On the heat equation for harmonic maps from non-compact manifolds
We study the heat equation for harmonic maps from \(M\) to \(N\) under the conditions (1) the domain manifold \(M\) is complete non-compact, (2) the target manifold \(N\) is compact and has non-positive sectional curvature, (3) the initial map \(h: M\to N\) has finite total energy. It is proved under somewhat weaker conditions that the total energy decreases in time \(t\). More precisely, let \(u: M\times (0,T)\to N\) be a smooth map satisfying the heat equation. Let \(P\in M\) be a fixed point. For \(0<t<T\) and \(R>0\), let \[ E(t,R)=\int_{B_ P(R)}e(u(\cdot,t))\,dV_ M, \] and \[ \bar E(T,R)=\sup_{0<\tau <t} E(\tau,R),\quad E(t)=\int_{M}e(u(\cdot,t))\,dV_ M. \] Suppose there exists a constant \(k>0\) such that \(\bar E(T,R)\leq \exp (k(1+R))\) for all \(R>0\). Then, for \(0<t_ 1<t_ 2<T\), if \(E(t_ 1)<\infty\), we have \[ E(t_ 2)+2\int^{t_ 2}_{t_ 1}\,dt\int_{M}| u_ t|^ 2 \,dV_ M\leq E(t_ 1)<\infty. \] The monotonicity result is used to prove the main result that under conditions (1), (2), and (3), the heat equation has a solution \(u\) in \(M\times (0,\infty)\) with initial condition \(h\) and there exists \(t_ i\to \infty\) such that \(u(\cdot,t_ i): M\to N\) converge together with their first and second derivatives in the space variable uniformly on compact subsets of \(M\) to a harmonic map. In doing so, we recover a result of Schoen and Yau that \(h: M\to N\) is homotopic on any compact subset of \(M\) to a harmonic map under the conditions (1), (2), and (3). There is also a uniqueness result in the paper.
scientific article; zbMATH DE number 4192696