Closed q-convex hypersurfaces in (n+1)-dimensional Riemannian manifolds satisfy vanishing theorems for Betti numbers under lower bounds on curvature operator eigenvalues, implying they are rational homology spheres with finite fundamental group when convex or pinched.
Milnor,Morse theory, Based on lecture notes by M
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Proves that equality in the sharp lower bound for the first p-eigenvalue of the Hodge Laplacian on closed submanifolds in space forms occurs only on topological spheres assuming positivity.
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On $q$-convex hypersurfaces in Riemannian manifolds
Closed q-convex hypersurfaces in (n+1)-dimensional Riemannian manifolds satisfy vanishing theorems for Betti numbers under lower bounds on curvature operator eigenvalues, implying they are rational homology spheres with finite fundamental group when convex or pinched.
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On the first eigenvalue of the Hodge Laplacian of submanifolds
Proves that equality in the sharp lower bound for the first p-eigenvalue of the Hodge Laplacian on closed submanifolds in space forms occurs only on topological spheres assuming positivity.