The heat flow and harmonic maps between complete manifolds
This paper consists of two parts. In the first one, the author extends the result of Li and Tam for heat equations of harmonic maps between complete noncompact manifolds. Let \(M\), \(N\) be complete noncompact Riemannian manifolds, and let \(\varphi\) be a \(C^1\)-map from \(M\) into \(N\). Consider the initial value problem for heat equations of harmonic maps: \(\frac{\partial u}{\partial t}(x,t) = \tau(u)(x,t)\), \(u(x,0) = \varphi(x)\) for \(x \in M\) and \(0 \leq t < \infty\), where \(\tau (u)\) is the tension field of \(u\). Li and Tam showed long-time existence of a unique solution for the above problem when the Ricci curvature \(\text{Ric}_M \geq -C\), the sectional curvature \(K_N \leq 0\), the map \(\varphi\) is bounded and \(\sup_M e(\varphi) < \infty\), where \(C\) is a nonnegative constant, and \(e(\varphi)\) is the energy density of the map \(\varphi\). The author proves a similar result under weaker conditions \(\text{Ric}_M \geq -C( 1+r(x))^{2-\varepsilon}\), \(K_N \leq 0\) and \(\sup_M e(\varphi) (x) \leq C \exp(C r(x)^{2-\varepsilon})\), where \(r(x)\) is the distance between \(x\) and the fixed point \(x_0\). He also solves the problem with some assumptions on tension fields of the initial map \(\varphi\). In the second part of this paper, the author constructs a solution of the Dirichlet problem at infinity for proper harmonic maps between hyperbolic spaces with the boundary map having finitely many points at which the energy density vanishes or the map being not differentiable.
scientific article; zbMATH DE number 1463782
scientific article