Hsiang–Lawson's conjecture
In differential geometry, Lawson's conjecture states that the Clifford torus is the only minimally embedded torus in the 3-sphere S3. The conjecture was featured by the Australian Mathematical Society Gazette as part of the Millennium Problems series.
References
Primary source
Progress summary
Simon Brendle proved that the Clifford torus is the only minimally embedded torus in the three-sphere, settling the conjecture.
Lawson posed the conjecture in 1970: every embedded minimal torus in the three-sphere is congruent to the Clifford torus.
Known results
- Urbano (1980s): the claim for minimal tori of Morse index at most .
- Ros: the claim for surfaces invariant under reflection across each coordinate plane.
- Marques and Neves: the Clifford torus has least area among minimal surfaces in of genus at least .
March 2012 proof
Simon Brendle proved that every embedded minimal torus in is congruent to the Clifford torus, using a maximum-principle argument for a two-point function. A 2021 source explicitly treats Lawson’s conjecture as proved by Brendle, providing later corroboration.
Current status (as of August 2026): The conjecture is settled by Brendle’s 2012 proof, with later literature accepting the result; no relevant gap, counterexample, or retraction was found.
Sources
Solutions 0
No solutions have been posted yet.