24 August 2022
Introductory notes on the short-time existence and uniqueness of the Ricci flow following Sheridan’s notes (Sheridan 2006). We will see that the Ricci flow is not strongly parabolic and therefore need to introduce a modification to the Ricci flow called the “DeTurck trick” and the Ricci-DeTurck flow to obtain existence results. Note that (Chow, Lu, and Ni 2006) Sections 2.2, 2.5-2.6 also provide an account of the Ricci-DeTurck flow.
Let \((M, g(t))\) be a Riemannian metric with a general time-dependent metric \(g(t)\). We will denote by \(h_{ij}(t)\) as
\[ h_{ij}(t) := \frac{\partial}{\partial t} g_{ij}(t). \]
We first establish the following lemma which contains some useful formulas.
Lemma 1. Let \(g^{ij}\) denote the metric inverse, then for the Christoffel symbols, we have
\[ \frac{\partial}{\partial t} \Gamma_{ij}^k = \frac{1}{2}g^{k\ell}(\nabla_ih_{j\ell} + \nabla_jh_{i\ell} - \nabla_\ell h_{ij}). \](2.1)
For the Riemann curvature tensor, we have
\[\begin{aligned} \frac{\partial}{\partial t}{R_{ijk}}^\ell &= \frac{1}{2}g^{\ell p}\left(\nabla_i\nabla_j h_{kp} + \nabla_i\nabla_k h_{jp} - \nabla_i\nabla_p h_{jk} \right. \\ &\qquad\qquad \left. - \nabla_j\nabla_i h_{kp} - \nabla_j\nabla_k h_{ip} + \nabla_j\nabla_p h_{ik}\right). \end{aligned}\](2.2)
Finally, for the Ricci tensor, we have
\[ \frac{\partial}{\partial t} R_{ij} = \frac{1}{2}g^{pq}(\nabla_q\nabla_i h_{jp} + \nabla_q\nabla_j h_{ip} - \nabla_q\nabla_p h_{ij} - \nabla_i\nabla_j h_{qp}). \](2.3)
Proof. To prove (2.1), recall that
\[ \Gamma^k_{ij} = \frac{1}{2}g^{k\ell}(\partial_i g_{j\ell} + \partial_j g_{i\ell} - \partial_\ell g_{ij}) \]
so differentiating with respect to \(t\), by the product rule, we have
\[\begin{aligned} \partial_t \Gamma^k_{ij} &= \frac{1}{2}\left(\partial_t g^{k\ell}\right)(\partial_i g_{j\ell} + \partial_j g_{i\ell} - \partial_\ell g_{ij}) \\ &\qquad + \frac{1}{2}g^{k\ell}(\partial_i\partial_t g_{j\ell} + \partial_j\partial_t g_{i\ell} - \partial_\ell\partial_t g_{ij}) \end{aligned}\]
Now we specialize to normal coordinates at a point \(p \in M\). In particular, in normal coordinates, we have that \(\partial_i g_{jk} = 0\) and \(\partial_i A = \nabla_i A\) for any tensor \(A\) at \(p\). So the first term above vanishes since each of the three summands in the first term vanishes, and we have
\[\begin{aligned} \partial_t \Gamma^k_{ij}(p) &= \frac{1}{2}g^{k\ell}(\partial_i h_{j\ell} + \partial_j h_{i\ell} - \partial_\ell h_{ij})(p) \\ &= \frac{1}{2}g^{k\ell}(\nabla_i h_{j\ell} + \nabla_j h_{i\ell} - \nabla_\ell h_{ij})(p) \end{aligned}\]
Even though the Christoffel symbols themselves are not tensorial, their derivatives are, so both sides are coordinates of tenrosial quantities, and hence the formula above is valid in any coordinates. To see why the derivative of the Christoffel symbols are tensorial, recall that the difference of Christoffel symbols are tensorial, so we take a fixed (meaning time-independent) connection with associated Christoffel symbols \(\tilde{\Gamma}^k_{ij}\), we have that \(\partial_t \tilde{\Gamma}^k_{ij} = 0\) and so we have
\[ \partial_t\left(\Gamma^k_{ij} - \tilde{\Gamma}^k_{ij}\right) = \partial_t \Gamma^k_{ij} - \partial_t \tilde{\Gamma}^k_{ij} = \partial_t \Gamma^k_{ij}. \]
and since the left hand side is a tensorial quantity (since the derivative of a tensor is a tensor), so is the right hand side above.
