On non-formal simply connected manifolds

Injectivity radius and Cartan polyhedron for simply connected symmetric spaces

We explore relationship between the cut locus of an arbitrary simply connected and compact Riemannian symmetric space and the Cartan polyhedron of corresponding restricted root system, and compute injectivity radius and diameter for every type of irreducible ones.

Simply connected Sasaki-Einstein rational homology 5-spheres

We completely determine which simply connected rational homology 5-spheres admit Sasaki-Einstein metrics.

The Yamabe invariant of simply connected manifolds

We prove that the Yamabe invariant of any simply connected smooth manifold of dimension n greater than four is non-negative. Equivalently that the infimum of the L^{n/2} norm of the scalar curvature, over the space of all Riemannian metrics on the manifold, is zero.

Left invariant generalized complex and Kähler structures on simply connected four dimensional Lie groups: classification and invariant cohomologies

We give a complete classification of left invariant generalized complex structures of type 1 on four dimensional simply connected Lie groups and we compute for each class its invariant generalized Dolbeault cohomology, its invariant generalized Bott-Chern cohomology and its invariant generalized Aeppli cohomology. We classify also left invariant generalized Kähler structures on four dimensional simply connected Lie groups.

Non-simply connected minimal planar domains in H^2 x R

We prove that any non-simply connected planar domain can be properly and minimally embedded in H^2 x R. The examples that we produce are vertical bi-graphs, and they are obtained from the conjugate surface of a Jenkins-Serrin graph.

Simply-connected K-contact and Sasakian manifolds of dimension 7

We construct a compact simply-connected 7-dimensional manifold admitting a K-contact structure but not a Sasakian structure. We also study rational homotopy properties of such manifolds, proving in particular that a simply-connected 7-dimensional Sasakian manifold has vanishing cup-product on the second cohomology and that it is formal if and only if all its triple Massey products vanish.

The classification of compact simply connected biquotients in dimension 6 and 7

We classify all compact simply connected biquotients of dimension 6 and 7. For each $6$-dimensional biquotient, all pairs of groups $(G,H)$ and homomorphisms $H\rightarrow G\times G$ giving rise to it are classified.

The classification of compact simply connected biquotients in dimension 4 and 5

We classify all compact simply connected biquotients of dimension 4 and 5. In particular, all pairs of groups $(G,H)$ and embeddings $H\rightarrow G\times G$ giving rise to a particular biquotient are classified.

一个具有闭测地线的单连通幂零李群

A simply connected nilpotent Lie group with a closed geodesic

给出一个六维单连通幂零李群及左不变黎曼度量，其存在非常值闭测地线，回答了Böhm--Lafuente关于同胚于欧氏空间的齐性黎曼流形能否有闭测地线的问题。

Ricci Limit Spaces Are Semi-locally Simply Connected

Let $(X,p)$ be a Ricci limit space. We show that for any $ε> 0$ and $x \in X$, there exists $r< ε$, depending on $ε$ and $x$, so that any loop in $B_{r}(x)$ is contractible in $B_ε(x)$. In particular, $X$ is semi-locally simply connected. Then we show that the generalized Margulis lemma holds for Ricci limit spaces of $n$-manifolds.

On simply connected K-contact non-Sasakian manifolds

We solve the problem posed by Boyer and Galicki about the existence of K-contact simply connected manifolds with no Sasakian structure. Although the result lies in the framework of metric contact geometry, our methods come from contact and symplectic geometry and are based on the method of fat bundles developed by Sternberg, Weinstein and Lerman.

The conformal group of a compact simply connected Lorentzian manifold

We prove that the conformal group of a closed, simply connected, real analytic Lorentzian manifold is compact. D'Ambra proved in 1988 that the isometry group of such a manifold is compact. Our result implies the Lorentzian Lichnerowicz Conjecture for real analytic Lorentzian manifolds with finite fundamental group. Third version includes corrections and clarifications, particularly in section 6.

