Hitchin Thorpe Inequality Research Articles

Harmonic Spinors in the Ricci Flow

This paper provides a new definition of the Ricci flow on closed manifolds admitting harmonic spinors. It is shown that Perelman’s Ricci flow entropy can be expressed in terms of the energy of harmonic spinors in all dimensions, and in four dimensions, in terms of the energy of Seiberg–Witten monopoles. Consequently, Ricci flow is the gradient flow of these energies. The proof relies on a weighted version of the monopole equations, introduced here. Further, a sharp parabolic Hitchin–Thorpe inequality for simply-connected, spin 4-manifolds is proven. From this, it follows that the normalized Ricci flow on any exotic K3 surface must become singular.

Improved oscillation estimates and the Hitchin–Thorpe inequality on compact Ricci solitons

Stimulated by improved oscillation estimates of the potential function and the scalar curvature on compact gradient Ricci solitons introduced in a recent study by Cheng, Ribeiro, and Zhou [Proc. Am. Math. Soc. Ser. B 10, 33–45 (2023)], we give several new sufficient conditions for compact four-dimensional normalized shrinking Ricci solitons to satisfy the Hitchin–Thorpe inequality. Our new conditions refine the validity of the Hitchin–Thorpe inequality obtained by Tadano [J. Math. Phys. 58, 023503 (2017)], Tadano [J. Math. Phys. 59, 043507 (2018)], and Tadano [Differ. Geom. Appl. 66, 231–241 (2019)].

On Euler characteristic and Hitchin-Thorpe inequality for four-dimensional compact Ricci solitons

In this article, we investigate the geometry of 4 4 -dimensional compact gradient Ricci solitons. We prove that, under an upper bound condition on the range of the potential function, a 4 4 -dimensional compact gradient Ricci soliton must satisfy the classical Hitchin-Thorpe inequality. In addition, some volume estimates are also obtained.

On the geometry of asymptotically flat manifolds

In this paper, we investigate the geometry of asymptotically flat manifolds with controlled holonomy. We show that any end of such manifold admits a torus fibration over an ALE end. In addition, we prove a Hitchin-Thorpe inequality for oriented Ricci-flat $4$-manifolds with curvature decay and controlled holonomy. As an application, we show that any complete asymptotically flat Ricci-flat metric on a $4$-manifold which is homeomorphic to $\mathbb R^4$ must be isometric to the Euclidean or the Taub-NUT metric, provided that the tangent cone at infinity is not $\mathbb R \times \mathbb R_+$.

Remark on a lower diameter bound for compact shrinking Ricci solitons

In this paper, inspired by Fernández-López and García-Río [11], we shall give a new lower diameter bound for compact non-trivial shrinking Ricci solitons depending on the range of the potential function, as well as on the range of the scalar curvature. Moreover, by using a universal lower diameter bound for compact non-trivial shrinking Ricci solitons by Chu and Hu [7] and by Futaki, Li, and Li [13], we shall provide a new sufficient condition for four-dimensional compact non-trivial shrinking Ricci solitons to satisfy the Hitchin–Thorpe inequality. Furthermore, we shall give a new lower diameter bound for compact self–shrinkers of the mean curvature flow depending on the norm of the mean curvature. We shall also prove a new gap theorem for compact self–shrinkers by showing a necessary and sufficient condition to have constant norm of the mean curvature.

An upper diameter bound for compact Ricci solitons with application to the Hitchin–Thorpe inequality. II

In this addendum to the author’s paper [H. Tadano, J. Math. Phys. 58, 023503 (2017)], by using an upper diameter bound for compact shrinking Ricci solitons by Wu [preprint arXiv:1706.07897 (2017)], we shall give a new sufficient condition for compact four-dimensional shrinking Ricci solitons to satisfy the Hitchin–Thorpe inequality. Our new condition sharpens the validity of the Hitchin–Thorpe inequality given by Tadano [J. Math. Phys. 58, 023503 (2017)].

An upper diameter bound for compact Ricci solitons with application to the Hitchin–Thorpe inequality

In this article, stimulated by Fernández-López and García-Río, we shall give an upper diameter bound for compact shrinking Ricci solitons in terms of the range of the scalar curvature, as well as in terms of the range of the potential function. As an application, we shall provide some new sufficient conditions for compact four-dimensional shrinking Ricci solitons to satisfy the Hitchin–Thorpe inequality.

On a Hitchin–Thorpe inequality for manifolds with foliated boundaries

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.

Remark on a diameter bound for complete Riemannian manifolds with positive Bakry–Émery Ricci curvature

In this paper, we shall give a new upper diameter estimate for complete Riemannian manifolds in the case that the Bakry–Émery Ricci curvature has a positive lower bound and the norm of the potential function has an upper bound. Our diameter estimate improves previous ones obtained by Wei and Wylie (2009) [12] and Limoncu (2012) [8]. As an application, we shall give an upper diameter bound for compact shrinking Ricci solitons in terms of the maximum value of the scalar curvature. By using such a diameter bound, we shall provide some new sufficient conditions for four-dimensional compact shrinking Ricci solitons to satisfy the Hitchin–Thorpe inequality.

