Let $(M,g)$ be a closed Riemannian manifold. Let $\lambda$ and $u$ be (the $k$-th) eigenvalue and eigenfunction, $$\Delta u=-\lambda u.$$

I was wondering under what condition (for example, spaces forms? Einstein manifolds? Kahler manifolds? with curvature constraints?) $$\int_M\frac{1}{u^2}dv<\infty,\quad\mathrm{or}\quad\int_M\frac{1}{u^{2k}}dv<\infty？$$ where $k$ is some positive real number.

I searched using google but didn't get any result. Thank you very much.

Let $(M,g)$ be a closed Riemannian manifold. Let $\lambda$ and $u$ be (the $k$-th) eigenvalue and eigenfunction, $$\Delta u=-\lambda u.$$

I was wondering under what condition (for example, spaces forms? Einstein manifolds? Kahler manifolds? with curvature constraints?) do we have $$\int_M\frac{1}{u^2}dv<\infty,\quad\mathrm{or}\quad\int_M\frac{1}{u^{2k}}dv<\infty？$$ where $k$ is some positive real number.

I searched using google but didn't get any result. Thank you very much.