Simply-connected open 3-manifolds with slow decay of positive scalar curvature

The goal of this paper is to investigate the topological structure of open simply-connected 3-manifolds whose scalar curvature has a slow decay at infinity. In particular, we show that the Whitehead manifold does not admit a complete metric, whose scalar curvature decays slowly, and in fact that any contractible complete 3-manifolds with such a metric is homeomorphic to $\mathbb{R}^{3}$. Furthermore, using this result, we prove that any open simply-connected 3-manifold $M$ with $π_{2}(M)=\mathbb{Z}$ and a complete metric as above, is homeomorphic to $\mathbb{S}^{2}\times \mathbb{R}$.

收缩梯度凯勒-里奇孤立子是单连通的

Shrinking gradient Kähler-Ricci solitons are simply-connected

本文通过证明光滑极化Fano纤维化不存在非平凡有限覆盖，并结合Sun-Zhang与Wylie的已有结果，得出收缩凯勒-里奇孤立子必为单连通的结论。

Bach Flow of Simply Connected Nilmanifolds

The Bach flow is a fourth order geometric flow defined on four manifolds. For a compact manifold, it is a conformally modified gradient flow for the $L^2$-norm of the Weyl curvature. In this paper we study the Bach flow on four-dimensional simply connected nilmanifolds whose Lie algebra is indecomposable. We show that the Bach flow beginning at an arbitrary left invariant metric exists for all positive times and after rescaling converges in the pointed Cheeger-Gromov sense to an expanding Bach soliton which is non-gradient. Combining our results with previous results of Helliwell gives a compl…

Minimal handle decomposition of smooth simply connected 5-manifolds

A theorem on the existence of the unique minimal topologic handle decomposition of differentiable simply connected five-dimensional manifolds is proved. For a decomposition of this sort, the number of handles of each index is given.

Szegö-Weinberger type inequalities for symmetric domains in simply connected space forms

We consider the Neumann eigenvalue problem for Laplacian on a bounded multi-connected domain contained in simply connected space forms. Under certain symmetry assumptions on the domain, we prove Szegö-Weinberger type inequalities for the first $n$ positive Neumann eigenvalues.

单连通流形上极作用的自相交截面

Self-intersecting sections of polar actions on simply connected manifolds

本文构造了SU(3)在闭单连通11维流形上的极作用，其紧致三维截面非单浸入，对Grove和Ziller的单射问题给出否定答案。

Basic gerbe over non simply connected compact groups

We present an explicit construction of the basic bundle gerbes with connection over all connected compact simple Lie groups. These are geometric objects that appear naturally in the Lagrangian approach to the WZW conformal field theories. Our work extends the recent construction of E. Meinrenken \cite{Meinr} restricted to the case of simply connected groups.

Circle actions on simply connected 5--manifolds

The aim of this paper is to study compact 5--manifolds which admit fixed point free circle actions. The first result implies that the torsion in the second homology and the second Stiefel--Whitney class have to satisfy strong restrictions. We then show that for simply connected 5--manifolds these restrictions are necessary and sufficient.

The classification of left-invariant para-Kähler structures on simply connected four-dimensional Lie groups

We give a complete classification of left invariant para-Kähler structures on four-dimensional simply connected Lie groups up to an automorphism. As an application we discuss some curvatures properties of the canonical connection associated to these structures as flat, Ricci flat and existence of Ricci solitons.

Heat Kernel for Simply-Connected Riemann Surfaces

From the uniformization theorem, we know that every Riemann surface has a simply-connected covering space. Moreover, there are only three simply-connected Riemann surfaces: the sphere, the Euclidean plane, and the hyperbolic plane. In this paper, we collect the known heat kernels, or Green's functions, for these three surfaces, and we derive the differential forms heat kernels as well. Then we give a method for using the heat kernels on the universal cover to generate the heat kernel on the underlying surface.

