Constant Mean Curvature Surfaces in Homology Classes

Systolic volume of homology classes

Given an integer homology class of a finitely presentable group, the systolic volume quantifies how tight could be a geometric realization of this class. In this paper, we study various aspects of this numerical invariant showing that it is a complex and powerful tool to investigate topological properties of homology classes of finitely presentable groups.

Scalar curvature and vector bundles

In the first part we use Gromov's K--area to define the K--area homology which stabilizes into singular homology on the category of pairs of compact smooth manifolds. The second part treats the questions of certain curvature gaps. For instance, the $L^\infty $--curvature gap of complex vector bundles on a compact manifold is positive if and only if the K--area homology coincides with the reduced singular homology in all even degrees. In the third part we give some upper bounds of the scalar curvature on compact manifolds. In particular, we generalize results by Llarull and Goette, Semmelmann.

扭曲悬架上的正 Ricci 曲率

Positive Ricci Curvature on Twisted Suspensions

本文证明正 Ricci 曲率度量可沿扭曲悬架提升，并据此确定 n≥4 维闭单连通正 Ricci 曲率流形的最大对称秩为 n−2，同时构造多类新例。

Rational Homology 5-Spheres with Positive Ricci Curvature

We prove that for every integer k>1 there is a simply connected rational homology 5-sphere $\scriptstyle{M^5_k}$ with spin such that $\scriptstyle{H_2(M^5_k,\bbz)}$ has order $\scriptstyle{k^2},$ and $\scriptstyle{M^5_k}$ admits a Riemannian metric of positive Ricci curvature. Moreover, if the prime number decomposition of $\scriptstyle{k}$ has the form $\scriptstyle{k=p_1... p_r}$ for distinct primes $\scriptstyle{p_i}$ then $\scriptstyle{M^5_k}$ is uniquely determined.

Anosov diffeomorphisms of products I. Negative curvature and rational homology spheres

We show that various classes of products of manifolds do not support transitive Anosov diffeomorphisms. Exploiting the Ruelle-Sullivan cohomology class, we prove that the product of a negatively curved manifold with a rational homology sphere does not support transitive Anosov diffeomorphisms. We extend this result to products of finitely many negatively curved manifolds of dimensions at least three with a rational homology sphere that has vanishing simplicial volume. As an application of this study, we obtain new examples of manifolds that do not support transitive Anosov diffeomorphisms, inc…

Positive scalar curvature and low-degree group homology

Let $Γ$ be a discrete group. Assuming rational injectivity of the Baum-Connes assembly map, we provide new lower bounds on the rank of the positive scalar curvature bordism group and the relative group in Stolz' positive scalar curvature sequence for $\mathrm{B} Γ$. The lower bounds are formulated in terms of the part of degree up to $2$ in the group homology of $Γ$ with coefficients in the $\mathbb{C}Γ$-module generated by finite order elements. Our results use and extend work of Botvinnik and Gilkey which treated the case of finite groups. Further crucial ingredients are a real counterpart t…

Upper bounds for the critical values of homology classes of loops

In this short note we discuss upper bounds for the critical values of homology classes in the based and free loop space of manifolds carrying a Riemannian or Finsler metric of positive Ricci curvature. In particular it follows that a shortest closed geodesic on a simply-connected $n$-dimensional manifold of positive Ricci curvature $\textrm{Ric} \ge n-1$ has length $\le n π.$

Floer homology of algebraically finite mapping classes

We compute the Floer homology of mapping classes which do not have any pseudo-Anosov components in the sense of Thurston's theory of surface diffeomorphisms. The formula for the Floer homology is obtained from a topological separation of fixed points and a separation mechanism for Floer connecting orbits. As examples, we consider the geometric monodromy of isolated plane curve singularities.

Homology and modular classes of Lie algebroids

For a Lie algebroid, divergences chosen in a classical way lead to a uniquely defined homology theory. They define also, in a natural way, modular classes of certain Lie algebroid morphisms. This approach, applied for the anchor map, recovers the concept of modular class due to S. Evans, J.-H. Lu, and A. Weinstein.

A characterization of real holomorphic chains and applications in representing homology classes by algebraic cycles

We show that a $2k$-current $T$ on a complex manifold is a real holomorphic $k$-chain if and only if $T$ is locally real rectifiable, $d$-closed and has $\mathcal{H}^{2k}$-locally finite support. This result is applied to study homology classes represented by algebraic cycles.

在给定同调类中等价min-max理论的单重性

Multiplicity one for equivariant min-max theory in prescribed homology classes

本文研究了在给定同调类中等价min-max理论的单重性，并证明了存在无限多个$G$-不变极小超曲面，同时建立了具有$G$-索引上界的等价min-max理论。

K-homology classes of elliptic uniform pseudodifferential operators

We show that an elliptic uniform pseudodifferential operator over a manifold of bounded geometry defines a class in uniform K-homology, and that this class only depends on the principal symbol of the operator.

