Scalar Curvature and Connected Sums Of Self-Dual 4-Manifolds

Open 3-manifolds which are connected sums of closed ones

We consider open, oriented 3-manifolds which are infinite connected sums of closed 3-manifolds. We introduce some topological invariants for these manifolds and obtain a classification in the case where there are only finitely many summands up to diffeomorphism. This result encompasses both the Kneser-Milnor Prime Decomposition Theorem for closed 3-manifolds and the Ker{é}kj{á}rt{ó}-Richards classification theorem for open surfaces.

P-spectrum and collapsing of connected sums, calculus of the limit

The goal of the paper is to calculate the limit sectrum of the Hodge-Laplace operator under the perturbation of collapse of one part of a connected sum. This gives some new results concerning the 'conformal spectrum' on differential forms.

Positive scalar curvature and connected sums

Let $N$ be a closed enlargeable manifold in the sense of Gromov-Lawson and $M$ a closed spin manifold of equal dimension, a famous theorem of Gromov-Lawson states that the connected sum $M\# N$ admits no metric of positive scalar curvature. We present a potential generalization of this result to the case where $M$ is nonspin. We use index theory for Dirac operators to prove our result.

Index theory with bounded geometry, the uniformly finite A-hat class, and infinite connected sums

We analyze the obstruction to metrics of positive scalar curvature within a given bounded distortion class of metrics. This obstruction lives in a non-Hausdorff cohomology group Poincare dual to the uniformly finite homology studied by Block and Weinberger. One of the applications is a converse to their theorem on infinite connected sums, giving a complete picture of when such manifolds have metrics of positive scalar curvature.

Curvature, Connected Sums, and Seiberg-Witten Theory

We consider several differential-topological invariants of compact 4-manifolds which directly arise from Riemannian variational problems. Using recent results of Bauer and Furuta, we compute these invariants in many cases that were previously intractable. In particular, we are now able to calculate the Yamabe invariant for certain connected sums of complex surfaces.

Mirror Symmetry for $G_2$-Manifolds: Twisted Connected Sums and Dual Tops

Recently, at least 50 million of novel examples of compact $G_2$ holonomy manifolds have been constructed as twisted connected sums of asymptotically cylindrical Calabi-Yau threefolds. The purpose of this paper is to study mirror symmetry for compactifications of Type II superstrings in this context. We focus on $G_2$ manifolds obtained from building blocks constructed from dual pairs of tops, which are the closest to toric CY hypersurfaces, and formulate the analogue of the Batyrev mirror map for this class of $G_2$ holonomy manifolds, thus obtaining several millions of novel dual superstring…

额外扭曲连通和的Nu不变量

Nu-invariants of extra-twisted connected sums

本研究推广了G₂流形的扭曲连通和构造，将扩展nu不变量分解为不动点与粘合贡献，结合数论论证证明了存在非G₂零配边的紧G₂流形。

Critical metrics on connected sums of Einstein four-manifolds

We develop a gluing procedure designed to obtain canonical metrics on connected sums of Einstein four-manifolds. The main application is an existence result, using two well-known Einstein manifolds as building blocks: the Fubini-Study metric on $\mathbb{CP}^2$ and the product metric on $S^2 \times S^2$. Using these metrics in various gluing configurations, critical metrics are found on connected sums for a specific Riemannian functional, which depends on the global geometry of the factors. Furthermore, using certain quotients of $S^2 \times S^2$ as one of the gluing factors, critical metrics o…

Connected sums with HP^n or CaP^2 and the Yamabe invariant

Let $M$ be a smooth closed $4k$-manifold whose Yamabe invariant $Y(M)$ is nonpositive. We show that $$Y(M\sharp l \Bbb HP^k\sharp m \bar{\Bbb HP^k})=Y(M),$$ where $l,m$ are nonnegative integers, and $\Bbb HP^k$ is the quaternionic projective space. When $k=4$, we also have $$Y(M\sharp l CaP^2\sharp m \bar{CaP^2})=Y(M),$$ where $CaP^2$ is the Cayley plane.

