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Strong maximum principle for mean curvature operators on subRiemannian manifolds

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Abstract

We study the strong maximum principle for horizontal (p-)mean curvature operator and p-(sub)Laplacian operator on subRiemannian manifolds including, in particular, Heisenberg groups and Heisenberg cylinders. Under a certain Hormander type condition on vector fields, we show the strong maximum principle holds in higher dimensions for two cases: (a) the touching point is nonsingular; (b) the touching point is an isolated singular point for one of comparison functions. For a background subRiemannian manifold with local symmetry of isometric translations, we have the strong maximum principle for associated graphs which include, among others, intrinsic graphs with constant horizontal (p-)mean curvature. As applications, we show a rigidity result of horizontal (p-)minimal hypersurfaces in any higher dimensional Heisenberg cylinder and a pseudo-halfspace theorem for any Heisenberg group.

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Acknowledgements

J.-H. Cheng, H.-L. Chiu, and J.-F. Hwang would like to thank the Ministry of Science and Technology of Taiwan for the support of the projects: MOST 104-2115-M-001-011-MY2, 104-2115-M-008-003-MY2, and 104-2115-M-001-009-MY2, resp. J.-H. Cheng is also grateful to the National Center for Theoretical Sciences of Taiwan for the constant support. P. Yang would like to thank the NSF of the US for the support of the project: DMS-1509505.

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Authors and Affiliations

  1. Institute of Mathematics, Academia Sinica, Taipei, Taiwan, ROC

    Jih-Hsin Cheng & Jenn-Fang Hwang

  2. Department of Mathematics, National Tsing Hua University, Hsinchu, Taiwan, ROC

    Hung-Lin Chiu

  3. Department of Mathematics, Princeton University, Princeton, NJ, 08544, USA

    Paul Yang

Authors
  1. Jih-Hsin Cheng
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  2. Hung-Lin Chiu
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  3. Jenn-Fang Hwang
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  4. Paul Yang
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Corresponding author

Correspondence to Paul Yang.

Additional information

Communicated by F. C. Marques.

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Cheng, JH., Chiu, HL., Hwang, JF. et al. Strong maximum principle for mean curvature operators on subRiemannian manifolds. Math. Ann. 372, 1393–1435 (2018). https://doi.org/10.1007/s00208-018-1700-1

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  • DOI: https://doi.org/10.1007/s00208-018-1700-1

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