Gilles Carron
We give a topological interpretation of the space of $L^2$-harmonic forms on Manifold with flat ends. It is an answer to an old question of J. Dodziuk. We also give a Chern-Gauss-Bonnet formula for the $L^2$-Euler characteristic of some of these Manifolds. These results are applications of general theorems on complete Riemannian Manifold whose Gauss-Bonnet operator is non-parabolic at infinity.
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In this paper we study the minimal and maximal $L^{2}$-cohomology of oriented, possibly not complete, Riemannian manifolds. Our focus will be on both the reduced and the unreduced $L^{2}$-cohomology groups. In particular we will prove that these groups are invariant under uniform homotopy equivalence quasi-isometric on the unbounded ends. A uniform map is a uniformly continuous map such that the diameter of the preimage of a subset is bounded in terms of the diameter of the subset itself. A map $f$ between two Riemannian manifolds $(X,g)$ and $(Y,h)$ is \textit{quasi-isometric on the unbounded…
Let $M$ be a smooth Riemannian manifold which is the union of a compact part and a finite number of Euclidean ends, $\RR^n \setminus B(0,R)$ for some $R > 0$, each of which carries the standard metric. Our main result is that the Riesz transform on $M$ is bounded from $L^p(M) \to L^p(M; T^*M)$ for $1 < p < n$ and unbounded for $p \geq n$ if there is more than one end. It follows from known results that in such a case the Riesz transform on $M$ is bounded for $1 < p \leq 2$ and unbounded for $p > n$; the result is new for $2 < p \leq n$. We also give some heat kernel estimates on such manifolds…
We study ends of an oriented, immersed, non-compact, complete Willmore surfaces, which are critical points of the integral of the square of the mean curvature, in asymptotically flat spaces of any dimension; assuming the surface has $L^2$-bounded second fundamental form and satisfies a weak power growth on the area. We give the precise asymptotic behavior of an end of such a surface. This asymptotic information is very much dependent on the way the ambient metric decays to the Euclidean one. Our results apply in particular to minimal surfaces.
Examples of Toric Scalar-flat Kähler Surfaces with Mixed-type Ends
该研究针对严格无界环面辛4-流形，在环面除子补集上显式构造完备环面数量平坦Kähler度量，并借鉴Abreu与Sena-Dias方法给出带锥奇点的同类度量的另一种构造。
Mass-Capacity Rigidity for Asymptotically Flat Manifolds with Arbitrary Ends
本文刻画了任意端渐近平坦流形质量-容量不等式等号成立的条件，证明等号迫使流形等距于欧氏空间去掉低维紧集，关键利用共形变换与Bakry-Émery曲率。
Motivated by the recent progress on positive mass theorem for asymptotically flat manifolds with arbitrary ends and the Gromov's definition of scalar curvature lower bound for continuous metrics, we start a program on the positive mass theorem for asymptotically flat manifolds with $C^0$ arbitrary ends. In this work as the first step, we establish the positive mass theorem of asymptotically flat manifolds with $C^0$ arbitrary ends when the metric is $W^{1,p}_{\mathrm{loc}}$ for some $p\in(n,\infty]$ and is smooth away from a non-compact closed subset with Hausdorff dimension $n-\frac{p}{p-1}$.…
On the first Neumann eigenvalue for critical points of a weighted area functional with asymptotically flat ends
本文研究带渐近平坦端的加权面积泛函临界曲面，通过积分 Bouchner 公式建立漂移 Laplace 算子第一 Neumann 特征值下界，并用于控制有限全曲率曲面的拓扑。
We prove that if the dimension of the first cohomology group of a $RCD^*(0,N)$ space is $N$, then the space is a flat torus. This generalizes a classical result due to Bochner to the non-smooth setting and also provides a first example where the study of the cohomology groups in such synthetic framework leads to geometric consequences.
This is the second part of the investigation started in [Stationary solutions and asymptotic flatness I]. We prove here that Strongly Stationary ends having cubic volume growth are Weakly Asymptotically Flat. Combined with the results of the previous paper this shows that Strongly Stationary ends are Asymptotically Flat with Schwarzschidian fall off.
