A four-dimensional sphere is the only geometric shape that flips from stable to unstable depending on how you measure its energy.
April 23, 2026
Original Paper
On the conformal-biharmonic stability of the identity map of Einstein manifolds
arXiv · 2604.20257
The Takeaway
Standard mathematical rules suggest that stable shapes should stay stable regardless of the energy formula used. The 4D sphere breaks this rule by being unstable under normal energy but stable under conformal-bienergy. This strange quirk does not occur in any other dimension, marking the number four as a bizarre geometric outlier. It reveals an inherent instability in the very dimensions that define our universe. Understanding these unique properties helps physicists model the fundamental fabric of spacetime more accurately.
From the abstract
The identity map of an Einstein manifold is a critical point of both the classical energy functional and the conformal-bienergy functional. In this paper, we investigate the conformal-biharmonic stability of the identity map of compact Einstein manifolds of dimension at least four and with nonnegative scalar curvature, and we compare it with the harmonic stability, when the identity map is considered as a harmonic map. Somewhat surprisingly, we show that the conformal-biharmonic index coincides