Equivariant K-Theory of Simply Connected Lie Groups

We compute the equivariant $K$-theory $K_G^*(G)$ for a simply connected Lie group $G$ (acting on itself by conjugation). We prove that $K_G^*(G)$ is isomorphic to the algebra of Grothendieck differentials on the representation ring. We also study a special example of a non-simply connected Lie group $G$, namely PSU(3), and compute the corresponding equivariant $K$-theory.

Global stability of the Pluriclosed flow on compact simply-connected simple Lie groups of rank two

We compute the (1,1)-Aeppli cohomology of compact simply-connected simple Lie groups of rank two. In particular, we verify that they are of dimension one and generated by the classes of the Bismut flat metrics coming from the Killing forms. This yields a result on the stability of the pluriclosed flow on these manifolds. Moreover, we show that for compact simply-connected simple Lie groups of rank two the Dolbeaut cohomology, as well as the Bott-Chern and the Aeppli cohomologies, arise from just the left-invariant forms and we computed the whole Bott-Chern diamonds of SU(3) and Spin(5) when th…

正数量曲率与单连通四流形上的奇异结构

Positive scalar curvature and exotic structures on simply connected four manifolds

本文针对单连通光滑四流形，证明了Gromov带宽不等式与Rosenberg的$S^1$-稳定性猜想在同胚意义下成立，并推广至非单连通情形。

Simply Connected Symplectic Calabi-Yau 6-Manifolds

In this paper, we construct simply connected symplectic Calabi-Yau 6-manifolds by applying Gompf's symplectic fiber sum operation along $T^4$. Using our construction, we also produce symplectic non-Kähler Calabi-Yau 6-manifolds with fundamental group $\Z$. In this paper, we also produce the first examples of simply connected symplectic Calabi-Yau and non-Calabi-Yau 6-manifolds via coisotropic Luttinger surgery.

Simply connected Alexandrov 4-manifolds with positive or nonnegative curvature and torus actions

We point out that a 4-dimensional topological manifold with an Alexandrov metric (of curvature bounded below) and with an effective, isometric action of the circle or the 2-torus is locally smooth. This observation implies that the topological and equivariant classifications of compact, simply connected Riemannian 4-manifolds with positive or nonnegative sectional curvature and an effective isometric action of a circle or a 2-torus also hold if we consider Alexandrov manifolds instead of Riemannian manifolds.

Formulas of Verlinde type for non-simply connected groups

We derive Verlinde's formula from the fixed point formula for loop groups proved in the companion paper "A fixed point formula for loop group actions", and extend it to compact, connected groups that are not necessarily simply-connected.

Simply-connected minimal surfaces with finite total curvature in $\H^2\times\R$

Laurent Hauswirth and Harold Rosenberg developed the theory of minimal surfaces with finite total curvature in $\H^2\times\R$. They showed that the total curvature of one such a surface must be a non-negative integer multiple of $-2π$. The first examples appearing in this context are vertical geodesic planes and Scherk minimal graphs over ideal polygonal domains. Other non simply-connected examples have been constructed recently. In the present paper, we show that the only complete minimal surfaces in $\H^2\times\R$ of total curvature $-2π$ are Scherk minimal graphs over ideal quadrilaterals. …

Negative Sasakian structures on simply-connected 5-manifolds

We study several questions on the existence of negative Sasakian structures on simply connected rational homology spheres and on Smale-Barden manifolds of the form $\#_k(S^2\times S^3)$. First, we prove that any simply connected rational homology sphere admitting positive Sasakian structures also admits a negative one. This result answers the question, posed by Boyer and Galicki in their book [BG], of determining which simply connected rational homology spheres admit both negative and positive Sasakian structures. Second, we prove that the connected sum $\#_k(S^2\times S^3)$ admits negative qu…

Self-Dual metrics on non-simply connected 4-manifolds

We construct self-dual(SD) but not locally conformally flat(LCF) metrics on families of non-simply connected 4-manifolds with small signature. We construct various sequences with bounded or unbounded Betti numbers and Euler characteristic. These metrics have negative scalar curvature. As an application, this addresses the Remark 4.79 of Besse.