To prove (2.2), recall that the Riemann curvature tensor can be explicitly written as
\[ {R_{ijk}}^\ell = \partial_i\Gamma^\ell_{jk} - \partial_j\Gamma^\ell_{ik} + \Gamma^p_{jk}\Gamma^\ell_{ip} - \Gamma^p_{ik}\Gamma^\ell_{jp}. \]
Then differentiating by \(t\), we have
\[\begin{aligned} \partial_t {R_{ijk}}^\ell &= \partial_i(\partial_t\Gamma^\ell_{jk}) - \partial_j(\partial_t\Gamma^\ell_{ik}) + (\partial_t\Gamma^p_{jk})\Gamma^\ell_{ip} + \Gamma^p_{jk}(\partial_t\Gamma^\ell_{ip}) \\ &\qquad - (\partial_t\Gamma^p_{ik})\Gamma^\ell_{jp} - \Gamma^p_{ik}(\partial_t\Gamma^\ell_{jp}) \\ \end{aligned}\]
Then we again specialize to normal coordinates about a point \(p\) so that \(\Gamma^k_{ij}(p) = 0\). Then evaluating the above at \(p\), the last four terms on the right hand side vanish and we have
\[ \partial_t {R_{ijk}}^\ell(p) = \nabla_i(\partial_t\Gamma^\ell_{jk})(p) - \nabla_j(\partial_t\Gamma^\ell_{ik})(p). \]
Clearly both sides are tensorial expressions, so we actually have
\[ \partial_t {R_{ijk}}^\ell = \nabla_i(\partial_t \Gamma^\ell_{jk}) - \nabla_j(\partial_t \Gamma^\ell_{ik}). \](2.4)
Then by (2.1), we can substitute expressions for \(\partial_t\Gamma^\ell_{jk}\) and continue
\[\begin{aligned} \partial_t {R_{ijk}}^\ell &= \nabla_i(\partial_t \Gamma^\ell_{jk}) - \nabla_j(\partial_t \Gamma^\ell_{ik}) \\ &= \nabla_i\left(\frac{1}{2}g^{\ell p}\left(\nabla_j h_{kp} + \nabla_k h_{jp} - \nabla_p h_{jk}\right)\right) \\ &\qquad - \nabla_j\left(\frac{1}{2}g^{\ell p}\left(\nabla_i h_{kp} + \nabla_k h_{ip} - \nabla_p h_{ik}\right)\right) \\ &= \frac{1}{2}(\nabla_i g^{\ell p})(\nabla_j h_{kp} + \nabla_k h_{jp} - \nabla_p h_{jk}) \\ &\qquad + \frac{1}{2}g^{\ell p}(\nabla_i\nabla_j h_{kp} + \nabla_i\nabla_k h_{jp} - \nabla_i\nabla_p h_{jk}) \\ &\qquad - \frac{1}{2}(\nabla_j g^{\ell p})(\nabla_i h_{kp} + \nabla_k h_{ip} - \nabla_p h_{ik}) \\ &\qquad - \frac{1}{2}g^{\ell p}(\nabla_i\nabla_i h_{kp} + \nabla_j\nabla_k h_{ip} - \nabla_j\nabla_p h_{ik}). \end{aligned}\]
By metric compatibility of \(\nabla\), we have that \(\nabla_i g^{\ell p} = 0 = \nabla_j g^{\ell p}\) we the first and third term above vanish and we are left with
\[\begin{aligned} \partial_t {R_{ijk}}^\ell &= \frac{1}{2}g^{\ell p}(\nabla_i\nabla_j h_{kp} + \nabla_i\nabla_k h_{jp} - \nabla_i\nabla_p h_{jk}) \\ &\qquad - \frac{1}{2}g^{\ell p}(\nabla_i\nabla_i h_{kp} + \nabla_j\nabla_k h_{ip} - \nabla_j\nabla_p h_{ik}) \\ &= \frac{1}{2}g^{\ell p}\left(\nabla_i\nabla_j h_{kp} + \nabla_i\nabla_k h_{jp} - \nabla_i\nabla_p h_{jk} - \nabla_j\nabla_i h_{kp} - \nabla_j\nabla_k h_{ip} + \nabla_j\nabla_p h_{ik}\right) \end{aligned}\]
which matches (2.2).
