The critical catenoid has Morse index equal to 4.

Minimal surfaces are critical points of the area functional, meaning that deformations do not decrease their area to first order. A minimal surface is considered unstable if certain deformations can decrease its area to second order. The Morse index quantifies the number of such deformations, making it a measure for the instability of a minimal surface.

References

• H. Tran, Index characterization for free boundary minimal surfaces, Comm. Anal. Geom. 28 (2020), 189–222.
• G. Smith and D. Zhou, The morse index of the critical catenoid, Geom. Dedicata 201 (2019), 13–19.
• B. Devyver, Index of the critical catenoid, Geom. Dedicata 199 (2019), 355–371.