Critical values of homology classes of loops and positive curvature

We study compact and simply-connected Riemannian manifolds with positive sectional curvature $K\ge 1.$ For a non-trivial homology class of lowest dimension in the space of loops based at a point $p$ or in the free loop space one can define a critical length ${\sf crl}_p\left(M,g\right)$ resp. ${\sf crl}\left(M,g\right).$ Then ${\sf crl}_p\left(M,g\right)$ equals the length of a geodesic loop and ${\sf crl}\left(M,g\right)$ equals the length of a closed geodesic. This is the idea of the proof of the existence of a closed geodesic of positive length presented by Birkhoff in case of a sphere and …

微分同胚群、嵌入与正标量曲率度量的族：基于Seiberg-Witten理论

Families of diffeomorphisms, embeddings, and positive scalar curvature metrics via Seiberg-Witten theory

本文通过Seiberg-Witten理论构造4维流形微分同胚群的高阶同伦和同调中的无限秩直和项，并应用于嵌入空间和正标量曲率度量空间的无限生成性。

Odd characteristic classes in entire cyclic homology and equivariant loop space homology

Given a compact manifold $M$ and $g\in C^{\infty}(M,U(l;\mathbb{C}))$ we construct a Chern character $\mathrm{Ch}^-(g)$ which lives in the odd part of the equivariant (entire) cyclic Chen-normalized bar complex $\underline{\mathscr{C}}(Ω_{\mathbb{T}}(M\times \mathbb{T}))$ of $M$, and which is mapped to the odd Bismut-Chern character under the equivariant Chen integral map. It is also shown that the assignment $g\mapsto \mathrm{Ch}^-(g)$ induces a well-defined group homomorphism from the $K^{-1}$ theory of $M$ to the odd homology group of $\underline{\mathscr{C}}(Ω_{\mathbb{T}}(M\times \mathbb{…

定向格拉斯曼包、法向曲率降维与持久同调

Oriented Grassmannian Bundle, Normal Curvature Reduction, and Persistent Homology

本文提出通过定向格拉斯曼包嵌入光滑闭合子流形，以降低法向曲率并增强持久同调的稳定性，从而在特定尺度下恢复流形的同调结构。

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.

Curvature bounds of subsets in dimension two

We show that closed subsets with vanishing first homology in two-dimensional spaces inherit the upper curvature bound from their ambient spaces and discuss topological applications.

Relative K-area homology and applications

We show an Uhlenbeck type estimate for closed simply connected manifolds which provides the existence of certain exact sequences in K-area homology. This leads to the behavior of the K-area homology under surgery. Moreover, we give an index theoretic obstruction to positive scalar curvature on compact spin manifolds with boundary.

Eternal forced mean curvature flows I - A compactness result

With a view to constructing a Morse/Floer homology theory for CMC hypersurfaces, we prove a compactness result modulo broken trajectories for eternal mean curvature flows with forcing term in compact, hyperbolic manifolds.

关于无球面流形中超曲面有理同调消失定理的一个注记

A note on rational homology vanishing theorem for hypersurfaces in aspherical manifolds

本文将Gromov约化方法推广至超曲面的Q-同调消失猜想，证明闭4维正数量曲率流形到无球面5维流形的映射在H_4上为零，并得到非负数量曲率两端的完备无球面5维流形分裂为闭平坦流形与实数直线的乘积。

On the quantization of conjugacy classes

Let G be a compact, simple, simply connected Lie group. A theorem of Freed-Hopkins-Teleman identifies the level k fusion ring R_k(G) of G with the twisted equivariant K-homology at level k+h, where h is the dual Coxeter number. In this paper, we review this result using the language of Dixmier-Douady bundles. We show that the additive generators of the group R_k(G) are obtained as K-homology push-forwards of the fundamental classes of conjugacy classes in G.

Optimal lower eigenvalue estimates for Hodge-Laplacian and applications

In this paper, we consider the eigenvalue problem for Hodge-Laplacian on a Riemannian manifold $M$ isometrically immersed into another Riemannian manifold $\bar M$ for arbitrary codimension. We first assume the pull back Weitzenböck operator (defined in Section 2) of $\bar M$ bounded from below, and obtain an extrinsic lower bound for the first eigenvalue of Hodge-Laplacian. As applications, we obtain some rigidity results and a homology sphere theorem. Second, when the pull back Weitzenböck operator of $\bar M$ bounded from both sides, we give a lower bound of the first eigenvalue by the Ricc…

Stiefel-Whitney Classes and the Conormal Cycle of a Singular Variety

A geometric construction of Sullivan's Stiefel-Whitney homology classes of a real analytic variety $X$ is given by means of the conormal cycle of an embedding of $X$ in a smooth variety. We prove that the Stiefel-Whitney classes define additive natural transformations from certain constructible functions to homology. We also show that, for a complex analytic variety, these classes are the mod 2 reductions of the Chern-MacPherson classes.

Nonrigidity of a class of two dimensional surfaces with positive curvature and planar points

Nontrivial infinitesimal bendings for a class of two-dimensional surfaces are constructed. The surfaces considered here are orientable; compact; with boundary; have positive curvature everywhere except at finitely many planar points; and have vanishing first homology group.As a consequence, a nonrigidity result for such surfaces is deduced.