Finally, to prove (2.3), we trace over the indices \(i\) and \(\ell\). From (2.4), we have
\[\begin{aligned} \frac{\partial}{\partial_t} R_{jk} &= \text{tr}_{i\ell}\left(\frac{\partial}{\partial_t}{R_{ijk}}^\ell\right) \\ &= \text{tr}_{i\ell}\left(\nabla_i\left(\partial_t\Gamma_{jk}^\ell\right) - \nabla_j\left(\partial_t\Gamma_{ik}^\ell\right)\right) \\ &= \nabla_m\left(\partial_t\Gamma_{jk}^m\right) - \nabla_j\left(\partial_t\Gamma_{mk}^m\right). \end{aligned}\]
Then using (2.1) we compute
\[\begin{aligned} \nabla_m\partial_t\Gamma_{jk}^m &= \nabla_m\left(\frac{1}{2}g^{mp}\left(\nabla_j h_{kp} + \nabla_k h_{jp} - \nabla_p h_{jk}\right)\right) \\ &= \frac{1}{2}g^{mp}\left(\nabla_m\nabla_j h_{kp} + \nabla_m\nabla_k h_{jp} - \nabla_m\nabla_p h_{jk}\right), \end{aligned}\]
and
\[\begin{aligned} \nabla_j\partial_t\Gamma_{mk}^m &= \nabla_j\left(\frac{1}{2}g^{mp}\left(\nabla_m h_{kp} + \nabla_k h_{mp} - \nabla_p h_{mk}\right)\right) \\ &= \frac{1}{2}g^{mp}\left(\nabla_j\nabla_m h_{kp} + \nabla_j\nabla_k h_{mp} - \nabla_j\nabla_p h_{mk}\right), \end{aligned}\]
where we have used the metric compatibility of \(\nabla\) with \(g\) in the second equalities above.
So we get that the difference is
\[\begin{aligned} \frac{\partial}{\partial_t} R_{jk} &= \frac{1}{2}g^{mp}\left(\nabla_m\nabla_j h_{kp} + \nabla_m\nabla_k h_{jp} - \nabla_m\nabla_p h_{jk} \right. \\ &\qquad \left. - \nabla_j\nabla_m h_{kp} - \nabla_j\nabla_k h_{mp} + \nabla_j\nabla_p h_{mk}\right). \end{aligned}\]
which, after renaming indices \(j,k\) to \(i,j\) and the dummy indices \(m,p\) to \(k,\ell\), becomes
\[\begin{aligned} \frac{\partial}{\partial_t} R_{ij} &= \frac{1}{2}g^{k\ell}\left(\nabla_k\nabla_i h_{j\ell} + \nabla_k\nabla_j h_{i\ell} - \nabla_k\nabla_\ell h_{ij} \right. \\ &\qquad \left. - \nabla_i\nabla_k h_{j\ell} - \nabla_i\nabla_j h_{k\ell} + \nabla_i\nabla_\ell h_{kj}\right). \end{aligned}\]
which matches the formula shown on page 261 of (Fong 2018). To match (2.3), note that since \(g^{k\ell}\) is symmetric, we can interchange \(k,\ell\) above and see that
\[ -\nabla_i\nabla_\ell h_{jk} + \nabla_i\nabla_\ell h_{kj} = -\nabla_i\nabla_\ell h_{jk} + \nabla_i\nabla_\ell h_{jk} = 0, \]
so the last and second-to-last term above cancels and we finally arrive at
\[ \frac{\partial}{\partial_t}R_{ij} = \frac{1}{2}g^{k\ell}\left(\nabla_k\nabla_i h_{j\ell} + \nabla_k\nabla_j h_{i\ell} - \nabla_k\nabla_\ell h_{ij} - \nabla_i\nabla_j h_{k\ell}\right), \]
and we are done.
