Symmetry of hypersurfaces with ordered mean curvature in one direction

Yan Yan Li, Xukai Yan, Yao Yao

Research output: Contribution to journalArticlepeer-review


For a connected n-dimensional compact smooth hypersurface M without boundary embedded in Rn+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) ∈ M with a< b has ordered mean curvature H(x, b) ≤ H(x, a) , then M is symmetric about some hyperplane xn+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].

Original languageEnglish (US)
Article number173
JournalCalculus of Variations and Partial Differential Equations
Issue number5
StatePublished - Oct 2021

All Science Journal Classification (ASJC) codes

  • Analysis
  • Applied Mathematics


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