Proof of Chern's conjecture on affine manifolds

一般仿射群上的Chern-Simons场论、3d引力与Cartan联络的延拓

Chern-Simons field theory on the general affine group, $3d$-gravity and the extension of Cartan connections

该研究从几何力学角度探究结构群为一般仿射群时3d引力与Chern-Simons场论的对应，通过推广Cartan联络下的联络延拓约化概念，建立了主纤维丛上Chern-Simons型变分问题与常规理论的关联。

Chern conjecture on minimal hypersurfaces

In this paper, we study $n$-dimensional complete minimal hypersurfaces in a unit sphere. We prove that an $n$-dimensional complete minimal hypersurface with constant scalar curvature in a unit sphere with $f_3$ constant is isometric to the totally geodesic sphere or the Clifford torus if $S\leq 1.8252 n-0.712898$, where $S$ denotes the squared norm of the second fundamental form of this hypersurface.

A note on the Chern Conjecture in dimension four

Let $M^4$ be a closed immersed minimal hypersurface with constant squared length of the second fundamental form $S$ and constant 3-mean curvature $H_3$ in $\mathbb{S}^{5}$. If $H_3^2\leq \frac{1}{2}.$ and Gauss-Kronecker curvature $K_M$ satisfies $K_M\leq1$ $($or $ K_M\leq\frac{S^2}{144}$$)$, then $M^4$ is isoparametric.

Chern猜想关于具有常标量曲率的极小Willmore超曲面

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。

具有常三次迹的陈猜想

Chern's Conjecture with Constant Cubic Trace

该研究证明单位球面中满足S与f₃均为常数的闭嵌入极小超曲面的S值集合局部有限，拓展了陈猜想的相关结论。

On analogues of Bäcklund theorem in affine differential geometry of surfaces

We recall the well-known Chern-Terng theorem concerning affine minimal surfaces. Next we formulate some complementary (with transversal fields necessarily not parallel) affine Bäcklund theorem. We describe some geometrical conditions which imply the local symmetry of both induced connections. We give also some necessary and sufficient conditions under which the affine fundamental forms are proportional.

The Auslander conjecture for NIL-affine crystallographic groups

Let N be a simply connected, connected real nilpotent Lie group of finite dimension n. We study subgroups $Γ$ in $\Aff (N)=N\rtimes \Aut (N)$ acting properly discontinuously and cocompactly on N. This situation is a natural generalization of the so-called affine crystallographic groups. We prove that for all dimensions $1\le n\le 5$ the generalized Auslander conjecture holds, i.e., that such subgroups are virtually polycyclic.

Bott-Chern homology, Bott-Chern differential cohomology and the Hodge conjecture

We propose a version of the Hodge conjecture in Bott-Chern cohomology and using results from characterizing real holomorphic chains by real rectifiable currents to provide a proof for this question. We define a Bott-Chern differential cohomology and use atomic section theory of Harvey and Lawson to construct refined Bott-Chern classes for holomorphic vector bundles in this differential cohomology. These refined Bott-Chern classes transform naturally to standard Chern classes, Bott-Chern classes and Cheeger-Simons' refined Chern classes.

通过Calabi仿射几何构造的新的非二次欧氏完备仿射极大型超曲面

New non-quadratic Euclidean complete affine maximal type hypersurfaces via Calabi affine geometry

本文从子流形理论与Calabi仿射几何视角研究仿射极大型曲面，刻画了已有显式欧氏完备反例的几何特征，并构造新的非二次欧氏完备仿射极大型超曲面，拓展了对应参数的取值范围。

New result on Chern conjecture for minimal hypersurfaces and its application

We verify that if $M$ is a compact minimal hypersurface in $\mathbb{S}^{n+1}$ whose squared length of the second fundamental form satisfying $0\leq |A|^2-n\leq\frac{n}{22}$, then $|A|^2\equiv n$ and $M$ is a Clifford torus. Moreover, we prove that if $M$ is a complete self-shrinker with polynomial volume growth in $\mathbb{R}^{n+1}$ whose equation is given by (\ref{selfshr}), and if the squared length of the second fundamental form of $M$ satisfies $0\leq|A|^2-1\leq\frac{1}{21}$, then $|A|^2\equiv1$ and $M$ is a round sphere or a cylinder. Our results improve the rigidity theorems due to Q. Di…

常三次迹条件下陈省身猜想的第二间隙与刚性

The Second Gap and Rigidity in Chern's Conjecture with Constant Cubic Trace

本文证明常三次迹条件下闭极小超曲面的第二间隙 $S\geq2n$，并刻画等号情形为三维、六维、十二维及二十四维中的极小 Cartan 等参超曲面。

Chern conjecture and isoparametric hypersurfaces

In this note we will review the most important results and questions related to Chern conjecture and isoparametric hypersurfaces, as well as their interactions and applications to various aspects in mathematics.

