Classification of isotropic Calabi hypersurfaces
Consider the graph \[ M^n:=\{(x_i,f(x_i))\mid x_{n+1}=f(x_1,\dots,x_n),(x_1,\dots,x_n)\in \Omega\}, \] where \(\Omega\) is a domain in \(\mathbb R^n\) and \(f\) a strictly convex \(C^{\infty}\)-function on \(\Omega\). Together with the affine \textit{Calabi normalization} \(Y:=(0,\dots,0,1)\in \mathbb R^{n+1}\) the graph \(M^n\) is called a \textit{Calabi hypersurface}. Denoting by \(A\) and \(G\) the Fubini-Pick tensor and the Calabi metric respectively, a Calabi hypersurface is called \textit{\(\lambda\)-isotropic}, if for some non-negative function \(\lambda\) it holds \[ G(A(X,X),A(X,X))=\lambda^2 G^2(X,X) \] for all \(X\in T_pM^n\) and for any point \(p\in M^n\). In the case when \(\lambda\) is a constant, \(M^n\) is called \textit{constant isotropic}. The main result of the paper is a complete classification of \(\lambda\)-isotropic Calabi hypersurfaces.
scientific article; zbMATH DE number 7622625
scientific article