[proofplan] We use short-time existence for the harmonic map heat flow and prove that no finite-time obstruction can occur. The key estimate is the parabolic Bochner inequality for the [energy density](/page/Energy%20Density) \begin{align*} e(u)=\frac{1}{2}|du|_{g,h}^2; \end{align*} the nonpositive [sectional curvature](/page/Sectional%20Curvature) of the target removes the dangerous target-curvature term, while compactness of $M$ bounds the Ricci term from the domain. The parabolic maximum principle then gives a uniform $C^1$ bound on every finite time interval. A standard continuation criterion for quasilinear parabolic systems extends the flow past any finite time, and iteration gives a smooth solution on $[0,\infty)$. [/proofplan]

[step:Start the flow by short-time parabolic existence] Let $u_0:M\to N$ be the given smooth initial map. By the [Short-Time Existence for Harmonic Map Heat Flow](/theorems/short-time-existence-for-harmonic-map-heat-flow), applied to the closed source manifold $(M,g)$, the compact target manifold $(N,h)$, and smooth initial data $u_0$, there exist a time $T_{\max} \in (0,\infty]$ and a smooth map $u:M \times [0,T_{\max})\to N$ such that $\partial_t u=\tau_g(u)$ on $M \times (0,T_{\max})$ and $u(\cdot,0)=u_0$, where $\tau_g(u)$ denotes the [tension field](/page/Tension%20Field) of $u$ with respect to $g$ and $h$. We take $T_{\max}$ to be the maximal existence time supplied by the usual continuation construction. For each $t\in[0,T_{\max})$, define $u_t:M\to N$ by $u_t(x)=u(x,t)$. Define the differential of the time-slice map as the vector-bundle morphism $du(\cdot,t):TM\to u_t^*TN$. For each $x\in M$, this morphism sends a tangent vector $X\in T_xM$ to $d(u_t)_x(X)\in T_{u(x,t)}N$. We write $|du(x,t)|_{g,h}$ for its Hilbert-Schmidt norm with respect to $g_x$ and $h_{u(x,t)}$. We use the following precise form of the [Continuation Criterion for Harmonic Map Heat Flow](/theorems/continuation-criterion-for-harmonic-map-heat-flow): if $T_{\max}<\infty$ and \begin{align*} \sup_{(x,t)\in M\times[0,T_{\max})}|du(x,t)|_{g,h}<\infty, \end{align*} then the compactness of the target and standard quasilinear parabolic estimates give uniform higher-derivative bounds up to $T_{\max}$, so $u$ has a smooth limit at time $T_{\max}$ and extends smoothly to some interval $[0,T_{\max}+\varepsilon)$ with $\varepsilon>0$. Therefore it remains to prove that $|du|_{g,h}$ is bounded on every finite time interval $M\times[0,T]$ with $T<T_{\max}$. [/step]

[step:Derive the parabolic Bochner inequality for the energy density]Define the [energy density](/page/Energy%20Density) $e:M\times[0,T_{\max})\to[0,\infty)$ by \begin{align*} e(x,t)=\frac{1}{2}|du(x,t)|_{g,h}^2. \end{align*} Let $\nabla du$ denote the covariant derivative of $du$ with respect to the Levi-Civita connection on $M$ and the pullback Levi-Civita connection on $u^*TN$. Let $\Delta_g:C^\infty(M)\to C^\infty(M)$ denote the Laplace-Beltrami operator associated to the Riemannian metric $g$. The [Bochner Formula for Harmonic Map Heat Flow](/theorems/bochner-formula-for-harmonic-map-heat-flow) gives, at each point $(x,t)\in M\times(0,T_{\max})$, \begin{align*} (\partial_t-\Delta_g)e=-|\nabla du|_{g,h}^2+\sum_{i=1}^{m} h\bigl(du(\operatorname{Ric}_g(e_i)),du(e_i)\bigr)+\sum_{i,j=1}^{m} h\bigl(R^N(du(e_i),du(e_j))du(e_i),du(e_j)\bigr), \end{align*} where $m=\dim M$, $(e_1,\dots,e_m)$ is any local $g$-orthonormal frame on $M$, $\operatorname{Ric}_g:TM\to TM$ is the Ricci endomorphism defined by $g(\operatorname{Ric}_g X,Y)=\operatorname{Ric}_g(X,Y)$, and $R^N$ is the Riemann curvature tensor of $(N,h)$ in the sign convention used by the cited Bochner formula. In that convention, for every $y\in N$ and every $\xi,\eta\in T_yN$ spanning a nonzero two-plane, \begin{align*} h(R^N(\xi,\eta)\xi,\eta)=K_N(\xi,\eta)\bigl(|\xi|_h^2|\eta|_h^2-h(\xi,\eta)^2\bigr). \end{align*} Because $K_N\leq 0$, the curvature term satisfies \begin{align*} h\bigl(R^N(du(e_i),du(e_j))du(e_i),du(e_j)\bigr)\leq 0 \end{align*} for every pair $i,j$. Hence the target-curvature contribution in the displayed Bochner formula is nonpositive: \begin{align*} \sum_{i,j=1}^{m} h\bigl(R^N(du(e_i),du(e_j))du(e_i),du(e_j)\bigr)\leq 0. \end{align*} Since $M$ is closed, the Ricci endomorphism has finite operator norm. Define \begin{align*} C_M:=2\sup_{x\in M}\|\operatorname{Ric}_g(x)\|_{\mathrm{op}}. \end{align*} Fix $(x,t)\in M\times(0,T_{\max})$. Since $\operatorname{Ric}_g(x):T_xM\to T_xM$ is self-adjoint with respect to $g_x$, choose a $g_x$-orthonormal eigenbasis $(e_1,\dots,e_m)$ of $T_xM$ and write $\lambda_i\in\mathbb{R}$ for the eigenvalue satisfying $\operatorname{Ric}_g(e_i)=\lambda_i e_i$. Then \begin{align*} \sum_{i=1}^{m} h\bigl(du(\operatorname{Ric}_g(e_i)),du(e_i)\bigr)=\sum_{i=1}^{m}\lambda_i |du(e_i)|_h^2 \leq \|\operatorname{Ric}_g(x)\|_{\mathrm{op}}\sum_{i=1}^{m}|du(e_i)|_h^2 \leq C_M e(x,t), \end{align*} because $e(x,t)=\frac{1}{2}\sum_{i=1}^{m}|du(e_i)|_h^2$. Discarding the nonpositive terms $-|\nabla du|_{g,h}^2$ and the target-curvature contribution gives \begin{align*} (\partial_t-\Delta_g)e\leq C_M e \end{align*} on $M\times(0,T_{\max})$.[/step]

