Cheeger’s inequality is a lower bound for the Laplacian’s first eigenvalue on a manifold depending only a geometric constant called Cheeger’s constant. I will present a higher-order Cheeger inequality for compact manifold without boundaries, giving a lower bound for higher eigenvalues using a natural generalisation of Cheeger’s constant. This result is adapted from the proof by Lee, Gharan and Trevissan  of this inequality for the discrete Laplacian on graphs. I will also apply their methods to prove a similar inequality for Steklov eigenvalues.
 Lee, James R., Shayan Oveis Gharan, and Luca Trevisan. “Multiway spectral partitioning and higher-order cheeger inequalities.” Journal of the ACM 61.6 (2014): 37.