Diameter estimates and Hitchin-Thorpe inequality for four-dimensional compact Quasi-Einstein manifolds

Abstract: We study compact $m$-quasi-Einstein manifolds and derive geometric estimates relating the oscillation of the potential function to the diameter of the manifold. We obtain lower bounds for the diameter in terms of the oscillation of the potential function. As an application in dimension four, we derive diameter conditions ensuring that compact $m$-quasi-Einstein manifolds satisfy the Hitchin--Thorpe inequality. Our results extend diameter estimates in smooth metric measure spaces and are consistent with known bounds in the limiting case corresponding to Ricci solitons. Finally, we provide a volume estimate involving the oscillation.

• The paper establishes oscillation-based Euler characteristic and Hitchin–Thorpe criteria, showing that sufficiently small potential oscillation forces the topological inequality for m>1.
• The authors derive three Sturm-comparison diameter estimates from minimum and maximum Ricci curvature, with formulas that converge to known gradient Ricci soliton bounds as m→∞.
• The paper proves volume and Yamabe invariant estimates with rigidity, attaining equality only for the round sphere or Einstein metrics, while leaving the unconditional Hitchin–Thorpe problem open.

Overview

This paper, by Samuel Belo, studies compact $m$-quasi-Einstein manifolds $(M^n,g,f,m)$ satisfying $$\operatorname{Ric} + \nabla^2 f - \frac{1}{m}df\otimes df = \lambda g$$ with $\lambda > 0$ and $m < \infty$, the regime in which compactness is guaranteed (2604.21002). The main contributions are: (i) an Euler characteristic estimate for compact four-dimensional quasi-Einstein manifolds expressed in terms of the oscillation $f_{osc} = f_{\max} - f_{\min}$ of the potential function; (ii) lower diameter bounds in terms of $f_{osc}$ and Ricci curvature extrema; (iii) diameter criteria ensuring the Hitchin–Thorpe inequality $2\chi(M) \pm 3\tau(M) \ge 0$; and (iv) a volume estimate with rigidity at equality. All results are consistent with the formal limit $m \to \infty$, recovering known bounds for compact gradient shrinking Ricci solitons.

The setting is motivated by the scarcity of examples: the Lü–Page–Pope family on $\mathbb{CP}^2 \# \overline{\mathbb{CP}}^{\,2}$ is, to the author's knowledge, the only explicit non-trivial compact quasi-Einstein family in dimension four, converging to the Koiso–Cao soliton as $(M^n,g,f,m)$0. Rigidity is severe — compact solutions are trivial when $(M^n,g,f,m)$1, when scalar curvature is constant, and in dimensions two and three — so dimension four is the first dimension where non-trivial compact examples exist and where topological obstructions such as Hitchin–Thorpe become meaningful.

Integral and oscillation criteria for the Hitchin–Thorpe inequality

The technical core is a pair of integral identities (Lemma 2 of the paper) expressing $(M^n,g,f,m)$2 via the Gauss–Bonnet–Chern formula combined with integrated curvature relations for quasi-Einstein manifolds. These identities express the Euler characteristic through $(M^n,g,f,m)$3, integrals involving $(M^n,g,f,m)$4, $(M^n,g,f,m)$5, $(M^n,g,f,m)$6, and $(M^n,g,f,m)$7.

Two sufficient conditions follow. First, if

$(M^n,g,f,m)$8

then the Hitchin–Thorpe inequality holds for $(M^n,g,f,m)$9. In the limit $$\operatorname{Ric} + \nabla^2 f - \frac{1}{m}df\otimes df = \lambda g$$0 this recovers the Li–Ma criterion for compact gradient shrinking solitons.

Second, the paper's central estimate states that for any compact four-dimensional quasi-Einstein manifold with $$\operatorname{Ric} + \nabla^2 f - \frac{1}{m}df\otimes df = \lambda g$$1,

$$\operatorname{Ric} + \nabla^2 f - \frac{1}{m}df\otimes df = \lambda g$$2

with equality if and only if $$\operatorname{Ric} + \nabla^2 f - \frac{1}{m}df\otimes df = \lambda g$$3 is constant. The proof combines a Colding–Minicozzi coarea-type lemma (applicable because $$\operatorname{Ric} + \nabla^2 f - \frac{1}{m}df\otimes df = \lambda g$$4 and $$\operatorname{Ric} + \nabla^2 f - \frac{1}{m}df\otimes df = \lambda g$$5 are real analytic, so the critical set of $$\operatorname{Ric} + \nabla^2 f - \frac{1}{m}df\otimes df = \lambda g$$6 has measure zero) with a Sturm comparison argument on sub-level sets of $$\operatorname{Ric} + \nabla^2 f - \frac{1}{m}df\otimes df = \lambda g$$7, completing the square in $$\operatorname{Ric} + \nabla^2 f - \frac{1}{m}df\otimes df = \lambda g$$8 to bound $$\operatorname{Ric} + \nabla^2 f - \frac{1}{m}df\otimes df = \lambda g$$9, and using the sharp lower bound $\lambda > 0$0 from Case–Shu–Wei to discard the gradient term.

