Abstract
For a connected n-dimensional compact smooth hypersurface M without boundary embedded in \({\mathbb {R}}^{n+1}\), a classical result of Aleksandrov shows that it must be a sphere if it has constant mean curvature. Li and Nirenberg studied a one-directional analog of this result: if every pair of points \((x',a), (x',b)\in M\) with \(a<b\) has ordered mean curvature \(H(x',b)\le H(x',a)\), then M is symmetric about some hyperplane \(x_{n+1}=c\) under some additional conditions. Their proof was done by the moving plane method and some variations of the Hopf Lemma. We obtain the symmetry of M under some weaker assumptions using a variational argument, giving a positive answer to the conjecture in [13].
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Communicated by X.Cabre.
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YYL is partially supported by NSF Grants DMS-1501004, DMS-2000261, and Simons Fellows Award 677077. XY is partially supported by AMS-Simons Travel Grant and AWM-NSF Travel Grant 1642548. YY is partially supported by NSF grants DMS-1715418 and DMS-1846745, and Sloan Research Fellowship .
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Li, Y.Y., Yan, X. & Yao, Y. Symmetry of hypersurfaces with ordered mean curvature in one direction. Calc. Var. 60, 173 (2021). https://doi.org/10.1007/s00526-021-02030-5
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DOI: https://doi.org/10.1007/s00526-021-02030-5