Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userStochastic half-space theorems for minimal surfaces and $H$-surfaces of $mathbb{R}^{3}$
221
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
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 $Mtimes R_+$ equals some $Mtimes {s}$ 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.
rate research
Read More
We establish curvature estimates and a convexity result for mean convex properly embedded $[varphi,vec{e}_{3}]$-minimal surfaces in $mathbb{R}^3$, i.e., $varphi$-minimal surfaces when $varphi$ depends only on the third coordinate of $mathbb{R}^3$. Led by the works on curvature estimates for surfaces in 3-manifolds, due to White for minimal surfaces, to Rosenberg, Souam and Toubiana, for stable CMC surfaces, and to Spruck and Xiao for stable translating solitons in $mathbb{R}^3$, we use a compactness argument to provide curvature estimates for a family of mean convex $[varphi,vec{e}_{3}]$-minimal surfaces in $mathbb{R}^{3}$. We apply this result to generalize the convexity property of Spruck and Xiao for translating solitons. More precisely, we characterize the convexity of a properly embedded $[varphi,vec{e}_{3}]$-minimal surface in $mathbb{R}^{3}$ with non positive mean curvature when the growth at infinity of $varphi$ is at most quadratic.
The main result of this paper is a discrete Lawson correspondence between discrete CMC surfaces in R^3 and discrete minimal surfaces in S^3. This is a correspondence between two discrete isothermic surfaces. We show that this correspondence is an isometry in the following sense: it preserves the metric coefficients introduced previously by Bobenko and Suris for isothermic nets. Exactly as in the smooth case, this is a correspondence between nets with the same Lax matrices, and the immersion formulas also coincide with the smooth case.
The Riemannian product of two hyperbolic planes of constant Gaussian curvature -1 has a natural Kahler structure. In fact, it can be identified with the complex hyperbolic quadric of complex dimension two. In this paper we study Lagrangian surfaces in this manifold. We present several examples and classify the totally umbilical and totally geodesic Lagrangian surfaces, the Lagrangian surfaces with parallel second fundamental form, the minimal Lagrangian surfaces with constant Gaussian curvature and the complete minimal Lagrangian surfaces satisfying a bounding condition on an important function that can be defined on any Lagrangian surface in this particular ambient space.
Let $M$ be a compact constant mean curvature surface either in $mathbb{S}^3$ or $mathbb{R}^3$. In this paper we prove that the stability index of $M$ is bounded below by a linear function of the genus. As a by product we obtain a comparison theorem between the spectrum of the Jacobi operator of $M$ and those of Hodge Laplacian of $1$-forms on $M$.
In this note, we study minimal Lagrangian surfaces in $mathbb{B}^4$ with Legendrian capillary boundary on $mathbb{S}^3$. On the one hand, we prove that any minimal Lagrangian surface in $mathbb{B}^4$ with Legendrian free boundary on $mathbb{S}^3$ must be an equatorial plane disk. One the other hand, we show that any annulus type minimal Lagrangian surface in $mathbb{B}^4$ with Legendrian capillary boundary on $mathbb{S}^3$ must be congruent to one of the Lagrangian catenoids. These results confirm the conjecture proposed by Li, Wang and Weng (Sci. China Math., 2020).
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