Consequently, the purely analytical condition

$\lambda > 0$1

forces the Hitchin–Thorpe inequality. As $\lambda > 0$2 this becomes $\lambda > 0$3, exactly the Cheng–Ribeiro–Zhou estimate for compact gradient Ricci solitons. Notably, the Lü–Page–Pope family satisfies this oscillation bound, so each member realizes the estimate explicitly — the criterion is not vacuous in the known non-trivial examples.

Diameter estimates via Sturm comparison

Let $\lambda > 0$4 and $\lambda > 0$5 denote the minimum and maximum of $\lambda > 0$6 over the unit tangent bundle. A preliminary lemma shows that for non-trivial compact quasi-Einstein manifolds, $\lambda > 0$7 strictly, by evaluating the traced equation at extrema of $\lambda > 0$8 and applying the strong maximum principle; this ensures the constants below are well-defined.

Along a minimizing geodesic $\lambda > 0$9 between the minimizer and maximizer of $m < \infty$0, the function $m < \infty$1 satisfies

$m < \infty$2

Sturm comparison against the model equations $m < \infty$3 ($m < \infty$4) and $m < \infty$5 ($m < \infty$6), using the vanishing gradient at the endpoints, yields the two lower bounds:

$m < \infty$7

A mixed estimate also holds: if $m < \infty$8, then

$m < \infty$9

obtained by splitting the geodesic at its midpoint and multiplying the two half-geodesic bounds. All three bounds converge to the Fernández-López–García-Río diameter estimates for compact Ricci solitons as $f_{osc} = f_{\max} - f_{\min}$0, confirming consistency with the limiting theory.

Diameter criteria for the Hitchin–Thorpe inequality

Combining the oscillation criterion with the diameter bounds gives explicit geometric hypotheses. Setting $f_{osc} = f_{\max} - f_{\min}$1, the Hitchin–Thorpe inequality holds whenever the diameter satisfies any of:

• $f_{osc} = f_{\max} - f_{\min}$2;
• $f_{osc} = f_{\max} - f_{\min}$3;
• $f_{osc} = f_{\max} - f_{\min}$4, where $f_{osc} = f_{\max} - f_{\min}$5 is the unique solution in $f_{osc} = f_{\max} - f_{\min}$6 of $f_{osc} = f_{\max} - f_{\min}$7.

Each case follows from monotonicity of the relevant comparison functions. In the soliton limit these recover the known diameter-based Hitchin–Thorpe criteria, and since $f_{osc} = f_{\max} - f_{\min}$8, the resulting combined bound improves the Fernández-López–García-Río diameter threshold for compact four-dimensional gradient shrinking solitons.

Volume and Yamabe invariant estimates

Using Gursky's inequality $f_{osc} = f_{\max} - f_{\min}$9 for positive-scalar-curvature four-manifolds, together with the Euler characteristic estimate, the paper derives

$f_{osc}$0

with equality if and only if $f_{osc}$1 is isometric to the round sphere of radius $f_{osc}$2. Similarly, via the Cheng–Ribeiro–Zhou Yamabe inequality, the Yamabe invariant obeys

$f_{osc}$3

with equality precisely when $f_{osc}$4 is Einstein. These results show that small oscillation of the potential forces quantitative volume and spectral rigidity, not merely topological constraints.

Limitations and open questions

Several caveats bear directly on the strength of the results. The diameter criteria require pointwise Ricci curvature bounds $f_{osc}$5 and $f_{osc}$6 satisfying $f_{osc}$7, which hold only for non-trivial solutions; trivial (Einstein) manifolds fall outside the strict inequalities and must be treated separately. The oscillation and diameter thresholds are sufficient conditions only — the paper does not establish that quasi-Einstein manifolds satisfy the Hitchin–Thorpe inequality unconditionally, which remains open even for compact gradient shrinking Ricci solitons (Cao's original question). The existence of a quasi-Einstein metric on $f_{osc}$8, analogous to the Wang–Zhu soliton, remains unresolved, so the scope of the criteria beyond the Lü–Page–Pope family cannot currently be tested against further examples. Finally, whether the constant $f_{osc}$9 or the diameter thresholds are sharp for general $2\chi(M) \pm 3\tau(M) \ge 0$0 is not addressed.

Conclusion

The paper extends the program initiated for compact Ricci solitons to compact $2\chi(M) \pm 3\tau(M) \ge 0$1-quasi-Einstein manifolds, providing oscillation-controlled Euler characteristic, volume, and Yamabe invariant estimates, together with diameter lower bounds derived from a clean Sturm comparison argument on the potential function. Every result degenerates correctly to the known soliton case as $2\chi(M) \pm 3\tau(M) \ge 0$2, and the criteria are realized by the only known non-trivial compact four-dimensional family. The principal open problem left by the work is whether the Hitchin–Thorpe inequality holds for all compact four-dimensional quasi-Einstein manifolds without additional hypotheses on $2\chi(M) \pm 3\tau(M) \ge 0$3, curvature, or diameter.