A variational problem related to conformal maps
For a smooth map \(f : (M^m, g) \to (N, h)\) between compact Riemannian manifolds, let us consider a covariant symmetric tensor \(T_f := f^* h - \frac{1}{m} |df|^2 g\). In this paper, the author considers the functional \[ \Phi(f) = \int_M \| T_f\|^2\, dv_g, \] where \(\| T_f\|^2 = \sum_{i,j} T_f(e_i, e_j)^2\) for an orthonormal frame \(\{e_i\}\) on \(M\). Note that \(f\) is conformal at a point if and only if \(T_f = 0\) at the point. The functional \(\Phi(f)\) measures a quantity of the conformality of \(f\). The author obtains the first variational formula and the second variational formula for the functional \(\Phi(f)\). The author also shows a kind of monotonicity formula and a Bochner type formula for \(T_f\). Finally, the author considers a variational problem of minimizing the functional \(\Phi(f)\) in each \(3\)-homotopy class of the Sobolev space. The author proves that for a given \(f_0 : M \to N\), there exists a minimizer of the functional \(\Phi(f)\) in the space of functions which are \(3\)-homotopic to \(f_0\) and have uniform \(L^{1,4}(M, N)\)-bound.
scientific article; zbMATH DE number 5969047