One of simple tools in estimating Dirichlet eigenvalues and the Dirichlet heat kernel is the monotonicity property for subdomains via the Rayleigh quotient. This fails even for convex domains for the Neumann kernel and in general for the Robin problem. In this talk I will discuss recent work for bounds of the Generalized Robin problem in terms of the underlying geometry of the domain. The work is motivated from problems in superconductivity and probability and extends work of Sperb among others. The notion of a Cheeger domain will be discussed as well as its relationship with domain monotonicity and when it holds.