The Euler characteristic and signature of four-dimensional closed manifolds and the normalized Ricci flow equation

Let M be a closed manifold of dimension four, and let [0, T) be the maximal time interval for the normalized Ricci flow equation. We prove that, if the normalized Ricci flow equation has a solution on the non-negative real line, i.e., \(T=\infty \), then the Euler characteristic \(\chi (M)\) of M is non-negative. Under suitable assumptions on the solution of the normalized Ricci flow equation on \(M\times [0,T)\), we prove one more theorem stating that the Hitchin-Thorpe type inequality \(2\chi (M)\ge 3|\sigma (M)|\) holds between the Euler characteristic \(\chi (M)\) and the signature \(\sigma (M)\) of M. To obtain these results, we utilize the Riccati comparison theorem. In this respect, we present a new application of the Riccati comparison theorem.

Hitchin–Thorpe inequality and Kaehler metrics for compact almost Ricci soliton

The purpose of this paper is to prove a Hitchin–Thorpe inequality for a four-dimensional compact almost Ricci soliton. Moreover, we prove that under a suitable integral condition, a four-dimensional compact almost Ricci soliton is isometric to standard sphere. Finally, we prove that under a simple condition, a four-dimensional compact Ricci soliton with harmonic self-dual part of Weyl tensor is either isometric to a standard sphere $$\mathbb S ^{4}$$ or is Kaehler–Einstein.

Einstein metrics and Yamabe invariants of weighted projective spaces

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.

A note on the Hitchin-Thorpe inequality and Ricci flow on 4-manifolds

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.

Einstein four-manifolds with skew torsion

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 S 4 for a family of metrics defined by Bonneau.

DIAMETER BOUNDS AND HITCHIN-THORPE INEQUALITIES FOR COMPACT RICCI SOLITONS

We give lower bounds for the diameter of a compact Ricci soliton depending on the scalar and Ricci curvatures as well as on the range of the potential function, which do not depend on the dimension of the manifold. As an application, sufficient conditions are provided for a four-dimensional compact Ricci soliton to satisfy the Hitchin-Thorpe inequality.

Hitchin–Thorpe inequality for noncompact Einstein 4-manifolds

We prove a Hitchin–Thorpe inequality for noncompact Einstein 4-manifolds with specified 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.

Sasakian-Einstein structures on $9\#(S^2\times S^3)$

We show that 9#(S 2 × S 3 ) admits an 8-dimensional complex family of inequivalent non-regular Sasakian-Einstein structures. These are the first known Einstein metrics on this 5-manifold. In particular, the bound b 2 (M)≤8 which holds for any regular Sasakian-Einstein M does not apply to the non-regular case. We also discuss the failure of the Hitchin-Thorpe inequality in the case of 4-orbifolds and describe the orbifold version.

Spin Manifolds, Einstein Metrics, and Differential Topology

We show that there exist smooth, simply connected, fourdimensional spin manifolds which do not admit Einstein metrics, but nevertheless satisfy the strict Hitchin-Thorpe inequality. Our construction makes use of the Bauer/Furuta cohomotopy re nement of the Seiberg-Witten invariant [4, 3], in conjunction with curvature estimates previously proved by the second author [17]. 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 in nitely many other smooth structures for which no Einstein metric can exist.

Hitchin–Thorpe-Type Inequalities for Pseudo-Riemannian 4-Manifolds of Metric Signature (++−−)

It is well known that the Euler characteristic χ and the Hirzebruch index τ of a compact Einstein 4-manifold satisfy the Hitchin–Thorpe inequality χ ≥ \(\tfrac{3}{2}\); |τ|, and that these topological invariants of a compact pseudo-Riemannian Einstein 4-manifold of metric signature (++−−), with certain curvature restrictions, also satisfy a similar inequality χ ≥ \(\tfrac{3}{2}\); |τ|, which is called the Hitchin–Thorpe-type inequality. This shows that the Euler characteristic χ is nonpositive for the indefinite case. In this paper, it is shown that for a compact pseudo-Riemannian 4-manifold of signature (++−−) the Hitchin–Thorpe-type inequality holds under a weaker condition, called the diagonal Einstein condition, than the Einstein condition. Our analysis is based on the fact that the existence of a metric of signature (++−−) on a 4-manifold with the structure group SO o (2,2) is equivalent to the existence of a pair of an almost complex structure and an opposite almost complex structure on the 4-manifold, which is also equivalent to the reduction of the structure group to the maximal compact subgroup SO(2) × SO(2) of SO o (2,2).

On the Gromov-Hitchin-Thorpe inequality

The purpose of this Note is to prove an inequality which 4-dimensional Einstein manifolds must satisfy, relating the Euler characteristic, the signature, and the simplicial volume. The inequality sharpens Gromov's generalization [3] of the Hitchin-Thorpe inequality [4].