Foliation of area-minimizing hypersurfaces in asymptotically flat manifolds of higher dimension
研究在高维渐近扁平流形中面积最小超曲面的分解及其全局行为
Asymptotically cylindrical Ricci-flat manifolds play a key role in constructing Topological Quantum Field Theories. It is particularly important to understand their behavior at the cylindrical ends and the natural restrictions on the geometry. In this paper we show that an orientable, connected, asymptotically cylindrical manifold (M,g) with Ricci-flat metric g can have at most two cylindrical ends. In the case where there are two such cylindrical ends then there is reduction in the holonomy group Hol(g) and (M,g) is a cylinder.
Taubes' gluing theorems establish the existence of ASD connections on closed, oriented 4-manifolds. We extend these gluing results to the mASD connections of Morgan-Mrowka-Ruberman on oriented 4-manifolds with cylindrical ends. As a corollary, we obtain an ASD-existence result in the presence of degenerate asymptotic flat connections.
This thesis studies moduli spaces of singular connections on 3-manifolds and manifolds with cylindrical ends. A Chern-Simons functional is defined for singular connections on 3-manifolds which are singular along a knot. The critical points of that Chern-Simons functional are flat singular connections. The Hodge-de Rham theory of singular flat connections is presented, yielding a topological condition for determining when a singular flat connection is isolated. Finally, moduli spaces of anti-self-dual singular connections are investigated on manifolds with cylindrical ends. The main analytical …
We show that a flat principal bundle with compact connected structure group and its adjoint bundles of Lie groups have the same cohomology as the trivial bundle, which is done by proving they satisfy the condition for the Leray-Hirsch theorem. This information has been used to construct a cohomology class of the adjoint bundle of a flat bundle whose restriction to each fiber is the class of the Maurer-Cartan 3-form. Then we use this result to define an invariant of a gauge transformation of a flat bundle which describes the effect of the gauge transformation on a Chern-Simons functional.
We show that any compact half-conformally flat manifold of negative type, with bounded $L^2$ energy, sufficiently small scalar curvature, and a non-collapsing assumption, has all betti numbers bounded. We show that this result is optimal from an analytic perspective by demonstrating singularity models that are 2-ended, and are asymptotically Kähler on both ends. We show that bounded self-dual solutions of $dω=0$ on ALE manifold ends are either asymptotically Kähler, or they have a decay rate of $O(r^{-4})$ or better.
We investigate asymptotically flat manifolds with cone structure at infinity. We show that any such manifold M has a finite number of ends. For simply connected ends we classify all possible cones at infinity, except for the 4-dimensional case where it remains open if one of the theoretically possible cones can actually arise. This result yields in particular a complete classification of asymptotically flat manifolds with nonnegative curvature: The universal covering of an asymptotically flat manifold with nonnegative sectional curvature is isometric to a product of Euclidean space and an asym…
In this article, we give a proof for positive mass theorem of asymptotically flat manifolds with arbitrary ends when the dimension is no greater than seven. As an application, we also show a positive mass theorem for asymptotically locally Euclidean manifolds with necessary incompressible conditions.
This paper constructs a Hodge theory of noncompact topologically tame manifolds $M$. The main result is an isomorphism between the de Rham cohomology with compact supports of $M$ and the kernel of the Hodge--Witten--Bismut Laplacian $\lap_μ$ associated to a measure $dμ$ which has sufficiently rapid growth at infinity on $M$. This follows from the construction of a space of forms associated to $\lap_μ$ which satisfy an ``extension by zero'' property. The ``extension by zero'' property is proved for manifolds with cylindrical ends possessing gaussian growth measures.
We prove a relation between the $\bar\partial_M$ cohomology of a minimal orbit $M$ of a real form $G_0$ of a complex semisimple Lie group $G$ in a flag manifold $G/Q$ and the Dolbeault cohomology of the Matsuki dual open orbit $X$ of the complexification $K$ of a maximal compact subgroup $K_0$ of $G_0$, under the assumption that $M$ is Levi-flat.