Now we quote some basic definitions related to parabolic PDEs and quote an existence and uniqueness result. Let \(M\) be a Riemannian manifold and consider a vector bundle \(\pi:\mathcal{E}\to M\) along with some bundle metric \(h\). Let \(u:M\times[0,T) \to \mathcal{E}\) be some time-dependent section of \(\mathcal{E}\). Then we are interested in the system
\[\begin{aligned} \frac{\partial u}{\partial t} &= L(u), \\ u(x,0) &= u_0(x), \end{aligned}\](3.1)
where \(L:C^\infty(\mathcal{E}) \to C^\infty(\mathcal{E})\) is some differential operator. Let us first consider the case where \(L\) is a linear differential operator, that is, let \(\alpha\) denote a multi-index, then we can write \(L(u)\) as
\[ L(u) = \sum\limits_{|\alpha|\leq k} L_\alpha \partial^\alpha u, \]
where
\[ \partial^\alpha f := \frac{\partial^{|\alpha|}f}{\partial x_1^{\alpha_1}\partial x_2^{\alpha_2}\cdots\partial x_n^{\alpha_n}}, \]
and \(L_\alpha \in \text{Hom}(\mathcal{E},\mathcal{E})\). Here, we say \(k\) is the order of \(L\). More concretely, suppose \(L\) is a second-order linear differential operator acting on scalar functions on \(\mathbb{R}^n\), then we can write
\[ \overline{L}(u) = \sum\limits_{i,j} a_{ij}\partial^i\partial^j u + \sum\limits_{i} b_i \partial^i u + cu. \]
If \(\varphi\) is some covector field, then the total symbol of \(L\) in the direction \(\varphi\) is the bundle homomorphism \(\sigma[L](\varphi):\mathcal{E} \to \mathcal{E}\) defined by
\[ \sigma[L](\varphi)(u) = \sum\limits_{|\alpha|\leq k} L_\alpha(u)\prod\limits_{j}\varphi^{\alpha_j}. \]
So in the example of the second-order linear differential operator \(\overline{L}\) above, we would have
\[ \sigma[\overline{L}](\varphi)(u) = \sum\limits_{i,j} a_{ij}\varphi^i\varphi^j u + \sum\limits_{i} b_i\varphi^i u + cu. \]
We obtain the principal symbol by only taking the highest order derivative terms of the total symbol, i.e.
\[\begin{aligned} &\hat{\sigma}[L](\varphi)(u) := \sum\limits_{|\alpha| = k}L_\alpha(u)\prod_j\varphi^{\alpha_j}, \\ &\hat{\sigma}[\overline{L}](\varphi)(u) = \sum\limits_{i,j} a_{ij}\varphi^i\varphi^j u. \end{aligned}\]
Then \(L\) is elliptic if its principal symbol \(\hat{\sigma}[L](\varphi)\) is a vector bundle isomorphism whenever \(\varphi \neq 0\). For \(\overline{L}\), this translates to the condition that
\[ \sum\limits_{i,j} a_{ij}\varphi^i\varphi^j \neq 0 \]
whenever \(\varphi \neq 0\), i.e. that the matrix \(a_{ij}\) is non-singular. Similarly, \(L\) is strongly parabolic/parabolic if there exists some \(\delta > 0\) such that at each point of \(M\), for all covectors \(\varphi \neq 0\) and elements \(u \neq 0\) of \(\mathcal{E}\), we have
\[ \langle \hat{\sigma}[\overline{L}](\varphi)(u), u\rangle > \delta |\varphi|^2|u|^2. \]
For \(\overline{L}\), this translates to the condition that
\[ \sum\limits_{i,j} a_{ij}\varphi^i\varphi^j > \delta |\varphi|^2 \]
for all \(\varphi \neq 0\), i.e. that the matrix \(a_{ij}\) is strictly positive-definite.