Almost complex structures on connected sums of complex projective spaces

We show that the m-fold connected sum $m\#\mathbb{C}\mathbb{P}^{2n}$ admits an almost complex structure if and only if m is odd.

Constant Scalar Curvature Metrics on Connected Sums

Let (M,g) be a compact Riemannian manifold with dimension n > 2. The Yamabe problem is to find a metric with constant scalar curvature in the conformal class of g, by minimizing the total scalar curvature. The proof was completed in 1984. Suppose (M',g') and (M'',g'') are compact Riemannian n-manifolds with constant scalar curvature. We form the connected sum M' # M'' of M' and M'' by removing small balls from M' and M'' and joining the S^{n-1} boundaries together. In this paper we use analysis to construct metrics with constant scalar curvature on M' # M''. Our description is quite explicit, …

Connected sums of constant mean curvature surfaces in Euclidean 3 space

We establish a general `gluing theorem', which states roughly that if two nondegenerate constant mean curvature surfaces are juxtaposed, so that their tangent planes are parallel and very close to one another, but oppositely oriented, then there is a new constant mean curvature surface quite near to this configuration (in the Hausdorff topology), but which is a topological connected sum of the two surfaces. Here nondegeneracy refers to the invertibility of the linearized mean curvature operator. This paper treats the simplest context for our result namely when the surfaces are compact with non…

$G_2$-instantons on twisted connected sums

We introduce a method to construct $G_2$-instantons over compact $G_2$-manifolds arising as the twisted connected sum of a matching pair of building blocks [Kov03,KL11,CHNP12]. Our construction is based on gluing $G_2$-instantons obtained from holomorphic bundles over the building blocks via the first named author's work [SE11]. We require natural compatibility and transversality conditions which can be interpreted in terms of certain Lagrangian subspaces of a moduli space of stable bundles on a K3 surface.

Ricci-Positive Metrics on Connected Sums of Projective Spaces

It is a well known result of Gromov that all manifolds of a given dimension with positive sectional curvature are subject to a universal bound on the sum of their Betti numbers. On the other hand, there is no such bound for manifolds with positive Ricci curvature: indeed, Perelman constructed positive Ricci metrics on arbitrary connected sums of complex projective planes. In this paper, we revisit and extend Perelman's techniques to construct positive Ricci metrics on arbitrary connected sums of complex, quaternionic, and octonionic projective spaces in every dimension.

纤维丛连通和上的正Ricci曲率

Positive Ricci curvature on connected sums of fibre bundles

本文基于core metrics概念，证明其可沿一般纤维丛提升，并适用于圆盘丛并集空间，从而证明无挠定向配边环的所有类均可用具有正Ricci曲率的连通流形表示。

正中间Ricci曲率在连通和上的研究

Positive intermediate Ricci curvature on connected sums

研究正$k^{th}$中间Ricci曲率下连通和的构造问题，提出$k$-核度量概念并证明线性球丛连通和在特定$k$值范围内允许该曲率，同时给出$\mathbb{H} P^n$和$\mathbb{O}P^2$的最优核度量实例。

Weyl能量与四维流形的连通和

Weyl energy and connected sums of four-manifolds

针对非局部共形平坦且非同时自对偶或反自对偶的闭定向黎曼四维流形，证明其连通和上存在度量使Weyl能量严格小于两流形Weyl能量之和。

Closed geodesics on connected sums and 3-manifolds

We study the asymptotics of the number N(t) of geometrically distinct closed geodesics of a Riemannian or Finsler metric on a connected sum of two compact manifolds of dimension at least three with non-trivial fundamental groups and apply this result to the prime decomposition of a three-manifold. In particular we show that the function N(t) grows at least like the prime numbers on a compact 3-manifold with infinite fundamental group. It follows that a generic Riemannian metric on a compact 3-manifold has infinitely many geometrically distinct closed geodesics. We also consider the case of a c…

Twisted connected sums and special Riemannian holonomy

We give a new, connected-sum-like construction of Riemannian metrics with special holonomy G_2 on compact 7-manifolds. The construction is based on a gluing theorem for appropriate elliptic partial differential equations. As a prerequisite, we also obtain asymptotically cylindrical Riemannian manifolds with holonomy SU(3) building up on the work of Tian and Yau. Examples of new topological types of compact 7-manifolds with holonomy G_2 are constructed using Fano 3-folds.