Fully nonlinear Yamabe-type problems on non-compact manifolds
本文研究非紧流形上的完全非线性Yamabe型问题，在正、负两种情形下证明了存在性结果，并构造了满足定理假设的扭曲积端和渐近平坦端流形的显式例子。
It has been showed by Byde that it is possible to attach a Delaunay-type end to a compact nondegenerate manifold of positive constant scalar curvature, provided it is locally conformally flat in a neighborhood of the attaching point. The resulting manifold is noncompact with the same constant scalar curvature. The main goal of this paper is to generalize this result. We will construct a one-parameter family of solutions to the positive singular Yamabe problem for any compact non-degenerate manifold with Weyl tensor vanishing to sufficiently high order at the singular point. If the dimension is…
The main goal of this article is to construct some geometric invariants for the topology of the set $\mathcal{F}$ of flat connections on a principal $G$-bundle $P\,\longrightarrow\, M$. Although the characteristic classes of principal bundles are trivial when $\mathcal{F}\neq \emptyset$, their classical Chern-Weil construction can still be exploited to define a homomorphism from the set of homology classes of maps $S\longrightarrow \mathcal{F}$ to the cohomology group $H^{2r-k}(M,\mathbb{R})$, where $S$ is null-cobordant $(k-1)$-manifold, once a $G$-invariant polynomial $p$ of degree $r$ on $\…
In this paper, we focus our study on the ends of a locally conformally flat complete manifold with finite total $Q$-curvature. We prove that for such a manifold, the integral of the $Q$-curvature equals an integral multiple of a dimensional constant $c_n$, where $c_n$ is the integral of the $Q$-curvature on the unit $n$-sphere. It provides further evidence that the $Q$-curvature on a locally conformally flat manifold controls geometry as the Gaussian curvature does in two dimension.
In this paper we consider the conformal type (parabolicity or non-parabolicity) of complete ends of revolution immersed in simply connected space forms of constant sectional curvature. We show that any complete end of revolution in the $3$-dimensional Euclidean space or in the $3$-dimensional sphere is parabolic. In the case of ends of revolution in the hyperbolic $3$-dimensional space, we find sufficient conditions to attain parabolicity for complete ends of revolution using their relative position to the complete flat surfaces of revolution.
In a previous article, we generalised the classical four-dimensional Chern-Gauss-Bonnet formula to a class of manifolds with finitely many conformally flat ends and singular points, in particular obtaining the first such formula in a dimension higher than two which allows the underlying manifold to have isolated conical singularities. In the present article, we extend this result to all even dimensions $n\geq 4$ in the case of a class of conformally flat manifolds.
In this thesis we prove analytic results about a cohomotopical Seiberg-Witten theory for a Riemannian, Spin$^c$(4), 4-manifold with periodic ends, $(X, g, τ)$ . Our results show that, under certain technical assumptions on $(X, g, τ)$, this new version is coherent and leads to Seiberg-Witten type invariants for this new class of 4-manifolds. In the first part, using Taubes criteria for end-periodic operators, we show that for a Riemannian 4-manifold with periodic ends, $(X, g)$, verifying certain topological conditions, the Laplacian, $Δ_+ : L^2_2({Λ^2}_+) \rightarrow L^2({Λ^2}_+)$, is a Fredh…
Let h^{*} be a multiplicative cohomology theory, h_{*} its dual homology theory and \hat{h}^{*} a differential refinement. We first construct the natural pairing between h_{*} and the flat part of \hat{h}^{*}, generalizing the holonomy of a flat Deligne cohomology class. Then, in order to generalize the holonomy of any Deligne cohomology class, we define the generalized Cheeger-Simons characters. The latter are functions from suitably defined differential cycles to the cohomology ring of the point, such that the value on a trivial cycle only depends on the curvature.
We consider the moduli space of flat $SO(2n+1)$-connections (up to gauge transformations) on a Riemann surface, with fixed holonomy around a marked point. There are natural line bundles over this moduli space; we construct geometric representatives for the Chern classes of these line bundles, and prove that the ring generated by these Chern classes vanishes below the dimension of the moduli space, generalising a conjecture of Newstead.
In this note, we extend the theory of Chern-Cheeger-Simons to construct canonical invariants for a one-parameter family of flat connections on a smooth manifold. These invariants lie in degrees $(2p-2)$-cohomology with $\C/\Z$-cohomology, for $p\geq 2$. Furthermore, they are shown to be rigid in a variation of paths (parametrising flat connections), in degrees at least three.