Note that the above definitions apply to the case where \(L\) is a linear differential operator. We now define analogous terms for when \(L\) is non-linear, which is the case for the Ricci flow. First we need to define what we mean by the linearization of a non-linear operator. For a non-linear operator \(L\), if \(u:[0,1] \to C^\infty(\mathcal{E})\) is a time-dependent section of \(\mathcal{E}\) such that
\[\begin{aligned} u(0) &= u_0, \\ u'(0) &= v, \end{aligned}\]
then we define the linearization of \(L\) at \(u_0\) to be the linear map \(D[L]:C^\infty(\mathcal{E})\to C^\infty(\mathcal{E})\) so that
\[ D[L](v) = \frac{d}{dt}L(u(t))\Big|_{t=0}. \]
Then we say the non-linear operator \(L\) is strongly parabolic at \(u_0\), if the linear system
\[\begin{aligned} \frac{\partial u}{\partial t} &= D[L](u), \\ u(x,0) &= u_0(x), \end{aligned}\]
is strongly parabolic in the sense for linear operators above. Then we have the following short-time existence and uniqueness result for strongly parabolic systems.
Theorem 2 (Short-time existence and uniqueness of strongly parabolic PDEs). If the system (3.1) is strongly parabolic at \(u_0\), then there exists a unique solution on some time interval \([0,T)\).
Given a Riemannian manifold \(M\) and a time-dependent metric \(g(t)\) with initial metric \(g_0\), the Ricci flow is the following PDE that evolves metric tensor
\[\begin{aligned} \partial_t g(t) &= -2\text{Rc}(g(t)), \\ g(0) &= g_0. \end{aligned}\]
Where \(\text{Rc}(g(t))\) denotes the Ricci curvature of the metric \(g(t)\). The idea is that if we can show this system to be strongly parabolic, then the existence theory for strongly parabolic PDE will give us a short-time existence result for the Ricci flow. To do this, our first step is to linearize the Ricci tensor. By (2.3) above, we have
\[ D[\text{Rc}](h)_{ij} = \frac{1}{2}g^{pq}\left(-\nabla_p\nabla_q h_{ij} - \nabla_i\nabla_j h_{pq} + \nabla_q\nabla_i h_{jp} + \nabla_q\nabla_j h_{ip}\right), \](4.1)
where we have interchanged the indices \(p\) and \(q\) in the last two terms which we can do because \(g^{pq}\) is symmetric.
Then the principal symbol of the Ricci flow operator is given by
\[ \hat{\sigma}[-2\text{Rc}](\varphi)(h)_{ij} = g^{pq}\left(\varphi_p\varphi_q h_{ij} + \varphi_i\varphi_j h_{pq} - \varphi_q\varphi_i h_{jp} + \varphi_q\varphi_j h_{ip}\right). \]
Note that this is strongly parabolic if there is some \(\delta > 0\) such that for all \(\varphi \neq 0\) and symmetric \(h_{ij} \neq 0\), then we need
\[ g^{pq}\left(\varphi_p\varphi_q h_{ij} + \varphi_i\varphi_j h_{pq} - \varphi_q\varphi_i h_{jp} + \varphi_q\varphi_j h_{ip}\right)h^{ij} > \delta\varphi_k\varphi^kh_{rs}h^{rs}. \]
However, choosing \(h_{ij} = \varphi_i\varphi_j\), we get on the left hand side that
\[ g^{pq}\left(\varphi_p\varphi_q\varphi_i\varphi_j + \varphi_i\varphi_j\varphi_p\varphi_q - \varphi_q\varphi_i\varphi_j\varphi_p + \varphi_q\varphi_j\varphi_i\varphi_p\right)\varphi^i\varphi^j = 0. \]
So we see that the Ricci flow is actually not strongly parabolic.
In this section we demonstrate “DeTurck’s trick” for turning the Ricci flow system into one that is strongly parabolic, from which we can obtain a short-time existence result. We start with the following lemma.
