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Generic scarring for minimal hypersurfaces along stable hypersurfaces
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Abstract We show that a complete, two-sided, stable immersed anisotropic minimal hypersurface in $$\mathbf {R}^4$$ has intrinsic cubic volume growth, provided the parametric elliptic integral is $C^2$ -close to the area functional. We also obtain an interior volume upper bound for stable anisotropic minimal hypersurfaces in the unit ball. We can estimate the constants explicitly in all of our results. In particular, this paper gives an alternative proof of our recent stable Bernstein theorem for minimal hypersurfaces in $$\mathbf {R}^4$$ . The new proof is more closely related to techniques from the study of strictly positive scalar curvature.more » « less
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Abstract We obtain a comparison formula for integrals of mean curvatures of Riemannian hypersurfaces via Reilly’s identities. As applications, we derive several geometric inequalities for a convex hypersurface Γ \Gamma in a Cartan-Hadamard manifold M M . In particular, we show that the first mean curvature integral of a convex hypersurface γ \gamma nested inside Γ \Gamma cannot exceed that of Γ \Gamma , which leads to a sharp lower bound for the total first mean curvature of Γ \Gamma in terms of the volume it bounds in M M in dimension 3. This monotonicity property is extended to all mean curvature integrals when γ \gamma is parallel to Γ \Gamma , or M M has constant curvature. We also characterize hyperbolic balls as minimizers of the mean curvature integrals among balls with equal radii in Cartan-Hadamard manifolds.more » « less
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1007/s00039-021-00571-7
