Obstruction to the existence of metric whose curvature has umbilical Hessian in a \(K\)-surface (Q1576540)

Let \(M\) be any compact, oriented smooth Riemannian surface without boundary; \(M_{\{\alpha_1, \dots ,\alpha_n\}}\) denote a \(K\)-surface associated with \(M\). In this paper, the author considers two metrics as the likely candidates of the ``best metric''. The first is the external metric, which is a critical point of an energy functional (\(L^2\) norm of the scalar curvature function). The second metric (refered to as ``HCMU'') has the property that its scalar curvature function has an umbilical Hessian, i.e., the 2nd covariant derivative tensor of the scalar curvature function of the metric is point-wise proportional to the metric tensor. The following theorem is the main result. Theorem 1. Let \(g\) be a HCMU metric in a \(K\)-surface \(M_{\{\alpha_1, \dots ,\alpha_n\}}\). Then the Euler-Lagrange character of the underlying surface should be determined by \(\chi(M) = \sum_{i=1}^j (1 - \alpha_i) + (n-j) + s\), where \(s\) is the number of critical points of the curvature \(K_g\) (excluding the singular points of \(g\)); \(\alpha_1, \dots ,\alpha_k\) (\(0 < k\leq n\)) are the only integers in the set of prescribed angles \(\{\alpha_1, \dots ,\alpha_n\}\); \(p_{j+1}, \dots ,p_k\) are the only local extremal points of \(K_g\) in the set of singular points \(\{p_j,\;0 < j\leq k\}\). Theorem 1 is proved by studying the structure of a special Killing vector field. The most important step in the proof of the theorem is to show that the Killing vector field has a finite number of singularities.