Two applications of the unit normal bundle of a minimal surface in \(R^ N\) (Q756141)

Let f be an eigenfunction of eigenvalue N on an open set U in \(S^ N(1)\) such that Hess f\(+f<, >\) has an eigenvalue 0 of multiplicity N-2, where \(<, >\) is the metric of \(S^ N(1)\). Then the map of U into \(R^{N+1}\) defined by \((*)f\eta +\text{grad} f\), where \(\eta\) is the identity map on \(S^ N(1)\), is a map of rank 2 and gives a minimal surface. Conversely, for a minimal surface M in \(R^{N+1}\), a neighborhood of each point of M without geodesic points has this representation (*). Moreover, if M is a complete orientable minimal surface of finite total curvature \(\alpha\) (M) then there is a global representation (*) of M and a positive real number c(N) depending on N such that index(M)\(\leq -c(N)\alpha (M)\).