[proofplan] Use the analytic heat-flow convergence theorem of Eells and Sampson as a separate preliminary result: for a closed Riemannian domain and compact nonpositively curved target, the harmonic map heat flow exists smoothly for all time, decreases the Dirichlet energy, has times along which the tension tends to zero, and subconverges smoothly to a harmonic map. Applying that result to the initial map $u_0:M\to N$ gives a smooth global deformation whose large-time limit $u_\infty$ is harmonic. The heat flow keeps each finite-time slice in the original homotopy class, and a smoothed geodesic interpolation from a sufficiently late slice to $u_\infty$ completes the homotopy without a nonsmooth concatenation point. [/proofplan]

[step:Start the harmonic map heat flow from the given smooth map]Let $E:C^\infty(M,N)\to [0,\infty)$ denote the Dirichlet energy functional defined by \begin{align*} E(v)=\frac{1}{2}\int_M |dv|_{g,h}^2\,d\operatorname{vol}_g(x). \end{align*} Here $d\operatorname{vol}_g$ is the Riemannian volume measure on $(M,g)$, and $|dv|_{g,h}$ is the Hilbert-Schmidt norm of the linear map $dv_x:T_xM\to T_{v(x)}N$ induced by $g$ and $h$. Let $\tau(v)\in \Gamma(v^*TN)$ denote the tension field of a smooth map $v:M\to N$; by definition, $v$ is harmonic exactly when $\tau(v)=0$. We invoke the [Harmonic Map Heat Flow Convergence Theorem](/theorems/harmonic-map-heat-flow-convergence) as a separate analytic input already established in the surrounding development; this proof uses it as a prerequisite and does not reprove its analytic content. In the form used below, it states that if the domain is closed, the target is compact with nonpositive sectional curvature, and the initial map is smooth, then the harmonic map heat flow exists smoothly for all time, the energy is nonincreasing, and the flow admits a smooth subsequential limit with zero tension field. Its hypotheses are satisfied because $(M,g)$ is closed, $(N,h)$ is compact with nonpositive sectional curvature, and $u_0:M\to N$ is smooth. Therefore there exists a smooth map $u:M\times [0,\infty)\to N$ solving \begin{align*} \partial_t u(x,t)=\tau(u(\cdot,t))(x), \qquad u(x,0)=u_0(x) \end{align*} for every $x\in M$ and $t\geq 0$.[/step]

[guided]The purpose of introducing the heat flow is to deform $u_0$ along the negative gradient direction of the Dirichlet energy. We define $E:C^\infty(M,N)\to [0,\infty)$ by \begin{align*} E(v)=\frac{1}{2}\int_M |dv|_{g,h}^2\,d\operatorname{vol}_g(x). \end{align*} The measure in this integral is the Riemannian volume measure $d\operatorname{vol}_g$ on $(M,g)$. The quantity $|dv|_{g,h}$ is the Hilbert-Schmidt norm of $dv_x:T_xM\to T_{v(x)}N$, computed using $g$ on $T_xM$ and $h$ on $T_{v(x)}N$. We also define $\tau(v)\in \Gamma(v^*TN)$ to be the tension field of $v:M\to N$; the Euler-Lagrange equation for the Dirichlet energy is $\tau(v)=0$, so vanishing of the tension field is exactly the harmonic map equation. Now we use the [Harmonic Map Heat Flow Convergence Theorem](/theorems/harmonic-map-heat-flow-convergence) as a separate input already established before this theorem. The version needed here has four conclusions: global smooth existence for the harmonic map [heat equation](/page/Heat%20Equation), monotonicity of the Dirichlet energy, large-time subsequential smooth convergence, and vanishing of the tension field at the subsequential limit. Its hypotheses are a closed Riemannian domain, a compact Riemannian target, nonpositive sectional curvature of the target, and a smooth initial map. We verify these one by one: $(M,g)$ is closed by hypothesis, $(N,h)$ is compact by hypothesis, the sectional curvature of $(N,h)$ is nonpositive by hypothesis, and $u_0:M\to N$ is smooth by hypothesis. Therefore there is a smooth map $u:M\times [0,\infty)\to N$ with initial value $u(\cdot,0)=u_0$ and satisfying \begin{align*} \partial_t u(x,t)=\tau(u(\cdot,t))(x) \end{align*} for every $x\in M$ and every $t\geq 0$. This supplies the global smooth deformation that will later be used both for homotopy and for extracting a harmonic limit.[/guided]

[step:Use the heat flow to stay inside the original homotopy class] For each $T>0$, define the map $H_T:M\times [0,1]\to N$ by \begin{align*} H_T(x,s)=u(x,sT). \end{align*} Because $u$ is smooth on $M\times [0,\infty)$, the map $H_T$ is smooth. It satisfies $H_T(x,0)=u_0(x)$ and $H_T(x,1)=u(x,T)$ for every $x\in M$, so $u(\cdot,T)$ is smoothly homotopic to $u_0$ for every $T>0$. [/step]

[step:Extract a smooth harmonic limit at large time] The large-time part of the separate Eells-Sampson heat-flow convergence theorem applies under the same verified hypotheses. It gives a sequence $(t_k)_{k\in\mathbb N}$ in $[0,\infty)$ with $t_k\to\infty$ and a smooth map $u_\infty:M\to N$ such that $u(\cdot,t_k)\to u_\infty$ in $C^\infty(M,N)$. The same analytic conclusion also gives \begin{align*} \tau(u_\infty)=0. \end{align*} Therefore $u_\infty$ is a smooth harmonic map. [/step]

[step:Identify the harmonic limit as a representative of the initial homotopy class] Since $(N,h)$ is compact, its injectivity radius is positive. Choose $k\in\mathbb N$ so large that $u(x,t_k)$ and $u_\infty(x)$ have Riemannian distance less than this injectivity radius for every $x\in M$; this is possible because $u(\cdot,t_k)\to u_\infty$ uniformly and $M$ is compact. Define the map $K:M\times [0,1]\to N$ by letting $K(x,s)$ be the unique constant-speed geodesic segment from $u(x,t_k)$ to $u_\infty(x)$ evaluated at parameter $s$. Equivalently, using the inverse of the exponential map on the injectivity-radius neighbourhood, \begin{align*} K(x,s)=\exp_{u(x,t_k)}\bigl(s\,\exp_{u(x,t_k)}^{-1}(u_\infty(x))\bigr). \end{align*} The smooth dependence of the exponential map and the chosen injectivity-radius neighbourhood imply that $K$ is smooth. The map $H_{t_k}$ is a smooth homotopy from $u_0$ to $u(\cdot,t_k)$, and $K$ is a smooth homotopy from $u(\cdot,t_k)$ to $u_\infty$. To make the concatenation smooth in the homotopy parameter, choose smooth maps $a:[0,1/2]\to[0,1]$ and $b:[1/2,1]\to[0,1]$ such that $a=0$ near $0$, $a=1$ near $1/2$, $b=0$ near $1/2$, and $b=1$ near $1$. Define $L:M\times[0,1]\to N$ as follows. For $0\leq s\leq 1/2$, set \begin{align*} L(x,s)=H_{t_k}(x,a(s)). \end{align*} For $1/2\leq s\leq 1$, set \begin{align*} L(x,s)=K(x,b(s)). \end{align*} Near $s=1/2$ both formulas are equal to $u(x,t_k)$, so $L$ is smooth across the joining time. Thus $L$ is a smooth homotopy from $u_0$ to $u_\infty$. Since $\tau(u_\infty)=0$, the map $u_\infty$ is the desired smooth harmonic representative of the homotopy class of $u_0$. [/step]