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.
Mazet,Stable minimal hypersurfaces inℝ6
13 Pith papers cite this work. Polarity classification is still indexing.
13
Pith papers citing it
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Entire finite-Morse-index solutions to the one-phase Bernoulli problem in R^3 are axially symmetric.
Every smooth complete connected embedded α-stationary hypersurface through the origin in R^{n+1} is a linear hyperplane for α > 0.
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.
Proves that connected mean convex regions in R^{n+1} with multiple components cannot have strictly positive mean curvature, answering Gromov, with decay estimates and hyperbolic generalization.
New proof of spectral splitting theorem plus proof that finite-index minimal hypersurfaces have finite ends under nonnegative biRic curvature, generalizing Li-Wang.
Intrinsic and extrinsic area density bounds are equivalent for complete connected smooth minimal immersions in Euclidean space of any dimension and codimension, enabling extension of Schoen-Simon-Yau estimates to n=6.
Complete two-sided stable minimal hypersurfaces in R^4 are hyperplanes, established via new gradient estimates for the Green kernel under spectral Ricci bounds.
Topological linking at infinity forces polynomial scalar curvature decay on weakly bounded non-compact manifolds and yields localized obstructions to uniformly positive scalar curvature via minimal hypersurface analysis.
Complete finite-index CMC hypersurfaces in 6D product manifolds (closed non-negative curvature factor times Euclidean) are either minimal or compact.
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.
Existence of index-one minimal hypersurfaces with unbounded volume in enlargeable manifolds (dims 3-7) plus 3D scalar curvature rigidity under area-nonincreasing maps.
A PDE-based improvement-of-flatness technique for annuli provides an alternative proof of the end-structure and asymptotics for finite Morse index minimal hypersurfaces with Euclidean area growth in low dimensions.
citing papers explorer
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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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Finite index solutions to the Bernoulli problem in three dimensions are axially symmetric
Entire finite-Morse-index solutions to the one-phase Bernoulli problem in R^3 are axially symmetric.
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Bernstein-type theorem for stationary hypersurfaces of the Euler-Dierkes-Huisken functional
Every smooth complete connected embedded α-stationary hypersurface through the origin in R^{n+1} is a linear hyperplane for α > 0.
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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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A Frankel type theorem in Euclidean and hyperbolic spaces
Proves that connected mean convex regions in R^{n+1} with multiple components cannot have strictly positive mean curvature, answering Gromov, with decay estimates and hyperbolic generalization.
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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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Equivalence of intrinsic and extrinsic area bounds for minimal surfaces
Intrinsic and extrinsic area density bounds are equivalent for complete connected smooth minimal immersions in Euclidean space of any dimension and codimension, enabling extension of Schoen-Simon-Yau estimates to n=6.
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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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Linking at Infinity and Scalar Curvature Decay on Non-Compact Manifolds
Topological linking at infinity forces polynomial scalar curvature decay on weakly bounded non-compact manifolds and yields localized obstructions to uniformly positive scalar curvature via minimal hypersurface analysis.
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Finite index constant mean curvature hypersurfaces in low dimensions
Complete finite-index CMC hypersurfaces in 6D product manifolds (closed non-negative curvature factor times Euclidean) are either minimal or compact.
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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.
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Closed minimal surfaces of index one in Riemannian manifolds
Existence of index-one minimal hypersurfaces with unbounded volume in enlargeable manifolds (dims 3-7) plus 3D scalar curvature rigidity under area-nonincreasing maps.
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Improvement of flatness in annuli
A PDE-based improvement-of-flatness technique for annuli provides an alternative proof of the end-structure and asymptotics for finite Morse index minimal hypersurfaces with Euclidean area growth in low dimensions.