Discrete flat surfaces and linear Weingarten surfaces in hyperbolic 3-space (Q2844722)

Away from umbilic points of a smooth minimal (or CMC) immersed surface \(f(x,y)\) into \(\mathbb{R}^3\) or \(\mathbb{H}^3\) one can take isothermic coordinates \((x,y)\), that are isothermal and diagonalize the second fundamental form for the induced metric. In the Euclidean case, the stereographic projection of the Gauss map defines a holomorphic function \(g\) on the complex plane that, with the Hopf differential, constitutes data to define the Weierstrass representation of \(f\), and consequent expressions of the derivatives \(f_x\) and \(f_y\) give rise to the definition of the difference \(f_q-f_p\) for \(p=(m,n)\) and \(q\) either \((m+1,n)\) or \((m, n+1)\) obtaining in this way a discretization of the surface \(f\) up to translations. A discrete minimal surface \(f\) is then defined (up to translations) from a holomorphic discrete map \(g:D\to \mathbb{C}\), \(D\subset\mathbb{Z}^2\), that is, the cross ratio (in quaternionic notation) \(\operatorname{cr}_{m, n}=(g_{m+1,n}-g_{m,n})(g_{m+1,n+1}-g_{m+1,n})^{-1}(g_{m,n+1}-g_{m+1,n+1})(g_{m,n}-g_{m,n+1})^{-1}\) can be described as \(\operatorname{cr}_{m, n}=\frac{\alpha_{(m,n)(m+1,n)}}{\alpha_{(m,n)(m,n+1)}}<0\) (not necessarily equal to \(-1\)) holding for all quadrilaterals, for some symmetric discrete real function \(\alpha\) satisfying \(\alpha_{(m,n)(m+1,n)}=\alpha_{(m,n+1)(m+1,n+1)}\), \(\alpha_{(m,n)(m,n+1)}=\alpha_{(m+1,n)(m+1,n+1)}\).NEWLINENEWLINEThe first proposal of the authors is to define discrete flat surfaces in the 3-dimensional hyperbolic space in the context of discrete integrable systems. Considering the 3-dimensional hyperbolic space as a hyperquadric of the Minkowski 4-space \(\mathbb{R}^{3,1}\) and using \( SL_2(\mathbb{C})\)-valued lifts \(F\) of a smooth isothermically-parameterized CMC 1 surface \(f_1:M\to\mathbb{H}^3\) and Bryant's equation on \(dF\), expressed in terms of an holomorphic function \(g\) (Weierstrass data) with non-zero derivative, one obtains \(f_1=F\cdot \bar{F}^T\) as shown by \textit{R. L. Bryant} in [Astérisque 154--155, 321--347 (1988; Zbl 0635.53047)]. NEWLINENEWLINENEWLINETo each \(f_1\) there is a related flat surface \(f_0\) with singularities, and a suitable change of coordinates leads to an equation satisfied by the components of the lift that in some particular cases is the Airy equation. One-parameter families of deformations \(f_t\) through linear Weingarten surfaces of Bryant-type between \(f_0\) and \(f_1\) are considered and non-uniqueness is shown for this type of deformations. A discretization of such surfaces can be built using the light cone model space of \(\mathbb{H}^3\) in \(\mathbb{R}^{4,1}\) and quaternionic notation, as shown by \textit{U. Hertrich-Jeromin} in [Manuscr. Math. 102, No. 4, 465--486 (2000; Zbl 0979.53008)]. In Theorem 4.2, the authors give an alternative way to define them by \(f_1=(1/\mathrm{det} F)F\bar{F}^T\) (up to rigid motion in \( \mathbb{H}^3 \)) following Bryant's formula \( F_q-F_p = F_p G(g_p,g_q) \frac{\lambda \alpha_{pq}}{g_q-g_p}\), where \(G(g_p,g_q)\) is a matrix depending only on \(g_p\) and \(g_q\) with a discrete holomorphic function \(g\), \(\alpha\) is the cross ratio factorization function and \(\lambda\) a free real parameter. Multiplication of \(F\) by a suitable matrix depending on \(g\) defines a new lift \(E\) that represents \(f_0\) discrete and flat in \(\mathbb{H}^3\), and it is proved in Theorem 4.6 that \(f_0\) has concircular quadrilaterals.NEWLINENEWLINEThe authors also describe a deformation family \(f_t\) of discrete linear Weingarten surfaces from \(f_0\) to \(f_1\) and with concircular quadrilaterals. A discrete caustic surface of a flat discrete one is also defined by following the smooth example in terms of the Weierstrass data. This requires to define a normal at vertices of a discrete flat surface and it is shown that negativeness of \(\lambda \alpha_{pq}\) is equivalent to the normal geodesics in \(\mathbb{H}^3\) emanating from two adjacent vertices \(f_p\) and \(f_q\) to intersect (in a unique point), or equivalently the edge \(pq\) is vertical. The set \(C_f\) of such intersection points is a discrete surface named the caustic or focal surface of \(f\), not necessarily flat. A lift \(E_{(C_f)}\) is defined for each vertical edge \(pq\) satisfying the formula \( C_f=(1/\mathrm{det}(E_{C_f})) E_{C_f}\cdot \overline{ E_{C_f}}^T\). Discrete caustics have properties similar to that of caustics in the smooth case. Many examples are described throughout the paper, with some beautiful figures, and special attention is given to the discrete flat surface for the case \(g=z^{4/3}\), related to the Airy equation.