Stability estimates for a non-linear partial differential equation are estimates for how much the solution can vary, given the size of the variation of the right hand side. Clearly, they are of great theoretical as well as practical importance. Such estimates had been obtained by Kolodziej [12] for the complex Monge-Ampère equation, by Dinew and Kolodziej [4] for complex Hessian equations, and by Dinew and Zhang [6] for Monge-Ampère equations when the background metric is not necessarily Kähler, but just big. In all cases, the proofs made extensive use of pluripotential theory and the background metric was fixed. It remained an open question whether these estimates can be established without pluripotential theory, and whether they can be made uniform under degenerations of the background metric, a situation which arises frequently in geometric applications.

In [9], the authors developed a method for obtaining sharp $L^{\infty}$ estimates for the complex Monge-Ampère equation without pluripotential theory. As explained in greater detail there, the method of [9] built on works of Wang, Wang, Zhou [14] and of Chen and Cheng [3], particularly on the last two authors’ idea of considering an associated complex Monge-Ampère equation. It achieved the stated goal of giving an alternate PDE proof of $L^{\infty}$ estimates for the complex Monge-Ampère equation, but it also went considerably beyond in, on one hand, applying to more general fully non-linear equations, and on the other hand, allowing the background metrics to degenerate. It can also give sharp gradient estimates [10], improving on the estimates in e.g. [13, 2, 8, 7, 11].

The main goal of the present paper is to obtain stability estimates for the complex Monge-Ampère and Hessian equations which are uniform under degenerations. We use the method of [9]. We recover in the process the stability estimates of [12, 4, 6], this time without pluripotential theory. Our estimates are also uniform under general degenerations of the background metric in the case of the Monge-Ampère equation, and under degenerations to a big class in the case of Hessian equations. Thus we answer in the positive both questions asked above.