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Register a new userDiscrete CMC surfaces in R^3 and discrete minimal surfaces in S^3. A discrete Lawson correspondence
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Added by
Alexander I. Bobenko
Publication date
2017
fields
Physics
and research's language is
English
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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.
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We discuss discretization of Koenigs nets (conjugate nets with equal Laplace invariants) and of isothermic surfaces. Our discretization is based on the notion of dual quadrilaterals: two planar quadrilaterals are called dual, if their corresponding sides are parallel, and their non-corresponding diagonals are parallel. Discrete Koenigs nets are defined as nets with planar quadrilaterals admitting dual nets. Several novel geometric properties of discrete Koenigs nets are found; in particular, two-dimensional discrete Koenigs nets can be characterized by co-planarity of the intersection points of diagonals of elementary quadrilaterals adjacent to any vertex; this characterization is invariant with respect to projective transformations. Discrete isothermic nets are defined as circular Koenigs nets. This is a new geometric characterization of discrete isothermic surfaces introduced previously as circular nets with factorized cross-ratios.
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.
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.
In this paper we are interested in defining affine structures on discrete quadrangular surfaces of the affine three-space. We introduce, in a constructive way, two classes of such surfaces, called respectively indefinite and definite surfaces. The underlying meshes for indefinite surfaces are asymptotic nets satisfying a non-degeneracy condition, while the underlying meshes for definite surfaces are non-degenerate conjugate nets satisfying a certain natural condition. In both cases we associate to any of these nets several discrete affine invariant quantities: a metric, a normal and a co-normal vector fields, and a mean curvature. Moreover, we derive structural and compatibility equations which are shown to be necessary and sufficient conditions for the existence of a discrete quadrangular surface with a given affine structure.
Asymptotic net is an important concept in discrete differential geometry. In this paper, we show that we can associate affine discrete geometric concepts to an arbitrary non-degenerate asymptotic net. These concepts include discrete affine area, mean curvature, normal and co-normal vector fields and cubic form, and they are related by structural and compatibility equations. We consider also the particular cases of affine minimal surfaces and affine spheres.
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