Prescribing nearly constant curvatures on balls Battaglia, Luca Cruz-Blázquez, Sergio Pistoia, Angela Prescribed curvature Conformal metrics Ljapunov-Schmidt construction S.C. acknowledges financial support from the Spanish Ministry of Universities and Next Generation EU funds, through a Margarita Salas grant from the University of Granada, by the FEDER-MINECO Grant PID2021-122122NB-I00 and by J. Andalucia (FQM-116). This work was carried out during his long visit to the University “Sapienza Universit`a di Roma”, to which he is grateful. The three authors are partially supported by the group GNAMPA of the Istituto Nazionale di Alta Matematica (INdAM). In particular, the first and second author are funded by the project “Fenomeni di blow-up per equazioni nonlineari”, project code CUP E55F22000270001. In this paper we address two boundary cases of the classical Kazdan- Warner problem. More precisely, we consider the problem of prescribing the Gaussian and boundary geodesic curvature on a disk of R2, and the scalar and mean curvature on a ball in higher dimensions, via a conformal change of the metric. We deal with the case of negative interior curvature and positive boundary curvature. Using a Ljapunov- Schmidt procedure, we obtain new existence results when the prescribed functions are close to constants. 2023-11-22T11:13:40Z 2023-11-22T11:13:40Z 2023-09-24 journal article Published version: Battaglia, L. et al. Prescribing nearly constant curvatures on balls. Proceedings of the Royal Society of Edinburgh Section A: Mathematics. 2023. [https://doi.org/10.1017/prm.2023.111] https://hdl.handle.net/10481/85818 10.1017/prm.2023.111 eng info:eu-repo/grantAgreement/EC/NextGenerationEU/PID2021-122122NB-I00 http://creativecommons.org/licenses/by/4.0/ open access Atribución 4.0 Internacional Cornell University