As shown by Gromov-Lawson and Stolz the only obstruction to the existence of positive scalar curvature metrics on closed simply connected manifolds in dimensions at least five appears on spin manifolds, and is given by the non-vanishing of the a-genus of Hitchin. [cite: https://www.google.com/search?q=20, 165] When unobstructed we shall realise a positive scalar curvature metric by an immersion into Euclidean space whose dimension is uniformly close to the classical Whitney upper bound for smooth immersions. [cite: https://www.google.com/search?q=21, 166] Our main tool is an extrinsic counterpart of the well-known Gromov-Lawson surgery procedure for constructing positive scalar curvature metrics. [cite: https://www.google.com/search?q=2https://www.google.com/search?q=2, 167] At this point we use the local flexibility lemma proven by Christian Bär and the speaker. [cite: https://www.google.com/search?q=23, 168] This is joint work with Luis Florit, IMPA (Rio de Janeiro). [cite: https://www.google.com/search?q=24, 169]