$\Delta f \le  - \lambda f$ then ${\lambda _1}\left( M \right) \ge \lambda$? Let M be a complete Riemannian manifold.If there exists a positive function defined on M satisfying$\Delta f \le  - \lambda f$ then ${\lambda _1}\left( M \right) \ge \lambda$?
 A: If you are looking for a simple proof, it is roughly the following.  (I will do the case of open domain in $R^N$).  Multiply both sides of $ \lambda f \le -\Delta f$  by $ \frac{\phi^2}{f}$ (here is where you use positivity of $f$)  where $ \phi$ is compactly supported.   Integrate over the domain and on the right hand side integrate by parts and then use Young's inequality to see $ \int \lambda \phi^2 \le \int | \nabla \phi|^2$. 
A: A positive function with $\Delta f \le -\lambda f$ is called $(-\lambda)$-superharmonic. Equivalent characterizations: on any bounded domain it exceeds the $(-\lambda)$-harmonic function with the same boundary values, or $P^t f \le e^{-\lambda t} f$ for any $t\ge 0$, where $P^t=e^{t\Delta}$ is the heat semigroup on the manifold. In addition to the usual $L^2$ spectrum of the Laplacian one can also define its positive spectrum, which is the set of values $\lambda$, for which there exists a positive $(-\lambda)$-harmonic (or superharmonic) function, so that the OP is just asking about the relationship between the positive and $L^2$ spectra. There is a Martin representation theory for positive $(-\lambda)$-superharmonic functions which generalizes the case of usual superharmonic functions; instead of the usual Green kernel it uses the $\lambda$-Green kernel
$$
G_\lambda (x,y) = \int_0^\infty e^{\lambda t} p_t(x,y) dt \;,
$$
where $p_t$ is the fundamental solution of the heat equation. Since $\lambda_1$ is the rate of decay of the heat kernel, the only intersection point of the positive and $L^2$ spectra is $\lambda_1$. By the way, it is pretty rare that positive (super)harmonic functions are square integrable (for $\lambda_1=0$ it is equivalent to the manifold having finite volume; in which case the corresponding function in constant). There is a good survey in an old paper of Sullivan (incidentally, he published it twice with almost the same title and completely identical content: MR0882827 and MR0849589).
