We prove that a compact, connected, and oriented 4-dimensional gradient $m$-quasi-Einstein manifold with $m\in [1, \infty]$ which is additionally a spin manifold must satisfy the Hitchin-Thorpe Inequality. We show further that the homeomorphism-type of the universal cover of such a manifold is either $S^4$ or a connected sum of some number of $S^2\times S^2$ when the potential function is nontrivial.
In this short paper, we prove a Hitchin-Thorpe type inequality for closed 4-manifolds with non-positive Yamabe invariant, and admitting long time solutions of the normalized Ricci flow equation with bounded scalar curvature.
In this article, we investigate the geometry of $4$-dimensional compact gradient Ricci solitons. We prove that, under an upper bound condition on the range of the potential function, a $4$-dimensional compact gradient Ricci soliton must satisfy the classical Hitchin-Thorpe inequality. In addition, some volume estimates are also obtained.
We prove a Hitchin-Thorpe inequality for noncompact Einstein 4-manifolds with asymptotic geometry at infinity. The asymptotic geometry at infinity is either a cusp bundle over a compact space (the fibered cusps) or a fiber bundle over a cone with a compact fiber (the fibered boundary). Many noncompact Einstein manifolds come with such a geometry at infinity.
In this article, stimulated by Fernandez-Lopez and Garcia-Rio, we shall give an upper diameter bound for compact Ricci solitons in terms of the range of the scalar curvature. As an application, we shall provide some sufficient conditions for four-dimensional compact Ricci solitons to satisfy the Hitchin-Thorpe inequality.
We prove a Hitchin-Thorpe inequality for noncompact 4-manifolds with foliated geometry at infinity by extending on previous work by Dai and Wei. After introducing the objects at hand, we recall some preliminary results regarding the $G$-signature formula and the rho invariant, which are used to obtain expressions for the signature and Euler characteristic in our geometric context. We then derive our main result, and present examples.
An orbifold version of the Hitchin-Thorpe inequality is used to prove that certain weighted projective spaces do not admit orbifold Einstein metrics. Also, several estimates for the orbifold Yamabe invariants of weighted projective spaces are proved.
We develop a notion of Einstein manifolds with skew torsion on compact, orientable Riemannian manifolds of dimension four. We prove an analogue of the Hitchin-Thorpe inequality and study the case of equality. We use the link with self-duality to study the moduli space of 1-instantons on the 4-sphere for a family of metrics defined by Bonneau.
It is shown that there are infinitely many compact orientable smooth 4-manifolds which do not admit Einstein metrics, but nevertheless satisfy the strict Hitchin-Thorpe inequality 2 chi > 3 |tau|. The examples in question arise as non-minimal complex algebraic surfaces of general type, and the method of proof stems from Seiberg-Witten theory.
We show that there exist smooth, simply connected, four-dimensional spin manifolds which do not admit Einstein metrics, but nonetheless satisfy the strict Hitchin-Thorpe inequality. Our construction makes use of the Bauer/Furuta cohomotopy refinement of the Seiberg-Witten invariant, in conjunction with curvature estimates previously proved by the second author. These methods also easily allow one to construct examples of topological 4-manifolds which admit an Einstein metric for one smooth structure, but which have infinitely many other smooth structures for which no Einstein metric can exist.
We introduce the notion of a special monopole class on a four-manifold. This is used to prove restrictions on the smooth structures of Einstein manifolds. As an application we prove that there are Einstein four-manifolds which are simply connected, spin, and satisfy the strict Hitchin--Thorpe inequality, and which are homeomorphic to manifolds without Einstein metrics. In the Appendix, written with J. Wehrheim, we use special monopole classes to discuss the classification of Ricci-flat four-manifolds.
We prove an analogue of the Kobayashi-Hitchin correspondence oncompact connected 3-folds that is fibered on orbifold Riemann surfaces and satisfy an integrability condition, which contains compact connected Sasakian 3-folds. We define mini-holomorphic bundles on such 3-folds and the algebraic Dirac-type singularities on mini-holomorphic bundles, and prove that there exists a special Hermitian metric (admissible BHE-metric) on a Dirac-type singular mini-holomorphic bundle if the bundle satisfies a slope stability.
We construct infinitely many examples of finite volume 4-manifolds with $T^3$ ends that do not admit any cusped asymptotically hyperbolic Einstein metrics yet satisfy a strict logarithmic version of the Hitchin-Thorpe inequality due to Dai-Wei. This is done by using estimates from Seiberg-Witten theory due to LeBrun as well as a method for constructing solutions to the Seiberg-Witten equations on noncompact manifolds due to Biquard. We also use constructions coming from the $Pin^-(2)$ monopole equations to obtain a larger class of manifolds where these techniques apply.
We show that the minimal volume entropy of closed manifolds remains unaffected when nonessential manifolds are added in a connected sum. We combine this result with the stable cohomotopy invariant of Bauer-Furuta in order to present an infinite family of four-manifolds with the following properties: 1) They have positive minimal volume entropy. 2) They satisfy a strict version of the Gromov-Hitchin-Thorpe inequality, with a minimal volume entropy term. 3) They nevertheless admit infinitely many distinct smooth structures for which no compatible Einstein metric exists.
We survey the definitions and some important properties of several asymptotic invariants of smooth manifolds, and discuss some open questions related to them. We prove that the (non-)vanishing of the minimal volume is a differentiable property, which is not invariant under homeomorphisms. We also formulate an obstruction to the existence of Einstein metrics on four-manifolds involving the volume entropy. This generalizes both the Gromov--Hitchin--Thorpe inequality and Sambusetti's obstruction.
We prove the existence of a Hermitian-Einstein metric on holomorphic vector bundles with a Hermitian metric satisfying the analytic stability condition, under some assumption for the underlying Kähler manifolds. We also study the curvature decay of the Hermitian-Einstein metrics. It is useful for the study of the classification of instantons and monopoles on the quotient of $4$-dimensional Euclidean space by some types of closed subgroups. We also explain examples of doubly periodic monopoles corresponding to some algebraic data.
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…
We prove the Kobayashi-Hitchin correspondence between good wild harmonic bundles and polystable good filtered $λ$-flat bundles satisfying a vanishing condition. We also study the correspondence for good wild harmonic bundles with the homogeneity with respect to a group action, which is expected to provide another way to construct Frobenius manifolds.
Simons Center for Geometry and Physics, Stony Brook University（石溪大学西蒙斯几何与物理中心）；Institute for Advanced Study in Mathematics, Zhejiang University（浙江大学数学高等研究院）
Necessary and sufficient conditions to the existence of a hermitian connection with totally skew-symmetric torsion and holonomy contained in SU(3) are given. Non-compact solution to the supergravity-type I equations of motion with non-zero flux and non-constant dilaton is found in dimensions 6. Non-conformally flat non-compact solutions to the supergravity-type I equations of motion with non-zero flux and non-constant dilaton are found in dimensions 7 and 8. A Riemannian metric with holonomy contained in $G_2$ arises from our considerations and Hitchin's flow equations, which seems to be new. …
In this article we introduce $A$-valued Einstein-Hilbert-Palatini functional ($A$-EHP) over a n-manifold $M$, where $A$ is an arbitrary graded algebra, as a generalization of the functional arising in the study of the first order formulation of gravity. We show that if $A$ is weak $(k,s)$-solvable, then $A$-EHP is non-null only if $n<k+s+3$. We prove that essentially all algebras modeling classical geometries (except semi-Riemannian geometries with specific signatures) satisfy this condition for $k=1$ and $s=2$, including Hitchin's generalized complex geometry, Pantilie's generalized quaternio…