[guided]The purpose of this step is to convert the geometric evolution equation into a scalar inequality for the size of the differential. Define $e:M\times[0,T_{\max})\to[0,\infty)$ by \begin{align*} e(x,t)=\frac{1}{2}|du(x,t)|_{g,h}^2. \end{align*} This scalar function records the pointwise energy density of the map $u(\cdot,t):M\to N$. Let $\Delta_g:C^\infty(M)\to C^\infty(M)$ denote the Laplace-Beltrami operator associated to $g$. The [Bochner Formula for Harmonic Map Heat Flow](/theorems/bochner-formula-for-harmonic-map-heat-flow) applies because $u$ is a smooth solution of $\partial_tu=\tau_g(u)$ on $M\times(0,T_{\max})$. It gives \begin{align*} (\partial_t-\Delta_g)e=-|\nabla du|_{g,h}^2+\sum_{i=1}^{m} h\bigl(du(\operatorname{Ric}_g(e_i)),du(e_i)\bigr)+\sum_{i,j=1}^{m} h\bigl(R^N(du(e_i),du(e_j))du(e_i),du(e_j)\bigr), \end{align*} where $m=\dim M$, $(e_1,\dots,e_m)$ is a local $g$-orthonormal frame, $\nabla du$ is the covariant derivative of $du$ using the Levi-Civita connection on $M$ and the pullback connection on $u^*TN$, $\operatorname{Ric}_g$ is the Ricci endomorphism of $M$, and $R^N$ is the curvature tensor of $N$ in the sign convention used by the cited formula. The Ricci contribution is summed over $i$ only because $\operatorname{Ric}_g(e_i)$ already includes the contraction of the Ricci tensor into an endomorphism. Now we use the curvature hypothesis. In this convention, for every $y\in N$ and every $\xi,\eta\in T_yN$ spanning a nonzero two-plane, \begin{align*} h(R^N(\xi,\eta)\xi,\eta)=K_N(\xi,\eta)\bigl(|\xi|_h^2|\eta|_h^2-h(\xi,\eta)^2\bigr). \end{align*} The factor $|\xi|_h^2|\eta|_h^2-h(\xi,\eta)^2$ is nonnegative by the [Cauchy-Schwarz inequality](/theorems/432) in $T_yN$. Since $K_N\leq 0$, this gives \begin{align*} h(R^N(\xi,\eta)\xi,\eta)\leq 0. \end{align*} Taking $\xi=du(e_i)$ and $\eta=du(e_j)$ gives \begin{align*} h\bigl(R^N(du(e_i),du(e_j))du(e_i),du(e_j)\bigr)\leq 0. \end{align*} Thus the target-curvature part cannot create positive growth in the energy density: \begin{align*} \sum_{i,j=1}^{m} h\bigl(R^N(du(e_i),du(e_j))du(e_i),du(e_j)\bigr)\leq 0. \end{align*} It remains to control the Ricci term from the domain. Since $M$ is closed, it is compact and the [continuous function](/page/Continuous%20Function) $x\mapsto \|\operatorname{Ric}_g(x)\|_{\mathrm{op}}$ has a finite supremum. Define \begin{align*} C_M:=2\sup_{x\in M}\|\operatorname{Ric}_g(x)\|_{\mathrm{op}}. \end{align*} Fix a point $(x,t)\in M\times(0,T_{\max})$. We should not compare $du(\operatorname{Ric}_g(e_i))$ directly with $du(e_i)$ for an arbitrary frame, because $du$ is not an isometry or even injective in general. Instead, use the symmetry of the Ricci tensor. The endomorphism $\operatorname{Ric}_g(x):T_xM\to T_xM$ is self-adjoint with respect to $g_x$, so the finite-dimensional spectral theorem gives a $g_x$-orthonormal eigenbasis $(e_1,\dots,e_m)$ of $T_xM$. Let $\lambda_i\in\mathbb{R}$ be defined by $\operatorname{Ric}_g(e_i)=\lambda_i e_i$. In this eigenbasis the Ricci term diagonalizes: \begin{align*} \sum_{i=1}^{m} h\bigl(du(\operatorname{Ric}_g(e_i)),du(e_i)\bigr)=\sum_{i=1}^{m}h\bigl(du(\lambda_i e_i),du(e_i)\bigr)=\sum_{i=1}^{m}\lambda_i |du(e_i)|_h^2. \end{align*} Since $|\lambda_i|\leq \|\operatorname{Ric}_g(x)\|_{\mathrm{op}}$ for each $i$, we obtain \begin{align*} \sum_{i=1}^{m} h\bigl(du(\operatorname{Ric}_g(e_i)),du(e_i)\bigr) \leq \|\operatorname{Ric}_g(x)\|_{\mathrm{op}}\sum_{i=1}^{m}|du(e_i)|_h^2 \leq C_M e(x,t), \end{align*} because $e(x,t)=\frac{1}{2}\sum_{i=1}^{m}|du(e_i)|_h^2$. This is exactly the estimate needed: the Ricci term can increase the energy density at most linearly in $e$, with coefficient depending only on the fixed source metric $g$. Combining the Bochner formula with the nonpositive target-curvature contribution and the Ricci estimate gives the scalar parabolic inequality \begin{align*} (\partial_t-\Delta_g)e\leq C_M e \end{align*} on $M\times(0,T_{\max})$.[/guided]

