Derivative of Mean Curvature and Scalar field

[darida]

Homework Statement

Page 16 (attached file)
\frac{dH}{dt}|_{t=0} = Δ_{Σ}φ + Ric (ν,ν)φ+|A|^{2}φ
\frac{d}{dt}(dσ_{t})|_{t=0} = - φHdσ
H = mean curvature of surface Σ
A = the second fundamental of Σ
ν = the unit normal vector field along Σ
φ = the scalar field on three manifold M
φ∈C^{∞}(Σ)

Homework Equations

Now I want to find \frac{dφ}{dt} = ...?
with φ≠\frac{1}{H}

The Attempt at a Solution

\frac{dH}{dt} = Δ_{Σ}φ + Ric (ν,ν)φ+|A|^{2}φ
\frac{1}{Δ_{Σ}+ Ric (ν,ν)+|A|^{2}} \frac{dH}{dt} = φ
\frac{d}{dt}\left ( \frac{1}{Δ_{Σ}+ Ric (ν,ν)+|A|^{2}} \frac{dH}{dt} \right )= \frac{dφ}{dt}
But I am not sure about this.

 

[darida]

Further information (file attached, Appendix A, page 99):
∂_{t} = φ\vec{ν}
So the derivation of φ with respect to t would be:
\frac{dφ}{dt} = \frac{d}{dt} \left (\frac{1}{ν} \frac{∂}{∂t} \right )
\frac{dφ}{dt} = \frac{1}{ν} \frac{∂}{∂t} \left ( \frac{∂}{∂t} \right ) + \frac{∂}{∂t} \frac{d}{dt} \frac{1}{ν}
And now after this I don't know what to do