A corollary of Li-Yau-Hamilton estimate

Question

Picture below is from the Hamilton's The Harnack estimate for the Ricci flow .How to get the corollary 1.2 by Theorem 1.1 ? It seemly be not immediately and hard to compute. Maybe just because I am rookie.

Answer 1

The trace of $M_{ij} W_i W_j$ over $W_i$ (in an orthonormal frame) is $$\frac 1 2 \left( \Delta R + 2 R_{kl}R_{kl} + \frac R t\right).$$

With the given $U$, $2P_{ijk} U_{ij} W_k$ is

$$ (D_i R_{jk} - D_j R_{ik})(V_i W_j W_k - V_j W_i W_k);$$

so tracing over $W$ and applying the twice-contracted second Bianchi identity $\rm{div Rc} = \frac12 \nabla \rm Scal$ we get

$$ V_i D_i R - V_i D_j R_{ij} - V_j D_i R_{ji} + V_j D_j R = 2\left(V_iD_iR-V_jD_jR_{ij}\right) = 2\left( V_i D_i R - \frac 1 2 V_j D_j R \right) = V_i D_i R.$$

Finally

$$R_{ijkl} U_{ij}U_{kl} = \frac14 R_{ijkl} (V_i W_j - W_i V_j)(V_k W_l - W_k V_l)$$

has trace $R_{ik} V_i V_k$ (all four terms in the expansion are the same due to the Riemann tensor symmetries), so putting this all together the conclusion of the theorem in this case is

$$\Delta R + 2 R_{kl} R_{kl} + \frac R t + 2V_i D_i R + 2 R_{ik}V_iV_k \ge 0.$$

Since $\partial_t R = \Delta R + 2 R_{ik} R_{ik}$ under Ricci flow, this is exactly the Corollary.