Hyperbolic linear Weingarten surfaces in \(\mathbb R^{3}\) (Q2644297)

A surface \(M\): \(x(u,v)\) in Euclidean space \(\mathbb{R}^3\) is called a hyperbolic linear Weingarten surface (shortly: HLW-surface) if its mean curvature \(H\) and its Gaussian curvature \(K\) satisfy the relationship \(2aH+bK=c\) for real numbers \(a,b,c\) such that \(a^2+bc<0\) (which implies \(bc<0)\). In Section 2, the authors obtain a representation for a HLW-surface \(M\): \(x(u,v)\) which is based on expressing the derivatives \(x_u,x_v\) in terms of the Gauss map \(N(u,v)\) of \(M\) and its derivatives \(N_u\), \(N_v\) when, more generally, \(a,b,c\) are functions on \(M\); hereby the coordinates \((u,v)\) on \(M\) are assumed to be (locally) isothermal for the Lorentzian metric \(\sigma:=a\cdot I+b\cdot II=a\langle dx,dx\rangle- b\langle dx,dN\rangle =\rho(u,v)\cdot(du^2-dv^2)\) (see Theorem 1) and additionally (see Theorem 2), isothermal with respect to the Gauss map \(N(u,v)\) of \(M\) \((\Leftrightarrow\langle N_u,N_v\rangle=0\) and \(\langle N_u,N_u\rangle=\langle N_v,N_v\rangle)\). Section 3 is devoted to complete HLW-surfaces. Each complete HLW-surface \(M\) endowed with the Lorentzian metric \(a\cdot I+b\cdot II\) is conformally diffeomorphic to the Lorentz-Minkowski plane and can be related with the sine-Gordon equation (Theorem 5). Conversely, Theorem 7 describes a procedure to construct complete HLW-surfaces from solutions \(\varphi(u, v)\) of the sine-Gordon type differential equation \(\varphi_{uv}=|b/c| \cdot\sin\varphi\). Finally, some examples of complete HLW-surfaces are given which are gained from solutions of the pendulum equation. Remark: Apparently, the method presented in this note yields (only) a special class of HLW-surfaces \(M\), since the coordinates \(u,v\) on \(M\) are assumed to be (locally) isothermal for both the Lorentzian metric \(\sigma\) and the Riemannian metric \(\langle dN,dN\rangle\); this is not stated explicitly by the authors.