Kähler和Riemannian流形的单连通性 via 谱估计

Simply connectedness of Kähler and Riemannian manifolds via spectral estimates

本文通过谱估计证明了Kähler和Riemannian流形的单连通性及相关拓扑性质。

Isospectral simply-connected homogeneous spaces and the spectral rigidity of group actions

We generalize Sunada's method to produce new examples of closed, locally non-isometric manifolds which are isospectral. In particular, we produce pairs of isospectral, simply-connected, locally non-isometric normal homogeneous spaces. These pairs also allow us to see that in general group actions with discrete spectra are not determined up to measurable conjugacy by their spectra. In particular, we show this for lattice actions.

Continuous families of isospectral metrics on simply connected manifolds

We construct continuous families of Riemannian metrics on certain simply connected manifolds with the property that the resulting Riemannian manifolds are pairwise isospectral for the Laplace operator acting on functions. These are the first examples of simply connected Riemannian manifolds without boundary which are isospectral, but not isometric. For example, we construct continuous isospectral families of metrics on the product of spheres S^4\times S^3\times S^3. The metrics considered are not locally homogeneous. For a big class of such families, the set of critical values of the scalar cu…

Simply Connected Manifolds with Infinitely Many Toric Contact Structures and Constant Scalar Curvature Sasaki Metrics

We study a class of simply connected manifolds in all odd dimensions greater than 3 that exhibit an infinite number of toric contact structures of Reeb type that are inequivalent as contact structures. We compute the cohomology ring of our manifolds by using the join construction for Sasaki manifolds and show that all such contact structures admit a ray of compatible Sasaki metrics of constant scalar curvature (CSC). Furthermore, infinitely many such structures admit at least 3 rays of constant scalar curvature Sasaki metrics.

A $6$-dimensional simply connected complex and symplectic manifold with no Kähler metric

We construct a simply connected compact manifold which has complex and symplectic structures but does not admit Kähler metrics, in the lowest possible dimension where this can happen, that is, dimension 6. Such a manifold is automatically formal and has even odd-degree Betti numbers but it does not satisfy the Lefschetz property for any symplectic form.

A Decomposition of Smooth Simply-connected h-Cobordant 4-Manifolds

We prove that any two smooth h-cobordant simply-connected 4-manifolds can be obtained by taking two manifolds with boundary, one of which is contractible, and gluing them along the boundary via two different attaching maps.

Concerning $L^p$ resolvent estimates for simply connected manifolds of constant curvature

We prove families of uniform $(L^r,L^s)$ resolvent estimates for simply connected manifolds of constant curvature (negative or positive) that imply the earlier ones for Euclidean space of Kenig, Ruiz and the second author \cite{KRS}. In the case of the sphere we take advantage of the fact that the half-wave group of the natural shifted Laplacian is periodic. In the case of hyperbolic space, the key ingredient is a natural variant of the Stein-Tomas restriction theorem.

非单连通流形上的 Floquet-Bloch 函数、Aharonov-Bohm 通量以及浸入曲面的共形不变量

Floquet-Bloch functions on non-simply connected manifolds, the Aharonov-Bohm fluxes, and conformal invariants of immersed surfaces

本文通过定义非单连通流形上的谱簇，刻画磁 Laplacian 谱对 Aharonov-Bohm 通量的解析依赖，并构造二维 Dirac 算子的谱曲线类似物，从而获得曲面浸入的新共形不变量。

Fast Disk Conformal Parameterization of Simply-connected Open Surfaces

Surface parameterizations have been widely used in computer graphics and geometry processing. In particular, as simply-connected open surfaces are conformally equivalent to the unit disk, it is desirable to compute the disk conformal parameterizations of the surfaces. In this paper, we propose a novel algorithm for the conformal parameterization of a simply-connected open surface onto the unit disk, which significantly speeds up the computation, enhances the conformality and stability, and guarantees the bijectivity. The conformality distortions at the inner region and on the boundary are corr…