Dupin 情形下的 Chern 猜想

Chern's Conjecture in the Dupin case

本文在 Dupin 超曲面情形下研究 Chern 猜想，证明了在 $g=3,4,6$ 等条件下具有常数曲率的闭正常 Dupin 超曲面是等参的，并运用拓扑与几何方法减少了代数论证所需的假设。

关于Chern猜想的一个改进结果

An improved result on Chern conjecture

本文针对Chern猜想，在中间问题（若S>n则S>n+1/2 n）上取得进展：对4≤n≤13给出肯定解，并对一般n证明S>1.4556737869 n，优于Chen的结果。

具有常Chern全纯截面曲率的Bismut挠平行埃尔米特流形

Bismut-Torsion-Parallel Hermitian Manifolds With Constant Chern Holomorphic Sectional Curvature

本文证明了所有复维数下平衡Bismut挠平行埃尔米特流形的非零Chern全纯截面曲率情形，确认该类紧流形若Chern全纯截面曲率非零则必为凯勒流形，完善了相关猜想的证明。

关于球的极小子流形的陈省身猜想

On Chern's conjecture for minimal submanifolds of the sphere

本文针对球的极小子流形的陈省身猜想，证明其在n≥3且m≥4、偶n≥4且m≥3的情形不成立，补充了该猜想的反例情形。

Affine $Λ$-buildings, ultrapowers of Lie groups and Riemannian symmetric spaces: an algebraic proof of the Margulis conjecture

In this paper, we give a general group-theoretic construction of affine $\RR$-buildings, and more generally, of affine $Λ$-buildings, associated to semisimple Lie groups over nonarchimedean real closed fields. The construction of Kleiner-Leeb using the asymptotic cone of a Riemannian symmetric space appears as a special case. The explicit knowledge of the building arising here as the asymptotic cone simplifies the proof of the Margulis conjecture due to Kleiner-Leeb.

Reciprocity of the Chern-Simons invariants of 3-manifolds

We pose reciprocity conjectures of the Chern-Simons invariants of 3-manifolds, and discuss some supporting evidence on the conjectures. Especially, we show that the conjectures hold if a Galois descent of a $K_3$-group is satisfied.

A note on Griffiths' conjecture about the positivity of Chern-Weil forms

Let $ (E,h) $ be a Griffiths semipositive Hermitian holomorphic vector bundle of rank $ 3 $ over a complex manifold. In this paper, we prove the positivity of the characteristic differential form $ c_1(E,h) \wedge c_2(E,h) - c_3(E,h) $, thus providing a new evidence towards a conjecture by Griffiths about the positivity of the Schur polynomials in the Chern forms of Griffiths semipositive vector bundles. As a consequence, we establish a new chain of inequalities between Chern forms. Moreover, we point out how to obtain the positivity of the second Chern form $ c_2(E,h) $ in any rank, starting …

Lectures on the Euler characteristic of affine manifolds

These are lecture notes prepared for the summer school "Geometric, algebraic and topological methods in quantum field theory", held in Villa de Leyva in July 2017. Our goal is to provide an introduction to a conjecture of Chern that states that the Euler characteristic of a closed affine manifold vanishes. We present part of the history and motivation for the conjecture as well as some recent developments. All comments and corrections are most welcome!

Chern class inequalities on polarized manifolds and nef vector bundles

This article is concerned with Chern class and Chern number inequalities on polarized manifolds and nef vector bundles. For a polarized pair $(M,L)$ with $L$ very ample, our first main result is a family of sharp Chern class inequalities. Among them the first one is a variant of a classical result and the equality case of the second one is a characterization of hypersurfaces. The second main result is a Chern number inequality on it, which includes a reverse Miyaoka-Yau type inequality. The third main result is that the Chern numbers of a nef vector bundle over a compact Kähler manifold are bo…

余辛陈-哈密顿猜想

Cosymplectic Chern--Hamilton conjecture

本文研究了紧致余辛流形上的陈-哈密顿能量泛函，证明了三维流形临界兼容度量的条件，并展示了某些流形不存在临界兼容度量的例子。

线性 Chern-Hopf-Thurston 猜想

Linear Chern-Hopf-Thurston conjecture

本文证明了当复射影流形的基本群在任意域上具有几乎忠实线性表示时的 Chern-Hopf-Thurston 猜想，通过引入多值一形式的消没圈函子及非阿贝尔 Hodge 理论技术，得出了关于反常层欧拉示性数的更强正定性结论。

Chern--Simons Perturbation Theory II

In a previous paper [\AS], we used superspace techniques to prove that perturbation theory (around a classical solution with no zero modes) for Chern--Simons quantum field theory on a general $3$-manifold $M$ is finite. We conjectured (and proved for the case of $2$-loops) that, after adding counterterms of the expected form, the terms in the perturbation theory define topological invariants. In this paper we prove this conjecture. Our proof uses a geometric compactification of the region on which the Feynman integrand of Feynman diagrams is smooth as well as an extension of the basic propagat…

On the Existence of Stable bundles with prescribed Chern classes on Calabi-Yau threefolds

We prove a case of the conjecture of Douglas, Reinbacher and Yau about the existence of stable vector bundles with prescribed Chern classes on a Calabi-Yau threefold. For this purpose we prove the existence of certain stable vector bundle extensions over elliptically fibered Calabi-Yau threefolds.