for uniqueness in the sense that if they are not satisfied then, in general, there can be infinitely many solutions of the Cauchy problem even for bounded data. Furthermore, under matching upper bounds on sectional curvatures, we give a precise esti…
Ricci pinching on complete Kahler manifolds
本文研究完备凯勒流形的里奇捏缩问题，证明了两类流形的平坦性/里奇平坦性、复维数≥2的特定凯勒流形的紧性，引入里奇形式积分估计等关键要素，完善了相关拓扑里奇质量定理的应用条件。
Pole Regularity of Green function and rigidity
本文研究格林函数的极点正则性与刚性，证明三维的相关结论无高维类比，对不同维数给出曲率条件或反例，揭示极点正则性与流形等距性质的关联。
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