We shall investigate flat surfaces in hyperbolic 3-space with admissible singularities, called `flat fronts'. An Osserman-type inequality for complete flat fronts is shown. When equality holds in this inequality, we show that all the ends are embedded. Moreover, we shall give new examples for which equality holds.
Let $π\colon P\to M$ be a principal bundle and $p$ an invariant polynomial of degree r on the Lie algebra of the structure group. The theory of Chern-Simons differential characters is exploited to define an homology map $χ^{k} : H_{2r-k-1}(M)\times H_{k}(\mathcal{F}/\mathcal{G})\to \mathbb{R}/\mathbb{Z}$, for $k<r-1$, where $\mathcal{F} /\mathcal{G}$ is the moduli space of flat connections of $π$ under the action of a subgroup $\mathcal{G}$ of the gauge group. The differential characters of first order are related to the Dijkgraaf-Witten action for Chern-Simons Theory. The second order charact…
It is classically known that complete flat surfaces in Euclidean 3-space are cylinders over space curves. This implies that the study of global behaviour of flat surfaces requires the study of singular points as well. If a flat surface $f$ admits singularities but its Gauss map $ν$ can be smoothly extended across the singular set, $f$ is called a frontal. In addition, if the pair $(f,ν)$ gives an immersion, $f$ is called a front. A front $f$ is called flat if the Gauss map degenerates everywhere. The parallel surfaces and the focal surface of a flat front $f$ are also flat fronts. In this pape…
On Kahler 4-manifolds, not necessarily compact or of finite topological type, we obtain relationships between the fundamental group of compact embedded Levi-flat or pseudoconvex submanifold and the fundamental group of the ambient manifold $M^4$. When a Levi-flat submanifold $V^3$ has finite fundamental group then $π_1(M^4)=ι_*π_1(V^3)$; when a non-separating pseudoconvex submanifold $V^3$ has finite fundamental group, then $π_1(M^4)=ι_*π_1(V^3)\rtimes\mathbb{Z}$. As applications, if a Kahler manifold (compact or not) has an embedded holomorphic $\mathbb{P}^1$ of positive self-intersection, it…
We prove a Riemannian positive mass theorem for manifolds with a single asymptotically flat end, but otherwise arbitrary other ends, which can be incomplete and contain negative scalar curvature. The incompleteness and negativity is compensated for by large positive scalar curvature on an annulus, in a quantitative fashion. In the complete noncompact case with nonnegative scalar curvature, we have no extra assumption and hence prove a long-standing conjecture of Schoen and Yau.
A famous result of Jurgen Moser states that a symplectic form on a compact manifold cannot be deformed within its cohomology class to an inequivalent symplectic form. It is well known that this does not hold in general for noncompact symplectic manifolds. The notion of Eliashberg-Gromov convex ends provides a natural restricted setting for the study of analogs of Moser's symplectic stability result in the noncompact case, and this has been significantly developed in work of Cieliebak-Eliashberg. Retaining the end structure on the underlying smooth manifold, but dropping the convexity and compl…
Symplectic Flatness and Twisted Primitive Cohomology
本文引入了辛流形上联络和纤维丛的辛平坦性概念，提出了基于$A_\infty$-代数的扭曲微分复形的平坦性条件，并证明辛平坦联络是Yang-Mills联络的特殊子类，同时研究了其上同调及消没定理。
$L^p$ cohomology and Hodge decomposition for ALE manifolds
研究ALE流形\(L^p\)约化上同调空间维数与衰减调和形式空间维数关系，证明特定\(k\)值下维数与\(p\)的关联及跳跃情况，还给出\(k\)形式的\(L^p\)霍奇分解及替代方法。
Tilted spacetime positive mass theorem with arbitrary ends
本文通过求解带有 Callias 势的 Dirac-Witten 算子的混合边值问题，证明了具有任意端和非紧边界的渐近平坦自旋初始数据集的时空正质量定理，并在倾斜边界主能量条件下证明了定量屏蔽定理。
Monotonicity of harmonic functions on $3$-manifolds with an asymptotically flat end
研究了三维流形上具有渐近平坦端的正调和函数的单调性，并探讨了具有两个端的Schwarzschild流形的刚性特征。
We study the Lagrangian antifield BRST formalism, formulated in terms of exterior horizontal forms on the infinite order jet space of graded fields for topological field theories associated to $Q$-bundles. In the case of a trivial Q-bundle with a flat fiber and arbitrary base, we prove that the BRST cohomology are isomorphic to the cohomology of the target space differential "twisted" by the de Rham cohomology of the base manifold. This generalizes the local result of G. Barnich and M. Grigoriev, computed for a flat base manifold.
