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arXiv 2104.01675math.DG
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$\mathbb{R}^{3}$的极小曲面和$H$ -曲面的随机半空间定理

Stochastic half-space theorems for minimal surfaces and $H$-surfaces of $\mathbb{R}^{3}$

G. Pacelli Bessa, Luquesio P. Jorge, Leandro Pessoa

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摘要：

证明了$\mathbb{R}^{3}_{\raisepunct{.}}$的递归极小曲面类与有界曲率完备极小曲面类之间的强半空间定理，并证明了只要$\mathbb{R}^{3}_{\raisepunct{.}}$是一个有$\text{Ric}_M \geq 0$的完备递归$n$维黎曼流形且其截面曲率自上而下有界，则$\mathbb{R}^{3}_{\raisepunct{.}}$中任何有界曲率浸入的极小超曲面都等于某个$M\times \{s\}$.对于$H$ -曲面，我们证明了当$\sup \vert H_{_M}\vert < H$或当$\sup \vert H_{_M}\vert = H$时，随机完备曲面$M$不可能在嵌入$\R^3$的具有有界曲率的$H$ -曲面${\rm dist}(M,N)=0$的平均凸边中.最后，在$M$具有非空边界的假设下，给出了无穷远处的极大值原理。

英文摘要：

We prove a version of the strong half-space theorem between the classes of recurrent minimal surfaces and complete minimal surfaces with bounded curvature of $\mathbb{R}^{3}_{\raisepunct{.}}$ We also show that any minimal hypersurface immersed with bounded curvature in $M\times \R_+$ equals some $M\times \{s\}$ provided $M$ is a complete, recurrent $n$-dimensional Riemannian manifold with $\text{Ric}_M \geq 0$ and whose sectional curvatures are bounded from above. For $H$-surfaces we prove that a stochastically complete surface $M$ can not be in the mean convex side of a $H$-surface $N$ embedded in $\R^3$ with bounded curvature if $\sup \vert H_{_M}\vert < H$, or ${\rm dist}(M,N)=0$ when $\sup \vert H_{_M}\vert = H$. Finally, a maximum principle at infinity is shown assuming $M$ has non-empty boundary.

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