In this paper, we give a new proof of the splitting theorem on manifolds with nonnegative spectral Ricci curvature proved in [APX24, CMMR24, HW26]. Furthermore, by constructing weighted minimizing geodesics at infinity, we show that minimal hypersurfaces with finite index in manifolds with nonnegative biRic curvature must have finite ends, generalizing the result of Li-Wang [LW04] on manifolds with nonnegative sectional curvature.
The paper develops a geodesic-line approach to spectral Ricci curvature and uses it to control the topology at infinity of minimal hypersurfaces.
Spectral splitting
If $\lambda_1(-\alpha\Delta+\operatorname{Ric})\geq0$ with $\alpha<4/(n-1)$ and the associated weighted metric admits a minimizing line, then $\operatorname{Ric}\geq0$ pointwise and the manifold splits off an $\mathbb R$ factor.
Ends at infinity
Nonnegative spectral Ricci curvature outside a compact set, together with infinitely many ends, forces the existence of a weighted minimizing geodesic line outside a larger compact set.
Minimal hypersurfaces
Every complete two-sided finite-index minimal hypersurface in an ambient manifold of dimension at most five with nonnegative biRic curvature has finitely many ends, without a properness assumption.
The splitting proof adapts the Cheeger–Gromoll Busemann-function argument to the weighted length functional determined by a positive solution of the spectral equation. For the application, infinitely many ends are used to construct a weighted line at infinity; the positive solution is then perturbed to make the spectral inequality strict outside a compact set. The second variation of weighted length produces the contradiction and hence finiteness of ends.
My research interests include Geometric Analysis and Partial Differential Equations.