Let $\mathcal{M}$ be a compact smooth hypersurface in $\mathbb{R}^{n}$ with affine metric $h$ and affine volume measure $d\mu_{h}$. If $$\int_{\mathcal{M}}|\nabla f|^2_{h}d\mu_{h}$$ is small what can I say about $f~$? Sort of Sobolev type inequalities? Here $|.|_{h}^2$ denotes the norm with respect to $h$.

Let $\mathcal{M}$ be a compact smooth hypersurface in $\mathbb{R}^{n}$ with affine metric $h$ and affine volume measure $d\mu_{h}$. If $$\int_{\mathcal{M}}|\nabla f|^2_{h}d\mu_{h}$$ is small what can I say about $f~$? Sort of Sobolev type inequalities? Here $|.|_{h}^2$ denotes the norm with respect to $h$.

Edit: I can let $f$ be affine invariant also.

Let $\mathcal{M}$ be a smooth hypersurface in $\mathbb{R}^{n}$ with affine metric $h$ and affine volume measure $d\mu_{h}$. If $$\int_{\mathcal{M}}|\nabla f|^2_{h}d\mu_{h}$$ is small what can I say about $f~$? Sort of Sobolev type inequalities? Here $|.|_{h}^2$ denotes the norm with respect to $h$.

Let $\mathcal{M}$ be a compact smooth hypersurface in $\mathbb{R}^{n}$ with affine metric $h$ and affine volume measure $d\mu_{h}$. If $$\int_{\mathcal{M}}|\nabla f|^2_{h}d\mu_{h}$$ is small what can I say about $f~$? Sort of Sobolev type inequalities? Here $|.|_{h}^2$ denotes the norm with respect to $h$.