In this paper, we apply the theory of Chern-Cheeger-Simons to construct canonical invariants associated to a $r$-simplex whose points parametrize flat connections on a smooth manifold $X$. These invariants lie in degrees $(2p-r-1)$-cohomology with $C/Z$-coefficients, for $p> r\geq 1$. In turn, this corresponds to a homomorphism on the higher homology groups of the moduli space of flat connections, and taking values in $C/Z$-cohomology of the underlying smooth manifold $X$.
We consider a compact manifold whose boundary is a locally trivial fiber bundle and an associated pseudodifferential algebra that models fibered cusps at infinity. Using trace-like functionals that generate the 0-dimensional Hochschild cohomology groups, we express the index of a fully elliptic fibered cusp operator as the sum of a local contribution from the interior and a term that comes from the boundary. This answers the index problem formulated by Mazzeo and Melrose. We give a more precise answer in the case where the base of the boundary fiber bundle is the circle. In particular, for Dir…
We study the de Rham 1-cohomology H^1_{DR}(M,G) of a smooth manifold M with values in a Lie group G. By definition, this is the quotient of the set of flat connections in the trivial principle bundle $M\times G$ by the so-called gauge equivalence. We consider the case when M is a compact Kähler manifold and G is a solvable complex linear algebraic group of a special class which contains the Borel subgroups of all complex classical groups and, in particular, the group $T_n(\Bbb C)$ of all triangular matrices. In this case, we get a description of the set H^1_{DR}(M,G) in terms of the 1-cohomolo…
In this paper, we focus on the moduli space of Seiberg-Witten equation on non-compact manifold with periodic end. Suppose that the scalar curvature on the periodic end is identically zero and the topological conditions: the first de-Rham cohomology and the self-dual cohomology restricting on the periodic end vanish. Then, we will show that the moduli space of the perturbed Seiberg-Witten equation is compact.
We consider a closed odd-dimensional oriented manifold $M$ together with an acyclic flat hermitean vector bundle $\cF$. We form the trivial fibre bundle with fibre $M$ over the manifold of all Riemannian metrics on $M$. It has a natural flat connection and a vertical Riemannian metric. The higher analytic torsion form of Bismut/Lott associated to the situation is invariant with respect to the connected component of the identity of the diffeomorphism group of $M$. Using that the space of Riemannian metrics is contractible we define continuous cohomology classes of the diffeomorphism group and i…
We use gauge theoretic and algebraic methods to examine sufficient conditions for smooth points on the moduli space of flat connections on a compact manifold and on the character variety of a finitely generated and presented group. We give a complete proof of the slice theorem for the action of the group of gauge transformations on the space of flat connections. Consequently, the slice is smooth if the second cohomology of the manifold with coefficients in the semisimple part of the adjoint bundle vanishes. On the other hand, we find that the smoothness of the slice for the character variety o…
We define exotic twisted $S^1$-equivariant cohomology for the loop space $LZ$ of a smooth manifold $Z$ via the invariant differential forms on $LZ$ with coefficients in the (typically non-flat) holonomy line bundle of a gerbe, with differential an equivariantly flat superconnection. We introduce the twisted Bismut-Chern character form, a loop space refinement of the twisted Chern character form, which represent classes in the completed periodic exotic twisted $S^1$-equivariant cohomology of $LZ$. We establish a localisation theorem for the completed periodic exotic twisted $S^1$-equivariant co…
Quantum cohomology gives a finite dimensional integrable system via the Dubrovin connection. Motivated by Givental's work on mirror symmetry, we use gauge theory techniques and the Frobenius Integrability Theorem to find flat sections for the Dubrovin connection. An explicit calculation is given for projective space.
