If a complete noncompact n-manifold with at least two ends satisfies lambda1(-gamma Delta + Ric) >= 0 for some gamma < 4/(n-1), then it splits isometrically as R x N with compact N and Ric_N >= 0; the constant is sharp.
Homological $n$-systole in $(n+1)$-manifolds and bi-Ricci curvature
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abstract
In this paper, we prove an optimal systolic inequality and the corresponding rigidity in the equality case on closed manifolds with positive bi-Ricci curvature, which generalizes the work of Bray-Brendle-Neves. The proof is given in all dimensions based on the method of minimal surfaces under the Generic Regularity Hypothesis, which is known to be true up to dimension ten.
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A sharp spectral splitting theorem
If a complete noncompact n-manifold with at least two ends satisfies lambda1(-gamma Delta + Ric) >= 0 for some gamma < 4/(n-1), then it splits isometrically as R x N with compact N and Ric_N >= 0; the constant is sharp.