Volume gap for minimal submanifolds in spheres

2022/10/10 by Jianquan Ge, Fagui Li, Ge, Jianquan +1
#math.DG

paper · pdf · doi:10.48550/arxiv.2210.04654

Abstract

For a closed minimal submanifold f:Mⁿ\looparrowright \mathbbS^N in the unit sphere (n<N), we prove \rm Vol(Mⁿ) ≥(n+1)/(n+2)∫_M( 1+φ_p²) ≥ m\rm Vol(\mathbbSⁿ), where φ_p(x):=⟨ f(x),p⟩ is the height function in direction p∈ f(M), m denotes the multiplicity of p∈ f(M) and \rm Vol denotes the Riemannian volume functional, and each equality holds if and only if M is totally geodesic. As an application, if the volume of Mⁿ is less than or equal to the volume of any n-dimensional minimal Clifford torus, then Mⁿ must be embedded, verifying the non-embedded case of Yau's conjecture. In addition, we also get volume gaps for minimal hypersurfaces with constant scalar curvature, improving Cheng-Li-Yau's classical volume gap in this case. Some other volume gaps and related pinching rigidities are also obtained.

Citations

• A lower bound for 𝐿₂ length of second fundamental form on minimal hypersurfaces
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• The Min-Max theory and the Willmore conjecture
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Related

• Volume gap between the minimal submanifold and the unit sphere
• Area of minimal hypersurfaces
• On Yau rigidity theorem for minimal submanifolds in spheres
• Gap theorems for complete submanifolds in the hyperbolic space
• Embeddedness of least area minimal hypersurfaces