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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Every smooth complete connected embedded α-stationary hypersurface through the origin in R^{n+1} is a linear hyperplane for α > 0.
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.
Derives new scalar curvature decay estimates for steady gradient Ricci solitons and diameter upper bounds using μ-bubbles from Gromov.
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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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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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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.
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Notes on scalar curvature lower bounds of steady gradient Ricci solitons
Derives new scalar curvature decay estimates for steady gradient Ricci solitons and diameter upper bounds using μ-bubbles from Gromov.