What are the parities of photon-ring images near a black hole?

Light that grazes a black-hole event horizon can loop around one or more times before escaping again, resulting for distance observers in an infinite sequence of ever fainter and more delayed images near the black hole shadow. In the case of the M87 and Sgr~A$^*$ back holes, the first of these so-called photon-ring images have now been observed. A question then arises: are such images minima, maxima, or saddle-points in the sense of Fermat’s principle in gravitational lensing? or more briefly, the title question above. In the theory of lensing by weak gravitational fields, image parities are readily found by considering the time-delay surface (also called the Fermat potential or the arrival-time surface). In this work, we extend the notion of the time delay surface to strong gravitational fields and compute the surface for a Schwarzschild black hole. The time-delay surface is the difference of two wavefronts, one travelling forward from the source and one travelling backwards from the observer. Image parities are read off from the topography of the surface, exactly as in the weak-field regime, but the surface itself is more complicated. Of the images, furthest from the black hole and similar to the weak-field limit, are a minimum and a saddle point. The strong field repeats the pattern, corresponding to light taking one or more loops around the back hole. In between, there are steeply-rising walls in the time-delay surface, which can be interpreted as maxima and saddle points that are infinitely delayed and not observable — these correspond to light rays taking a U-turn around the black hole.