The critical catenoid has Morse index equal to 4.

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.

first eigenvector

second eigenvector

third eigenvector

fourth eigenvector