A conformal invariant and its application to the nonexistence of minimal submanifolds

2023/10/15 by Chen, Hang
#53C18 #53C40 #53C42 #Differential Geometry (math.DG) #FOS: Mathematics

paper · doi:10.48550/arxiv.2310.09724

Abstract

Let (M^m,g) be an m-dimensional closed Riemannian manifold with non-negative sectional curvatures, m≥ 3. We define a conformal invariant and prove that, if the conformal invariant is bounded from above by a constant depending only on m, then there are no closed n-dimensional stable minimal submanifolds in M for all ξ(m)≤ n≤ m-2, where ξ(m)=1 when 3≤ m≤ 5 and ξ(m)=2 when m≥ 6. In particular, a conformal m-sphere with non-negative sectional curvatures does not admit any closed n-dimensional stable minimal submanifold for all ξ(m)≤ n≤ m-2.

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