As we know , a curve $\gamma$ is geodesic iff $\nabla_\dot{\gamma}\dot{\gamma}=0$. The energy of curve is defined $$ E(t)=\frac{1}{2}\int |\dot{\gamma}|^2 $$ I don't know why define the energy so. I just think it is for the monotonicity in heat flow .
So I want to define suitable energy of other curve for monotonicity.For example, we consider the curve such that $$ \nabla_{\dot{\gamma}}\dot{\gamma}+Ric(\dot{\gamma},\cdot)=0 $$ then how to define a suitable energy such that in evolving equation $$ \partial_tu(\theta,t)=\nabla_{u_\theta}u_\theta+Ric(u_\theta,\cdot) $$ the energy is monotonicity ? $\theta$ is the parameter of curve, $t$ is the time in evolving equation. I think no matter how define the energy , not always ,we can get the monotonicity. It is needed adding some condition of curvature.