TY - JOUR

T1 - Symmetry of hypersurfaces with ordered mean curvature in one direction

AU - Li, Yan Yan

AU - Yan, Xukai

AU - Yao, Yao

N1 - Funding Information:
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 .
Publisher Copyright:
© 2021, The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature.

PY - 2021/10

Y1 - 2021/10

N2 - 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].

AB - 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].

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U2 - 10.1007/s00526-021-02030-5

DO - 10.1007/s00526-021-02030-5

M3 - Article

AN - SCOPUS:85111328676

VL - 60

JO - Calculus of Variations and Partial Differential Equations

JF - Calculus of Variations and Partial Differential Equations

SN - 0944-2669

IS - 5

M1 - 173

ER -