On the growth of torsion in the cohomology of some arithmetic groups of $\mathbb{Q}$-rank one
本文研究了某些有理数秩为一的算术群的同调中 torsion 的增长，通过解析 torsion 与 Reidemeister torsion 的关系，证明了同余子群序列中 torsion 的指数增长。
Positive mass theorem for initial data sets with arbitrary ends
本文通过引入谱PSC概念，证明了渐近平坦初值集的正能量定理，并建立了渐近双曲流形的能量动量向量因果性质的定量屏蔽定理，从而得出完整渐近双曲流形在主导能量条件下的正质量定理。
For any $n\geq 7$, $k\geq 3$, we give pairs of compact flat $n$-manifolds $M, M'$ with holonomy groups $\mathbb Z_2^k$, that are strongly isospectral, hence isospectral on $p$-forms for all values of $p$, having nonisomorphic cohomology rings. Moreover, if $n$ is even, $M$ is Kähler while $M'$ is not. Furthermore, with the help of a computer program we show the existence of large Sunada isospectral families; for instance, for $n=24$ and $k=3$ there is a family of eight compact flat manifolds (four of them Kähler) having very different cohomology rings. In particular, the cardinalities of the s…
In this paper, we investigate the asymptotic behavior of regular ends of flat surfaces in the hyperbolic 3-space H^3. Galvez, Martinez and Milan showed that when the singular set does not accumulate at an end, the end is asymptotic to a rotationally symmetric flat surface. As a refinement of their result, we show that the asymptotic order (called "pitch" p) of the end determines the limiting shape, even when the singular set does accumulate at the end. If the singular set is bounded away from the end, we have -1<p<=0. If the singular set accumulates at the end, the pitch p is a positive ration…
Positive Mass Theorem with Arbitrary Ends and Noncompact Boundary
该研究证明了带非紧边界及特殊渐近平坦半空间端的黎曼流形正质量定理，结合多种方法完成证明，还得到了对应的精确黎曼 Penrose 不等式。
We study complete, finite volume $n$-manifolds $M$ of bounded nonpositive sectional curvature. A classical theorem of Gromov says that if such $M$ has negative curvature then it is homeomorphic to the interior of a compact manifold-with-boundary, and we denote this boundary $\partial M$. If $n\geq 3$, we prove that the universal cover of the boundary $\widetilde{\partial M}$ and also the $π_1M$-cover of the boundary $\partial\widetilde M$ have vanishing $(n-2)$-dimensional homology. For $n=4$ the first of these recovers a result of Nguyen Phan saying that each component of the boundary $\parti…
We show that the moduli space of Ricci flat metrics of unit volume (including orbifold metrics) on a K3 surface is simply connected and that it has the same rational cohomology as the automorphism group of the K3 lattice $(-E_8)^{\oplus 2}\oplus U^{\oplus 3}$.
We give a presentation for the Floer cohomology ring $HF^*(Σ\times S^1)$, where $Σ$ is a Riemann surface of genus bigger than one, which coincides with the conjectural presentation for the quantum cohomology ring of the moduli space of flat SO(3)-connections of odd degree over $Σ$. We study the spectrum of the action of the homology of $Σ$ on the Floer cohomology and prove a physical assumption made by Vafa et al.
We prove the vanishing of a certain characteristic class of flat vector bundles when the structure groups of the bundles are contained in GL(N,Z). We do so by explicitly writing the characteristic class as an exact form on the base of the bundle.
The purpose of this article is to give an interpretation of real projective structures and associated cohomology classes in terms of connections, sections, etc. satisfying elliptic partial differential equations in the spirit of Hodge theory. We shall also give an application of these results as the uniqueness of a minimal surface in a symmetric space.
We introduce the notion of cofoliation on a stack. A cofoliation is a change of the differentiable structure which amounts to giving a full representable smooth epimorphism. Cofoliations are uniquely determined by their associated Lie algebroids. Cofoliations on stacks arise from flat connections on groupoids. Connections on groupoids generalize connections on gerbes and bundles in a natural way. A flat connection on a groupoid is an integrable distribution of the morphism space compatible with the groupoid structure and complementary to both source and target fibres. A cofoliation of a stack …
$\del\delbar$-Lemma and Bott-Chern cohomology of twistor spaces
本文研究了twistor空间的Bott-Chern上同调和$\partial \overline{\partial}$-lemma的有效性，并计算了平坦4维torus twistor空间的Dolbeault上同调。
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