Connected sums of closed Riemannian manifolds and fourth order conformal invariants

In this note we take some initial steps in the investigation of a fourth order analogue of the Yamabe problem in conformal geometry. The Paneitz constants and the Paneitz invariants considered are believed to be very helpful to understand the topology of the underlined manifolds. We calculate how those quantities change, analogous to how the Yamabe constants and the Yamabe invariants do, under the connected sum operations.

Lorentzian Sasaki-Einstein Metrics on Connected Sums of $S^{2}\times S^{3}$

Negative Sasakian manifolds, where the first Chern class of the contact subbundle is a torsion class, can be viewed as Seifert-$S^1$ bundles where the base orbifold has an ample orbifold canonical class. We use this framework to settle completely an open problem formulated by C.Boyer and K.Galicki which asks whether or not arbitrary connected sums of $S^2\times S^3$ are negative Sasakian manifolds. As a consequence of the affirmative answer to this problem, there exists so-called Sasaki eta-Einstein and Lorentzian Sasaki-Einstein metrics on all of these five-manifolds and moreover all of these…

Current progress on $G_2$--instantons over twisted connected sums

We review a method to construct $\rm{G}_2$--instantons over compact $\rm{G}_2$--manifolds arising as the twisted connected sum of a matching pair of Calabi-Yau $3$-folds with cylindrical end, based on the series of articles [SE15, SEW15, JMPSE17, MNSE17] by the author and others. The construction is based on gluing $\rm{G}_2$--instantons obtained from holomorphic bundles over such building blocks, subject to natural compatibility and transversality conditions. Explicit examples are obtained from matching pairs of semi-Fano $3$-folds by an algorithmic procedure based on the Hartshorne-Serre cor…

具有正数量曲率的完备黎曼流形的体积增长函数

Volume growth functions of complete Riemannian manifolds with positive scalar curvature

该研究针对维数不小于3的开流形是否存在与给定函数同增长类型体积增长的正数量曲率度量这一问题，利用Gromov-Lawson和Grimaldi-Pansu构造，证明了闭流形无穷连通和情形下结论成立，并推广至沿低维球面的无穷连通和情形。

Ricci-positive metrics on connected sums of products with arbitrarily many spheres

In this paper we construct Ricci-positive metrics on the connected sum of products of arbitrarily many spheres provided the dimensions of all but one sphere in each summand are at least 3. There are two new technical theorems required to extend previous results on sums of products of two spheres. The first theorem is a gluing construction for Ricci-positive manifolds with corners generalizing a gluing construction of Perelman for Ricci-positive manifolds with boundaries. Our construction gives a sufficient condition for gluing together two Ricci-positive manifolds with corners along isometric …

Stable Cohomotopy Seiberg-Witten Invariants of Connected Sums of Four-Manifolds with Positive First Betti Number

We shall prove a new non-vanishing theorem for the stable cohomotopy Seiberg-Witten invariant of connected sums of 4-manifolds with positive first Betti number. The non-vanishing theorem enables us to find many new examples of 4-manifolds with non-trivial stable cohomotopy Seiberg-Witten invariants and it also gives a partial, but strong affirmative answer to a conjecture concerning non-vanishing of the invariant. Various new applications of the non-vanishing theorem are also given. For example, we shall introduce variants $\barλ_k$ of Perelman's $\barλ$ invariants for real numbers $k$ and com…

