English

Minimal graphs of arbitrary codimension in Euclidean space with bounded 2-dilation

Differential Geometry 2021-09-21 v1

Abstract

For any Λ>0\Lambda>0, let Mn,Λ\mathcal{M}_{n,\Lambda} denote the space containing all locally Lipschitz minimal graphs of dimension nn and of arbitrary codimension mm in Euclidean space Rn+m\mathbb{R}^{n+m} with uniformly bounded 2-dilation Λ\Lambda of their graphic functions. In this paper, we show that this is a natural class to extend structural results known for codimension one. In particular, we prove that any tangent cone CC of MMn,ΛM\in\mathcal{M}_{n,\Lambda} at infinity has multiplicity one. This enables us to get a Neumann-Poincareˊ\mathrm{\acute{e}} inequality on stationary indecomposable components of CC. A corollary is a Liouville theorem for MM. For small Λ>1\Lambda>1(we can take any Λ<2\Lambda<\sqrt{2}), we prove that (i) for n7n\leq7, MM is flat; (2) for n>8n>8 and a non-flat MM, any tangent cone of MM at infinity is a multiplicity one quasi-cylindrical minimal cone in Rn+m\mathbb{R}^{n+m} whose singular set has dimension n7\leq n-7.

Keywords

Cite

@article{arxiv.2109.09383,
  title  = {Minimal graphs of arbitrary codimension in Euclidean space with bounded 2-dilation},
  author = {Qi Ding and J. Jost and Y. L. Xin},
  journal= {arXiv preprint arXiv:2109.09383},
  year   = {2021}
}

Comments

56 pages

R2 v1 2026-06-24T06:07:49.082Z