Constrained matrix Li-Yau-Hamilton estimates on Kähler manifolds (Q2339331)

Let \((M,g_{i\bar j}(t))\) be a compact solution of the \(\epsilon\)-Kähler-Ricci flow \[ \frac{\partial}{\partial t}g_{i\bar j}= -\epsilon R_{i\bar j} \] with nonnegative holomorphic bisectional curvature. If \(u\) and \(v\) are two solutions of the equations \[ \frac{\partial}{\partial t}u=\Delta u+ \epsilon R u, \quad \frac{\partial}{\partial t}v=\Delta v+ \epsilon R v \] with \(|v|<u\), then the following holds. \[ \nabla_i \nabla_{\bar j}\ln\, u+\frac 1t g_{i\bar j}+\epsilon R_{i\bar j}> \frac{\nabla_i h\nabla_{\bar j}h}{1-h^2} \] where \(\displaystyle h=\frac vu\). As a special case when \(\epsilon=0\), the constrained Li-Yau-Hamilton estimate is obtained on a Kähler manifold with a fixed metric. {Corollary:} Let \(M\) be a compact Kähler manifold with nonnegative holomorphic bisectional curvature. If \(u\) and \(v\) are two solutions of the heat equations \(\displaystyle \frac{\partial}{\partial t}u=\Delta u,\;\frac{\partial}{\partial t}v=\Delta v\) with \(|v|<u\), then the following holds. \[ \nabla_i \nabla_{\bar j}\ln\, u+\frac 1t g_{i\bar j}> \frac{\nabla_i h\nabla_{\bar j}h}{1-h^2}\;\text{with}\;h=\frac vu. \] When \(\epsilon =1\), the constrained Li-Yau-Hamilton estimate is obtained on Kähler manifolds under the Kähler Ricci-flow. {Corollary:} Let \((M,g_{i\bar j}(t))\) be a compact solution of Kähler-Ricci flow \[ \frac{\partial}{\partial t}g_{i\bar j}= -R_{i\bar j} \] with nonnegative holomorphic bisectional curvature. If \(u\) and \(v\) are two solutions of the forward conjugate heat equations \[ \frac{\partial}{\partial t}u=\Delta u+ R u,\;\frac{\partial}{\partial t}v=\Delta v+ R v \] with \(|v|<u\), then the following holds: \[ \nabla_i \nabla_{\bar j}\ln\, u+\frac 1t g_{i\bar j}+ R_{i\bar j}> \frac{\nabla_i h\nabla_{\bar j}h}{1-h^2},\;\text{where}\;h=\frac vu. \] The proof of Theorem 1 is achieved by a series of calculations.