在 $\mathbb{T}^k\times \mathbb{S}^m$ 上具有正 Ricci 特征值和的度量

Metrics with positive Ricci eigenvalue sums on $\mathbb{T}^k\times \mathbb{S}^m$

本文构造了环面与球面乘积上任意接近标准度量的正 Ricci 特征值和度量，否证了 Wolfson 猜想，并给出了二正 Ricci 曲率的局部障碍。

Gluing formulae for Donaldson invariants for connected sums along surfaces

We solve a conjecture of Morgan and Szabo (Embedded genus 2 surfaces in four-manifolds, Preprint) about the relationship of the basic classes of two four-manifolds $X_i$ of simple type with $b_1=0$, $b^+>1$, such that there are embedded Riemann surfaces of genus $g \geq 2$ and self-intersection zero (and representing odd homology classes) with the basic classes of the manifold X which appears as a connected sum along the surfaces (supposing this latter one is of simple type). This is also expressed as constraints in the basic classes of X. The result is in accordance with the results on Seiber…

Spin(7)-Manifolds as Generalized Connected Sums and 3d N=1 Theories

M-theory on compact eight-manifolds with $\mathrm{Spin}(7)$-holonomy is a framework for geometric engineering of 3d $\mathcal{N}=1$ gauge theories coupled to gravity. We propose a new construction of such $\mathrm{Spin}(7)$-manifolds, based on a generalized connected sum, where the building blocks are a Calabi-Yau four-fold and a $G_2$-holonomy manifold times a circle, respectively, which both asymptote to a Calabi-Yau three-fold times a cylinder. The generalized connected sum construction is first exemplified for Joyce orbifolds, and is then used to construct examples of new compact manifolds…

Compact Kähler manifolds with quasi-positive second Chern-Ricci curvature

Let $X$ be a compact Kähler manifold. We prove that if $X$ admits a smooth Hermitian metric $ω$ with quasi-positive second Chern-Ricci curvature $\mathrm{Ric}^{(2)}(ω)$, then $X$ is projective and rationally connected. In particular, $X$ is simply connected.

Connected sums of almost complex manifolds

In this paper, firstly, for some $4n$-dimensional almost complex manifolds $M_{i}, ~1\le i \le α$, we prove that $\left(\sharp_{i=1}^α M_{i}\right) \sharp (α{-}1) \mathbb{C} P^{2n}$ must admits an almost complex structure, where $α$ is a positive integer. Secondly, for a $2n$-dimensional almost complex manifold $M$, we get that $M\sharp \overline{\mathbb{C} P^{n}}$ also admits an almost complex structure. At last, as an application, we obtain that $α\mathbb{C} P^{2n}\sharp β\overline{\mathbb{C} P^{2n}}$ admits an almost complex structure if and only if $α$ is odd.

Connected sums of almost complex manifolds, products of rational homology spheres, and the twisted spin^c Dirac operator

We record an answer to the question "In which dimensions is the connected sum of two closed almost complex manifolds necessarily an almost complex manifold?". In the process of doing so, we are naturally led to ask "For which values of l is the connected sum of l closed almost complex manifolds necessarily an almost complex manifold?". We answer this question, along with its non-compact analogue, using obstruction theory and Yang's results on the existence of almost complex structures on (n-1)-connected 2n-manifolds. Finally, we partially extend Datta and Subramanian's result on the nonexisten…

Path-connectedness of the moduli spaces of metrics with positive isotropic curvature on four-manifolds

In this paper we prove the path connectedness of the moduli spaces of metrics with positive isotropic curvature on certain compact four-dimensional manifolds.

Donaldson invariants for connected sums along surfaces of genus 2

We relate the Donaldson invariants of two four-manifolds $X_i$ with embedded Riemann surfaces of genus 2 and self-intersection zero with the invariants of the manifold X which appears as a connected sum along the surfaces. When the original manifolds are of simple type with $b_1=0$ and $b^+>1$, X is of simple type with $b_1=0$ and $b^+>1$ as well, and the relationship between the invariants is expressed as constraints in the basic classes for X. Also we give some applications. For instance, if $X_i$ have both $b_1=0$ then X is of simple type with $b_1=0$, $b^+>1$, and has no basic classes eval…

通过局部共形平坦流形的连通和降低Weyl能量

Decreasing Weyl's energy by connected sums with locally conformally flat manifolds

研究两个四维流形连通和的Weyl函数，当其中一个为Bach平坦且另一个为局部共形平坦时，证明在特定条件下可构造更低Weyl能量的度量。

Ricci curvature and monopole classes on 3-manifolds

We prove an L^2-estimate involving Ricci curvature and a harmonic 1-form on a closed oriented Riemannian 3-manifold admitting a solution of any rescaled Seiberg-Witten equations. We also give a necessary condition to be a monopole class on some special connected sums.

