Euler Characteristic of Closed Manifolds with Almost Nonnegative Curvature Operator

Abstract: We study closed manifolds with almost nonnegative curvature operator and address a question of Herrmann--Sebastian--Tuschmann concerning the sign of their Euler characteristic. Our main result shows that if a closed $2n$-dimensional manifold admits an almost nonnegative curvature operator together with a uniform upper bound on the curvature operator, then its Euler characteristic is nonnegative. In addition, under an ANCO-type condition and assuming that the fundamental group is infinite, we prove vanishing results for the Euler characteristic, the signature, and, in the spin case, the $\widehat{A}$-genus, extending recent work of Chen--Ge--Han from almost nonnegative Ricci curvature to the curvature-operator setting.

• The paper proves that every closed even-dimensional manifold with sufficiently small negative and uniformly bounded scaled curvature-operator eigenvalues has nonnegative Euler characteristic.
• The paper uses volume lower bounds, Cheeger–Gromov compactness, curvature convergence, and Chern–Gauss–Bonnet to transfer nonnegative curvature results to an almost nonnegative setting.
• The paper shows that an ANCO-type condition with infinite fundamental group forces chi and signature to vanish, and also forces the chat A-genus to vanish for spin manifolds, while the unrestricted problem remains open.

Background and motivation

The paper addresses a question posed by Herrmann, Sebastian, and Tuschmann in [HST13]: whether closed manifolds admitting an almost nonnegative curvature operator (ANCO) have nonnegative Euler characteristic. A closed smooth manifold $X$ is ANCO if it carries a sequence of metrics $g_i$ with all eigenvalues $\lambda_j(g_i)$ of the curvature operator bounded below by $-1/i$ and diameter at most 1. This notion interpolates between almost flat manifolds—where it should play a role analogous to that predicted by Lott's collapsing theory with a lower curvature-operator bound [Lot14]—and manifolds with genuinely nonnegative curvature operator, whose classification was completed by Böhm–Wilking [BW08].

For metrics with pointwise nonnegative curvature operator, nonnegativity of $\chi$ is classical (Chern–Gauss–Bonnet combined with Kulkarni's observation that the Euler polynomial is nonnegative when the curvature operator is nonnegative; see also Bourguignon–Karcher [BK78]). The question is whether this persists under "almost" hypotheses. Huang–Tan [HT23] gave partial affirmative answers: for a closed $2n$-manifold with $b_1 > 0$, if $$(\lambda_1 + \cdots + \lambda_n)\cdot \mathrm{diam}^2 \ge -c(n)$$ then $\chi = 0$. In dimension 4 this yields a complete answer to Question 4.6: either $b_1 = 0$, giving $g_i$0, or $g_i$1, giving $g_i$2. The present paper extends these partial results to all even dimensions under an additional hypothesis.

Main result: nonnegativity under a uniform upper bound

The central theorem states: let $g_i$3 be a closed smooth $g_i$4-manifold. For any $g_i$5 there exists $g_i$6 such that if the scaled eigenvalues satisfy

$g_i$7

then $g_i$8. In particular, Question 4.6 has an affirmative answer for ANCO manifolds whose curvature operators admit such a uniform upper bound along the approximating sequence. The upper bound is essential to the method: without it, no volume lower bound is available and the compactness argument fails.

The proof is by contradiction and compactness. Assuming counterexamples with $g_i$9 and $\lambda_j(g_i)$0, one rescales to unit diameter. A key lemma shows that any closed $\lambda_j(g_i)$1-manifold with $\lambda_j(g_i)$2 and curvature-operator eigenvalues bounded by $\lambda_j(g_i)$3 has volume bounded below by $\lambda_j(g_i)$4: since $\lambda_j(g_i)$5 with $\lambda_j(g_i)$6 uniformly bounded by $\lambda_j(g_i)$7 and $\lambda_j(g_i)$8, nonzero Euler characteristic forces large volume. This supplies the missing hypothesis in Kasue's Cheeger–Gromov convergence theorem [Kas89], yielding diffeomorphisms $\lambda_j(g_i)$9 with $-1/i$0 in $-1/i$1 and, via harmonic-coordinate elliptic estimates (following Anderson [And90] and Lott [Lot00]), convergence of curvature tensors in $-1/i$2 for all $-1/i$3.

Weyl's perturbation theorem for Hermitian matrices then gives $-1/i$4-convergence of each ordered eigenvalue function $-1/i$5, so the limit metric has almost-everywhere nonnegative curvature operator. Since sectional curvature bounds control the curvature-operator norm pointwise, the Chern–Gauss–Bonnet integral passes to the limit, and the classical nonnegativity result contradicts $-1/i$6.

Vanishing of genera under infinite fundamental group

The second main theorem adapts Chen–Ge–Han's vanishing results [CGH24] from almost nonnegative Ricci curvature to the curvature-operator setting. If $-1/i$7 is a closed $-1/i$8-manifold with infinite fundamental group and admits metrics $-1/i$9 with $\chi$0, then $\chi$1 and $\chi$2; if moreover $\chi$3 is spin, $\chi$4.

The argument follows [CGH24] closely. Applying the Petersen–Wink Bochner estimate [PW21] with $\chi$5, $\chi$6, one obtains $\chi$7 for forms of degree $\chi$8 or $\chi$9, which makes the Weitzenböck curvature term of the relevant Hodge-type Dirac operators almost nonnegative. Combined with infiniteness of $2n$0, the $2n$1-index argument on the universal cover forces vanishing of the indices, hence of $2n$2 and $2n$3. For spin manifolds, the estimate on 1-forms yields $2n$4, i.e., almost nonnegative Ricci curvature, and the $2n$5-vanishing follows directly from [CGH24, Theorem 1.2]. Note that an $2n$6-ANCO condition implies almost nonnegative Ricci curvature, so this theorem applies to the natural class of interest.

Limitations and open questions

Two caveats are explicit in the paper. First, the main theorem requires the uniform upper bound $2n$7 on the scaled curvature operator; the general form of Question 4.6—without any upper bound—remains open outside dimension 4. Second, the Petersen–Wink estimate is invoked only in the range $2n$8, which is appropriate here but means the argument does not extend verbatim to other sign regimes. Third, the vanishing theorem assumes infinite fundamental group; whether the conclusion holds for finite $2n$9 under the same curvature sequence is not addressed. Finally, the compactness proof relies on Kasue's theorem and $b_1 > 0$0 curvature convergence, so the regularity of the limit metric ($b_1 > 0$1, with curvature only in $b_1 > 0$2) is a structural constraint on the method rather than a defect of the statement.

Conclusion

The paper establishes two results constraining the topology of manifolds with almost nonnegative curvature operator: nonnegativity of the Euler characteristic in even dimensions under a uniform upper bound on the scaled curvature operator, and vanishing of $b_1 > 0$3, $b_1 > 0$4, and (in the spin case) $b_1 > 0$5 under an ANCO-type condition with infinite fundamental group. Both results supply evidence for the Herrmann–Sebastian–Tuschmann question, while leaving its unrestricted form—the case with no upper bound on the curvature operator—as the principal open problem.