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Hopf type theorem for surfaces of constant weighted mean curvature in Riemannian manifolds

  • Jiaming Chen
  • Shanze Gao
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Abstract

We consider compact immersed surfaces of genus zero in a three-dimensional smooth metric measure space (M3,g,e−fdVg). We introduce a new weighted mean curvature and prove that any such surface whose weighted mean curvature is constant must be totally umbilical, provided that a condition relating the Ricci curvature of M and the weight function f is satisfied. We also give an application of this result to surfaces in ℝ3 satisfying |x|2 H + 2⟨ x,ν⟩ = c.

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Cite

BibTeXRIS
BibTeX
@misc{chen2026hopf,
  title = {{Hopf type theorem for surfaces of constant weighted mean curvature in Riemannian manifolds}},
  author = {Jiaming Chen and Shanze Gao},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2608.23036},
  eprint = {2608.23036},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2608.23036},
}
RIS
TY  - UNPB
TI  - Hopf type theorem for surfaces of constant weighted mean curvature in Riemannian manifolds
AU  - Jiaming Chen
AU  - Shanze Gao
PY  - 2026
DA  - 2026/10/21
PB  - arXiv
DO  - 10.48550/arxiv.2608.23036
UR  - https://arxiv.org/abs/2608.23036
AB  - We consider compact immersed surfaces of genus zero in a three-dimensional smooth metric measure space (M^3,g,e^−fdV_g). We introduce a new weighted mean curvature and prove that any such surface whose weighted mean curvature is constant must be totally umbilical, provided that a condition relating the Ricci curvature of M and the weight function f is satisfied. We also give an application of this result to surfaces in ℝ^3 satisfying |x|^2 H + 2⟨ x,ν⟩ = c.
ER  -