Half-Space Theorems for 1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{1}$$\end{document}-Surfaces of H3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {H}^3$$\end{document}

被引：0
作者：
G. Pacelli Bessa [1 ]
Tiarlos Cruz [2 ]
Leandro F. Pessoa [3 ]
机构：
[1] Universidade Federal do Ceará,Departamento de Matemática
[2] Universidade Federal de Alagoas,Instituto de Matemática
[3] Universidade Federal do Piauí,Departamento de Matemática
来源：
The Journal of Geometric Analysis | 2024年 / 34卷 / 11期
关键词：
1-surfaces; Half-space theorems; Intersection problem; Maximum principle at infinity; parabolicity; Stochastic completeness; 53A10; 53C21; 53C42;
D O I：
10.1007/s12220-024-01799-z
中图分类号：
学科分类号：
摘要：
We study strong half-space theorems for the classes of complete 1-surfaces with bounded curvature, parabolic 1-surfaces, and stochastically complete H-surfaces with H<1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H<1$$\end{document} immersed in the hyperbolic space H3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {H}^3$$\end{document}. As a by-product of the techniques we obtain a Maximum Principle at Infinity for 1-surfaces in H3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {H}^3$$\end{document}. We also address the intersection problem for 1-surfaces immersed in a complete Riemannian three-manifold P with Ricci curvature bounded from below by -2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$-2$$\end{document}. We establish a splitting result provided the distance between the 1-surfaces is realized and RicP≥-2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\text {Ric}_{P} \ge -2$$\end{document}, and a Frankel’s type theorem for 1-surfaces with bounded curvature immersed in P when RicP>-2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\text { Ric}_{P} > -2$$\end{document}.
引用
收藏
相关论文
共 103 条
  • [1] Barbosa J.L(1988)Stability of hypersurface of constant mean curvature in Riemannian manifolds Math. Z 197 123-138
  • [2] do Carmo M(2001)Half-space theorems for minimal surfaces with bounded curvature J. Differ. Geom. 57 493-508
  • [3] Eschenburg J(2024)Half-space theorems for recurrent minimal and H-surfaces of Rev. Mat. Iberoam. 40 1609-1630
  • [4] Bessa GP(1987)Surfaces of mean curvature one in hyperbolic space Astérisque No. 154–155 321-347
  • [5] Jorge L(2018)Mean curvature in manifolds with Ricci curvature bounded from below Comment. Math. Helv. 93 55-69
  • [6] Oliveira-Filho G(2001)The geometry of finite topology Bryant surfaces Ann. Math. 153 623-659
  • [7] Bessa GP(2017)Embeddedness of the solutions to the H-Plateau problem Adv. Math. 317 553-574
  • [8] Jorge L(1992)A warped product splitting theorem Duke Math. J. 67 571-574
  • [9] Pessoa L(2009)Half-space theorem, embedded minimal annuli and minimal graphs in the Heisenberg group Proc. Lond. Math. Soc. 98 445-470
  • [10] Bryant R (2011)Half-space theorems for minimal surfaces in J. Differ. Geom. 88 41-59
12345678910