Lemma 3 ((Sheridan 2006), Lemma 5.3). Let
\[ V_i = g^{pq}\left(\frac{1}{2}\nabla_i h_{pq} - \nabla_q h_{pi}\right), \]
then the linearization of the \(-2\text{Rc}\) operator can be rewritten as
\[ D[-2\text{Rc}](h)_{ij} = g^{pq}\nabla_p\nabla_q h_{ij} + \nabla_i V_j + \nabla_j V_i + l.o.t.(h). \](4.2)
Proof. By the Ricci identity, commuting covariant derivatives, we have
\[\begin{aligned} \nabla_q\nabla_i h_{jp} &= \nabla_i\nabla_q h_{jp} - R^r_{qij}h_{rp} - R^r_{qip}h_{jr} \\ &= \nabla_i\nabla_qh_{jp} - l.o.t.(h). \end{aligned}\]
So for in the principal symbol for the Ricci flow operator, which only contains the highest order term, we can rearrange (4.1) by commuting covariant derivatives to obtain
\[\begin{aligned} D[-2\text{Rc}](h)_{ij} &= g^{pq}\nabla_p\nabla_qh_{ij} + g^{pq}\nabla_i\left(\frac{1}{2}\nabla_jh_{pq}-\nabla_qh_{jp}\right) \\ &\qquad + g^{pq}\nabla_j\left(\frac{1}{2}\nabla_ih_{pq} - \nabla_qh_{ip}\right) + l.o.t.(h) \\ &= g^{pq}\nabla_p\nabla_qh_{ij} + \nabla_iV_j + \nabla_jV_i + l.o.t.(h), \end{aligned}\]
where we used the metric compatibility of the connection with \(g\) to commute \(g^{pq}\) with \(\nabla_i\) and with \(\nabla_j\) above.
The first term above \(g^{pq}\nabla_p\nabla_q h_{ij} = \Delta h_{ij}\) is good since the Laplacian has a strictly positive principal symbol, so it is not the term contributing to non-parabolicity. Rather, the terms involving \(V\) are the ones contributing to non-parabolicity, so we need to deal with those terms.
Theorem 4. Given a closed Riemannian manifold \(M\) with initial metric \(g_0\), there exists a \(T > 0\) such that a unique \(g(t)\) that satisfies
\[\begin{aligned} \partial_t g(t) &= -2\text{Rc}(g(t)), \\ g(0) &= g_0, \end{aligned}\]
exists and is smooth on \([0,T)\).
Proof. Our strategy to force a parabolic system is to introduce a time-dependent reparametrization of the manifold. We seek a differential operator \(P\) such that the system
\[ \partial_t \bar{g}(t) = P(\bar{g}(t)) \]
is parabolic, and a time-dependent diffeomorphism \(\varphi_t: M\to M\) which has \(\varphi_0 = \text{id}\) and so that
\[ g(t) = \varphi_t^\ast\bar{g}(t) \](4.3)
is a solution to the Ricci flow. By the relationship (4.3), we have that
\[\begin{aligned} \partial_t g_t &= \partial_t(\varphi_t^\ast\bar{g}_t) \\ &= \partial_s(\varphi_{t+s}^\ast\bar{g}_{t+s})|_{s=0} \\ &= \left(\varphi_t^\ast\partial_s\bar{g}_{t+s}\right)|_{s=0} + \left(\partial_s(\varphi_{t+s}\bar{g}_t)\right)|_{s=0} \\ &= \varphi_t^\ast P(\bar{g}_t) + \varphi_t^\ast\mathcal{L}_{\partial_t\varphi_t}\bar{g}_t, \end{aligned}\]
by the product rule in the third equality and the rule for the Lie derivative on tensors in the last equality above. Choosing \(\varphi_t\) to take the form
\[\begin{aligned} \partial_t \varphi_t &= W(t), \\ \varphi_0 &= \text{id}, \end{aligned}\]
for some time-dependent vector field \(W(t)\), reduces the problem to finding a differential operator \(P\) that is strongly parabolic, and a time-dependent vector field \(W(t)\) such that if \(\varphi_t\) satisfies (4.3), then
\[ \varphi_t^\ast P(\bar{g}_t) + \varphi_t^\ast \mathcal{L}_{W(t)}\bar{g}_t = -2\text{Rc}(\varphi_t^\ast\bar{g}_t) = -2\varphi_t^\ast\text{Rc}(\bar{g}_t), \]
where we have used the diffeomorphism invariance of the Ricci tensor. The above is then equivalent to
