I am trying to understand Lemma 3.5.4 in Jost's book Riemannian Geometry and Geometric Analysis, 7-ed and running into some troubles.
It seems that the notation is not consistent. In the Lemma, I think $H\beta_0$ is the harmonic p-form of the limit of the heat flow with initial condition $\beta(x,0) = \beta_0(x)$. Then, my question comes from the first sentence of the proof.
Given $t>0$, we seek $\beta$ with $\left\Vert \beta \right\Vert = 1$ and $H\beta = 0$ for which for the solution $\beta$ of (3.5.1) $$ \frac{\partial\beta(x,t)}{\partial t} + \Delta \beta(x,t) = 0 $$ with the initial value $\beta(x,0)=\beta(x)$, $\left\Vert \beta(.,t) \right\Vert$ is maximal.
What is the $\beta$ we seek in the beginning? $\beta$ is a solution of heat equation or an initial condition?
For a given $t>0$, if we can find such $\beta$, then $\beta$ depends on $t$? To avoid abuse of notation, if we call $t'$ for the given time parameter, we should write $\beta^{(t')}(x)$ or $\beta^{(t')}(x,t)$.
Maxmial is in what sense? Among all choice of $\beta$?