On projective Kähler manifolds of partially positive curvature and rational connectedness

In a previous paper, we proved that a projective Kähler manifold of positive total scalar curvature is uniruled. At the other end of the spectrum, it is a well-known theorem of Campana and Kollár-Miyaoka-Mori that a projective Kähler manifold of positive Ricci curvature is rationally connected. In the present work, we investigate the intermediate notion of $k$-positive Ricci curvature and prove that for a projective $n$-dimensional Kähler manifold of $k$-positive Ricci curvature the MRC fibration has generic fibers of dimension at least $n-k+1$. We also establish an analogous result for projec…

Highly connected 7-manifolds and non-negative curvature

In this article, a six-parameter family of highly connected 7-manifolds which admit an SO(3)-invariant metric of non-negative sectional curvature is constructed and the Eells-Kuiper invariant of each is computed. In particular, it follows that all exotic spheres in dimension 7 admit an SO(3)-invariant metric of non-negative curvature.

Connected sum construction of constant Q-curvature manifolds in higher dimensions

For a compact Riemannian manifold $(M, g_2)$ with constant $Q$-curvature of dimension $n\geq 6$ satisfying nondegeneracy condition, we show that one can construct many examples of constant $Q$-curvature manifolds by gluing construction. We provide a general procedure of gluing together $(M,g_2)$ with any compact manifold $(N, g_1)$ satisfying a geometric assumption. In particular, we can prove that there exists a metric with constant $Q$-curvature on the connected sum $N #M$.

Legendrian嵌入空间中的参数化卫星与连通和

Parametric satellites and connected-sums in the space of Legendrian embeddings

本文在Legendrian嵌入空间中引入参数化卫星与连通和构造，生成了高阶同伦新不变量，并构造出具有非平凡LCH monodromy不变量的Legendrian嵌入环的新无限族。

平均曲率正性与有理连通性

Mean curvature positivity and rational connectedness

该研究借助Uhlenbeck-Yau连续性方法，建立紧埃尔米特流形上全纯向量丛的平均曲率正性与HN-正性的对应，进而给出紧凯勒流形射影且有理连通的微分几何等价判据。

Intermediate curvatures and highly connected manifolds

We show that after forming a connected sum with a homotopy sphere, all (2j-1)-connected 2j-parallelisable manifolds in dimension 4j+1, j > 0, can be equipped with Riemannian metrics of 2-positive Ricci curvature. The condition of 2-positive Ricci curvature is defined to mean that the sum of the two smallest eigenvalues of the Ricci tensor is positive at every point. This result is a counterpart to a previous result of the authors concerning the existence of positive Ricci curvature on highly connected manifolds in dimensions 4j-1 for j > 1, and in dimensions 4j+1 for j > 0 with torsion-free co…

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 $\mathbb{S}^{2}\times \mathbb{R}$.

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. …

Connected sums of special Lagrangian submanifolds

Let $M_1$ and $M_2$ be special Lagrangian submanifolds of a compact Calabi-Yau manifold $X$ that intersect transversely at a single point. We can then think of $M_1\cup M_2$ as a singular special Lagrangian submanifold of $X$ with a single isolated singularity. We investigate when we can regularize $M_1\cup M_2$ in the following sense: There exists a family of Calabi-Yau structures $X_α$ on $X$ and a family of special Lagrangian submanifolds $M_α$ of $X_α$ such that $M_α$ converges to $M_1\cup M_2$ and $X_α$ converges to the original Calabi-Yau structure on $X$. We prove that a regularization …

Self-Dual Manifolds with Positive Ricci Curvature

We prove that the connected sums CP_2 # CP_2 and CP_2 # CP_2 # CP_2 admit self-dual metrics with positive Ricci curvature. Moreover, every self-dual metric of positive scalar curvature on CP_2 # CP_2 is conformal to a metric with positive Ricci curvature.

