Hi. I ran into the following situation and it turned out to be more subtle than it looked.

I have a complete Riemannian manifold $M$ and my objective is to construct a sequence of functions $f:M \to [0,1]$ which have compact support, $L^2$ norm bounded away from zero, but all third derivatives converge uniformly to zero.

Trivial case: if the manifold is compact I can take the constant function $1$.

If the manifold is $\mathbb{R}$ I take a smooth bump function $f:\mathbb{R} \to [0,1]$ and then rescale it to $f_\lambda(x) = f(\lambda x)$. The third derivatives are $f_\lambda'''(x) = \lambda^3 f'''(\lambda x)$ and for the $L^2$ norm you get: $$|f_\lambda|^2 = \int f(\lambda x)^2 \mathrm{d} x = \frac{1}{\lambda}\int f(y)^2 \mathrm{d} y = \frac{1}{\lambda}|f|^2$$

So that if $\lambda \to 0$ I get what I want.

The same idea works for $\mathbb{R}^k$.

Can this be done on any complete Riemannian manifold $M$?

Idea: Embed the manifold isometrically in $\mathbb{R}^k$ using Nash's theorem and use the restriction of a sequence of functions constructed for $\mathbb{R}^k$.

Problem: Maybe the embedding has very large (second and third order) derivatives. This can be a problem since $M$ is non-compact and the support of the sequence of functions has to grow. Also this seems too high-tech for the problem at hand.

Any suggestions?

I ran into the following situation and it turned out to be more subtle than it looked.

I have a complete Riemannian manifold $M$ and my objective is to construct a sequence of functions $f:M \to [0,1]$ which have compact support, $L^2$ norm bounded away from zero, but all third derivatives converge uniformly to zero.

Trivial case: if the manifold is compact I can take the constant function $1$.

If the manifold is $\mathbb{R}$ I take a smooth bump function $f:\mathbb{R} \to [0,1]$ and then rescale it to $f_\lambda(x) = f(\lambda x)$. The third derivatives are $f_\lambda'''(x) = \lambda^3 f'''(\lambda x)$ and for the $L^2$ norm you get: $$|f_\lambda|^2 = \int f(\lambda x)^2 \mathrm{d} x = \frac{1}{\lambda}\int f(y)^2 \mathrm{d} y = \frac{1}{\lambda}|f|^2$$

So that if $\lambda \to 0$ I get what I want.

The same idea works for $\mathbb{R}^k$.

Can this be done on any complete Riemannian manifold $M$?

Idea: Embed the manifold isometrically in $\mathbb{R}^k$ using Nash's theorem and use the restriction of a sequence of functions constructed for $\mathbb{R}^k$.

Problem: Maybe the embedding has very large (second and third order) derivatives. This can be a problem since $M$ is non-compact and the support of the sequence of functions has to grow. Also this seems too high-tech for the problem at hand.

Any suggestions?