Complex, multi-holed surfaces have been discovered hidden within five-dimensional spheres for the first time.
April 24, 2026
Original Paper
Embedded special Legendrian surfaces in $\mathbb S^5$
arXiv · 2604.21521
The Takeaway
Special Legendrian surfaces with complex shapes exist within the geometry of a 5-sphere. Mathematicians previously struggled to prove that such smooth, embedded shapes with a genus greater than one could even be constructed. These shapes are fundamental to understanding the symplectic geometry of higher dimensions. This discovery provides the first concrete examples of these manifolds, filling a gap that has existed in the field for years. Solving these geometric existence problems helps physicists model how strings behave in the extra dimensions of the universe.
From the abstract
We construct the first smooth embedded compact special Legendrian surfaces in \(\mathbb S^5\) of genus greater than one. More precisely, for every sufficiently large integer \(k\), we construct an embedded special Legendrian surface whose conformal structure is the Fermat curve of degree \(k\) and genus \(\tfrac12(k-1)(k-2)\). Our approach combines an elementary implicit function theorem with the description of special Legendrian surfaces via loop algebra-valued meromorphic connections and a cha