Existence of Riemannian metrics with positive biorthogonal curvature on simply connected 5-manifolds

Using recent work of Bettiol, we show that a first-order conformal deformation of Wilking's metric of almost-positive sectional curvature on $S^2\times S^3$ yields a family of metrics with strictly positive average of sectional curvatures of any pair of 2-planes that are separated by a minimal distance in the 2-Grassmanian. A result of Smale's allows us to conclude that every closed simply connected 5-manifold with torsion-free homology and trivial second Stiefel-Whitney class admits a Riemannian metric with a strictly positive average of sectional curvatures of any pair of orthogonal 2-planes…

$\mathbb Z_2$-调和旋量与1-形式在三维流形的连通和与环面和上的构造

$\mathbb Z_2$-Harmonic Spinors and 1-forms on Connected sums and Torus sums of 3-manifolds

通过粘合论证，在连通和与环面和上构造$\mathbb Z_2$-调和旋量与1-形式，并证明每个闭三维流形存在无穷多个具有不同奇异链的$\mathbb Z_2$-调和旋量。

极小超曲面何时能通过平均曲率流连接？

When can minimal hypersurfaces be connected by mean curvature flow?

本文从莫尔斯理论视角研究极小超曲面之间是否存在连接平均曲率流，并通过构造例子揭示了除已知条件外还存在拓扑与变分层面的额外阻碍。

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.

Highly connected manifolds with positive Ricci curvature

We prove the existence of Sasakian metrics with positive Ricci curvature on certain highly connected odd dimensional manifolds. In particular, we show that manifolds homeomorphic to the 2k-fold connected sum of S^{2n-1} x S^{2n} admit Sasakian metrics with positive Ricci curvature for all k. Furthermore, a formula for computing the diffeomorphism types is given and tables are presented for dimensions 7 and 11.

Nonconnected Moduli Spaces of Nonnegative Sectional Curvature Metrics on Simply Connected Manifolds

We show that in each dimension $4n+3$, $n\ge 1$, there exist infinite sequences of closed smooth simply connected manifolds $M$ of pairwise distinct homotopy type for which the moduli space of Riemannian metrics with nonnegative sectional curvature has infinitely many path components. Closed manifolds with these properties were known before only in dimension seven, and our result does also hold for moduli spaces of Riemannian metrics with positive Ricci curvature. Moreover, in conjunction with work of Belegradek, Kwasik and Schultz, we obtain that for each such $M$ the moduli space of complete…

Generalized connected sum construction for constant scalar curvature metrics

We consider the problem of constructing solutions to the Yamabe equation (i.e. conformal constant scalar curvature metrics) on the generalized connected sum M = (M_1) #_K (M_2) of two compact Riemannian manifolds (M_1,g_1) and (M_2,g_2) along a common (isometrically embedded) submanifold (K,g_K) of codimension greater or equal than 3.

Deforming 3-manifolds of bounded geometry and uniformly positive scalar curvature

We prove that the moduli space of complete Riemannian metrics of bounded geometry and uniformly positive scalar curvature on an orientable 3-manifold is path-connected. This generalizes the main result of the fourth author [Mar12] in the compact case. The proof uses Ricci flow with surgery as well as arguments involving performing infinite connected sums with control on the geometry.

拟正混合曲率、消灭定理与有理连通性

Quasi-positive mixed curvature, vanishing theorems, and rational connectedness

本文研究了紧复流形上的拟正混合曲率，证明了在特定参数条件下流形具有射影性和有理连通性，并将相关消灭定理与结论推广至$k$-Ricci曲率、2-标量曲率及Hermitian情形。