New spectral Bishop--Gromov and Bonnet--Myers theorems and applications to isoperimetry

We show a sharp and rigid spectral generalization of the classical Bishop--Gromov volume comparison theorem: if a closed Riemannian manifold (M, g) of dimension n3 satisfies \ ₁ (-+Ric) n-1 \ for some 0-1n-2, then vol (M) (S^n), and ₁ (M) is finite. Moreover, the bound on is sharp for this result to hold. A generalization of the Bonnet--Myers theorem is also shown under the same spectral condition. The proofs involve the use of a new unequally weighted isoperimetric problem and unequally warped -bubbles. As an application, in dimensions 3 n 5, we infer sharp results on the isoperimetric structure at infinity of complete manifolds with nonnegative Ricci curvature and uniformly positive biRicci curvature.