Parametrization of level sets of a smooth function

Question

Let $H:\mathbb{R}^2\rightarrow\mathbb{R}$ be given by $H(q,p)=p^2/2+3q^2/2$ (single-well potential). This function has a critical point at $(0,0)$. Define $T:\mathbb{R}^+\rightarrow \mathbb{R}$ by,

$T(h)=\int\limits_{H^{-1}(h)}\frac{dx}{|\nabla H(x)|}$.

Question: Is $T$ continuous? And if possible can this be shown by the intuitive idea below?

My intuition suggests that I should somehow convert this integral over a level set into an integral over a fixed surface by creating some sort of parametrization of level sets. This seems plausible because $H$ is smooth enough. Searching over internet esp. in Differential Geometry based references I have found the idea of a tubular neighbourhood (I don't have much background in geometry). Can this be used somehow to convert the integral above to an integral over a fixed domain?

Apologies if the question and the idea is too vague. Would be grateful for any references in this direction as well.

PS. Not too sure about tags for this question.

Answer 1

Here's a relevant result you can extract from the proof of Theorem 3.1 in Milnor's Morse Theory:

Let $H$ be a smooth real-valued function on a Riemannian manifold $M$ (e.g. $\mathbb R^n$ with Euclidean metric) such that $H^{-1}\left(\left[a,b\right]\right)$ is compact and contains no critical points of $H$. Then there is a unique solution to the modified gradient flow equation $$ \psi_{0}\left(x\right)=x,\quad\frac{d\psi_{t}\left(x\right)}{dt}=\frac{\nabla H}{\left|\nabla H\right|^{2}}\bigg|_{\psi_{t}\left(x\right)}\tag{1} $$ with $\psi_{t}$ sending level sets $H^{-1}\left(h\right)$ to level sets $H^{-1}\left(h+t\right)$ diffeomorphically.

Now let's try to show continuity of $T$ for $H$ satisfying the above the conditions. We can apply a change of variables to make the domain of integration fixed: $$ T\left(a+t\right)=\int_{\psi_{t}H^{-1}\left(a\right)}\frac{dx}{\left|\nabla H\right|}=\int_{H^{-1}\left(a\right)}\frac{\det\left(d\psi_{t}\left(x\right)\right)}{\left|\nabla H\left(\psi_{t}x\right)\right|}dx. $$

From here note that $\psi_{t}$ is $C^{1}$ in the $t$ direction (since $(1)$ says that its $t$ derivative is a continuous function evaluated along a continuous curve), and thus the Jacobian determinant term is continuous; and very similarly we have continuity of $\left|\nabla H\circ\psi_{t}\right|^{-1}$. (Recall throughout that $\left|\nabla H\right|\ne0$.) So the integrand is continuous in $t$. Since the integrand is continuous on a compact set $\left[a,b\right]\times H^{-1}\left(a\right)$, we can exchange limits with integrals (e.g. by the Dominated Convergence theorem) and thus the integral itself is continuous on $\left[a,b\right]$.

For your example, any $b>a>0$ satisfy the conditions; so $T$ is continuous on $\left(0,\infty\right)$.