# Derivative of Mean Curvature and Scalar field

## 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

$$\frac{dH}{dt} = Δ_{Σ}φ + Ric (ν,ν)φ+|A|^{2}φ$$
$$\frac{1}{Δ_{Σ}+ Ric (ν,ν)+|A|^{2}} \frac{dH}{dt} = φ$$
$$\frac{d}{dt}\left ( \frac{1}{Δ_{Σ}+ Ric (ν,ν)+|A|^{2}} \frac{dH}{dt} \right )= \frac{dφ}{dt}$$

#### Attachments

• D Maximo - Hawking mass and local rigidity of minimal two-spheres.pdf
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• D Maximo - Hawking mass and local rigidity of minimal two-spheres.pdf
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Thanks for the post! This is an automated courtesy bump. Sorry you aren't generating responses at the moment. Do you have any further information, come to any new conclusions or is it possible to reword the post?

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

#### Attachments

• HL Bray Thesis.pdf
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