[step:Apply the maximum principle to bound the differential on finite intervals] Fix $T<T_{\max}$. Define $w_T:M\times[0,T]\to[0,\infty)$ by \begin{align*} w_T(x,t)=e^{-C_Mt}e(x,t). \end{align*} Using the product rule and the inequality from the previous step, \begin{align*} (\partial_t-\Delta_g)w_T=e^{-C_Mt}\bigl((\partial_t-\Delta_g)e-C_Me\bigr)\leq 0 \end{align*} on $M\times(0,T]$. Since $M$ is closed, there is no boundary term. By the [Parabolic Maximum Principle on Closed Manifolds](/theorems/parabolic-maximum-principle-on-closed-manifolds), applied to the smooth function $w_T$ satisfying $(\partial_t-\Delta_g)w_T\leq 0$ on the compact boundaryless manifold $M$, \begin{align*} \sup_{x\in M}w_T(x,t)\leq \sup_{x\in M}w_T(x,0) \end{align*} for every $t\in[0,T]$. Since $w_T(x,0)=e(x,0)$, this gives \begin{align*} \sup_{x\in M}e(x,t)\leq e^{C_Mt}\sup_{x\in M}e(x,0) \end{align*} for every $t\in[0,T]$. Equivalently, \begin{align*} \sup_{(x,t)\in M\times[0,T]}|du(x,t)|_{g,h}^2 \leq e^{C_MT}\sup_{x\in M}|du_0(x)|_{g,h}^2. \end{align*} Thus $|du|_{g,h}$ is bounded on every finite interval $M\times[0,T]$ with $T<T_{\max}$. [/step]

[step:Use the continuation criterion to rule out finite-time breakdown] Assume, for contradiction, that $T_{\max}<\infty$. Applying the estimate from the previous step with $T<T_{\max}$ gives \begin{align*} \sup_{(x,t)\in M\times[0,T]}|du(x,t)|_{g,h}^2 \leq e^{C_MT_{\max}}\sup_{x\in M}|du_0(x)|_{g,h}^2. \end{align*} Taking the supremum over all $T<T_{\max}$ yields \begin{align*} \sup_{(x,t)\in M\times[0,T_{\max})}|du(x,t)|_{g,h}<\infty. \end{align*} The [Continuation Criterion for Harmonic Map Heat Flow](/theorems/continuation-criterion-for-harmonic-map-heat-flow), in the precise form stated at the start of the proof, therefore extends $u$ smoothly beyond $T_{\max}$, contradicting the maximality of $T_{\max}$. Hence $T_{\max}=\infty$. [/step]

[step:Bootstrap regularity and obtain the global smooth solution] The solution already produced by short-time existence is smooth on $M\times[0,T)$ for every $T<T_{\max}$. Since the previous step proves $T_{\max}=\infty$, the map $u:M\times[0,\infty)\to N$ is smooth on every compact time interval and solves \begin{align*} \partial_t u=\tau_g(u) \end{align*} on $M\times(0,\infty)$ with $u(\cdot,0)=u_0$, where $\tau_g(u)$ is the [tension field](/page/Tension%20Field). This proves the asserted long-time smooth existence for harmonic map heat flow. [/step]