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