\[ P(\bar{g}_t) = -2\text{Rc}(\bar{g}_t) - \mathcal{L}_{W(t)}\bar{g}_t. \]
Recall on \((M,g)\) that for a vector field \(X\), the Lie derivative with respect to \(X\) of the metric is given in coordinates by
\[ \left(\mathcal{L}_Xg\right)_{ij} = \nabla_iX_j + \nabla_jX_i, \]
so we have
\[ \left(\mathcal{L}_{W(t)}\bar{g}_t\right)_{ij} = \nabla_iW_j + \nabla_jW_i. \]
We can use lemma 3 to write the linearization of \(P\) as
\[\begin{aligned} D[P](h)_{ij} &= g^{pq}\nabla_p\nabla_qh_{ij} + \nabla_iV_j + \nabla)jV)i \\ &\qquad + l.o.t.(h) - D[\nabla_iW_j + \nabla_jW_i](h)_{ij}. \end{aligned}\]
Our goal is to do away with the terms involving \(V\), that is, we want to choose \(W\) so that the expression \(\nabla_iV_j + \nabla_jV_i - D[\nabla_iW_j + \nabla_jW_i](h)\) cancels out. Since for vector field \(Y\) we have that
\[ \nabla_i Y = \partial_i Y + \Gamma^k_{ij}Y^j, \]
the highest order term is \(\partial_i Y\), so to a principal symbol, \(\nabla_i\) can be replaced by \(\partial_i\). In that case, the principal part of \(D[\nabla_iW_j](h)\) is given by
\[ \partial_t\partial_iW_j(g(t)) = \partial_i\partial_tW_j(g(t)) = \partial_iD[W_j](h) \]
So the requirement that
\[\begin{aligned} \nabla_iV_j + \nabla_jV_i &= D[\nabla_iW_j + \nabla_jW_i](h) \\ &= D[\nabla_iW_j](h) + D[\nabla_jW_i](h) \\ &= \nabla_iD[W_j](h) + \nabla_jD[W_i](h) \end{aligned}\]
up to highest order term is equivalent to the requirement that
\[ V_i = D[W_i](h) \]
up to highest order term.
Remember that \(V_i\) is given by
\[\begin{aligned} V_i &= g^{pq}\left(\frac{1}{2}\nabla_ih_{pq} - \nabla_ph_{qi}\right) \\ &= -\frac{1}{2}g^{pq}\left(\nabla_ph_{qi} + \nabla_qh_{pi} - \nabla_ih_{pq}\right), \end{aligned}\]
and that by (2.1), we have
\[ D[\Gamma^k_{ij}](h) = \partial_t\left(\Gamma^k_{ij}(g(t))\right) = \frac{1}{2}g^{k\ell}\left(\nabla_ih_{j\ell} + \nabla_jh_{i\ell} - \nabla_\ell h_{ij}\right). \]
This suggests we try defining \(W\) by
\[ W_i = -g^{pq}g_{ij}\Gamma^j_{pq}, \]
however, this is not tensorial since Christoffel symbols are not tensors. We can amend this by remembering that the difference of Christoffel symbols is a tensor, so we introduce \(\tilde{\Gamma}^j_{pq}\) the Christoffel symbols associated with a fixed constant connection independent of the metric, and we instead define \(W\) by
\[ W_i = -g^{pq}g_{ij}\left(\Gamma^j_{pq} - \tilde{\Gamma}^j_{pq}\right), \]
which produces the vector field \(W(t)\) as required. Then by engineering \(W(t)\), we have that
\[ \hat{\sigma}(D[P])(\varphi)(h)_{ij} = g^{pq}\varphi_p\varphi_qh_{ij} \Rightarrow \langle\hat{\sigma}(D[P])(\varphi)(h),h\rangle = |\varphi|^2|h|^2 > 0 \]
Then the Ricci-DeTurck flow defined by
\[ \partial_t \bar{g}_{ij} = P(\bar{g}) = -2\bar{R}_{ij} + \nabla_iW_j + \nabla_jW_i \]
is strictly parabolic and therefore a short-time solution exists by Theorem 2.
Then as long as the unique short-time solution \(\bar{g}_t\) to the Ricci-DeTurck flow exists, \(W(t)\) will exist and we can obtain the diffeomorphisms \(\varphi_t\) by solving the ODE
\[\begin{aligned} \partial_t \varphi_t &= W(t) \\ \varphi_0 &= \text{id}. \end{aligned}\]
Then we can obtain the metrics \(g_t = \varphi_t^\ast\bar{g}_t\) by pulling back by \(\varphi_t\) to obtain a unique solution to the Ricci flow as required.