Complete 3-manifolds with scalar curvature lower bound, finitely many ends, and finite first Betti number satisfy a sharp bottom spectrum upper bound and are parabolic under positive scalar curvature.
A sharp spectral splitting theorem
7 Pith papers cite this work. Polarity classification is still indexing.
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Pith papers citing it
abstract
We prove a sharp spectral generalization of the Cheeger--Gromoll splitting theorem. We show that if a complete non-compact Riemannian manifold $M$ of dimension $n\geq 2$ has at least two ends and \[ \lambda_1(-\gamma\Delta+\mathrm{Ric})\geq 0, \] for some $\gamma<\frac{4}{n-1}$, then $M$ splits isometrically as $\mathbb R\times N$ for some compact manifold $N$ with nonnegative Ricci curvature. We show that the constant $\frac{4}{n-1}$ is sharp, and the multiple-end assumption is necessary for any $\gamma>0$.
fields
math.DG 7representative citing papers
Under spectral Ricci bounds and mean-convex boundary, complete manifolds split isometrically as products or admit positive sectional curvature metrics in dimensions other than 4.
Rigidity theorems establish that nonnegative m-intermediate curvature forces product splitting with Euclidean space in dimensions 3-7 for restricted m, with constructions proving the condition m² - mn + m + n > 0 is sharp.
Establishes new Lipschitz and width lower bounds plus rigidity theorems for bands in 3-manifolds with negative Ricci bounds via μ-bubbles, including a sharp boundary-area estimate for certain noncompact manifolds with scalar curvature ≥ −6.
New proof of spectral splitting theorem plus proof that finite-index minimal hypersurfaces have finite ends under nonnegative biRic curvature, generalizing Li-Wang.
Complete two-sided stable minimal hypersurfaces in R^4 are hyperplanes, established via new gradient estimates for the Green kernel under spectral Ricci bounds.
Proves new criticality and splitting theorems for operators with spectral Ricci bounds, then classifies 1/3-stable minimal hypersurfaces in R^4 as one-ended or catenoids and δ-stable ones with δ>1/3 as hyperplanes.
citing papers explorer
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Bottom spectrum and parabolicity of 3-manifolds with scalar curvature lower bound
Complete 3-manifolds with scalar curvature lower bound, finitely many ends, and finite first Betti number satisfy a sharp bottom spectrum upper bound and are parabolic under positive scalar curvature.
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Rigidity and flexibility under spectral Ricci lower bounds and mean-convex boundary
Under spectral Ricci bounds and mean-convex boundary, complete manifolds split isometrically as products or admit positive sectional curvature metrics in dimensions other than 4.
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Intermediate curvature and splitting theorem
Rigidity theorems establish that nonnegative m-intermediate curvature forces product splitting with Euclidean space in dimensions 3-7 for restricted m, with constructions proving the condition m² - mn + m + n > 0 is sharp.
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Band Width Estimates and Rigidity of Manifolds with Negative Curvature
Establishes new Lipschitz and width lower bounds plus rigidity theorems for bands in 3-manifolds with negative Ricci bounds via μ-bubbles, including a sharp boundary-area estimate for certain noncompact manifolds with scalar curvature ≥ −6.
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Spectral splitting theorem and ends of minimal hypersurfaces
New proof of spectral splitting theorem plus proof that finite-index minimal hypersurfaces have finite ends under nonnegative biRic curvature, generalizing Li-Wang.
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Gradient estimates for the Green kernel under spectral Ricci bounds, and the stable Bernstein theorem in $\mathbb{R}^4$
Complete two-sided stable minimal hypersurfaces in R^4 are hyperplanes, established via new gradient estimates for the Green kernel under spectral Ricci bounds.
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Criticality, splitting theorems under spectral Ricci bounds and the topology of stable minimal hypersurfaces
Proves new criticality and splitting theorems for operators with spectral Ricci bounds, then classifies 1/3-stable minimal hypersurfaces in R^4 as one-ended or catenoids and δ-stable ones with δ>1/3 as hyperplanes.