Generalized torus in \(R^ 3\) with prescribed mean curvature
The general problem concerning the existence of surfaces in Euclidean space with prescribed genus g and mean curvature H was suggested by \textit{S. T. Yau} [Semin. differential geometry, Ann. Math. Stud. 102, 669-706 (1982; Zbl 0479.53001)]. In 1983 \textit{A. E. Treibergs} and \textit{S. W. Wei} [J. Differ. Geom. 18, 513-521 (1984; Zbl 0529.53043)] gave a partial answer proving that under suitable conditions on a given function H, there exists a closed surface in \({\mathbb{R}}^{n+1}\) of genus 0, with mean curvature H. In this paper an existence theorem of such a closed surface in \({\mathbb{R}}^ 3\) of genus 1 is established. More precisely the following main result is proved: Let \(T_ s: \bar y_ s={\bar \rho}+s\bar r\) be a family of tori in \({\mathbb{R}}^ 3\), where s is a constant, \(0<s<b\), b constant, \(b>1\), \({\bar \rho}=(b \cos \theta,b \sin \theta,0)\), \(\bar r=(\cos \phi \cos \theta,\cos \phi \sin \theta,\sin \phi)\), \(0\leq \phi,\theta <2\pi\), and \(N=\cup_{s\in (0,b)}T_ s\). An arbitrary point X in N has a unique expression \({\bar \rho}+s\bar r\). Suppose \(H_{Ts}\) is the mean curvature of \(T_ s\) and H(X): \(N\to {\mathbb{R}}\), \(H\in C^{k,\alpha}\) \((k\geq 1,k\in {\mathbb{Z}},0<\alpha <1)\), is a given function which satisfies the following conditions: There are two constants \(s_ 1,s_ 2\), \(b>s_ 1\geq b/2\geq s_ 2>0\), such that (1) \(H(X)<H_{Ts}(X)\) when \(s>s_ 1\), \(H(X)>H_{Ts}(X)\) when \(0<s<s_ 2\) and \[ (2)\quad -s(b+s \cos \phi)(\partial /\partial s)H(X)\geq (b/2)(b+s \cos \phi)^{-1}+(2b+s)| H(X)| \] when \(s_ 2\leq s\leq s_ 1\). Then there exists a closed surface which is homeomorphic to a torus \(T_ 1\) and its mean curvature is equal to H(X).
scientific article; zbMATH DE number 4082411