Hsiang–Lawson's conjecture

{{short description|Theorem that the Clifford torus is the only minimally embedded torus in the 3-sphere}}

In mathematics, Lawson's conjecture states that the Clifford torus is the only minimally embedded torus in the 3-sphere S³.{{cite journal |first=H. Blaine Jr. |last=Lawson |title=The unknottedness of minimal embeddings |journal=Invent. Math. |volume=11 |year=1970 |issue=3 |pages=183–187 |doi=10.1007/BF01404649 |bibcode=1970InMat..11..183L |s2cid=122740925 }}{{cite journal |first=H. Blaine Jr. |last=Lawson |title=Complete minimal surfaces in S³ |journal=Ann. of Math. |volume=92 |year=1970 |issue=3 |pages=335–374 |doi=10.2307/1970625 |jstor=1970625 }} The conjecture was featured by the Australian Mathematical Society Gazette as part of the Millennium Problems series.{{cite journal

| last = Norbury

| first = Paul

| title = The 12th problem

| journal = The Australian Mathematical Society Gazette

| volume = 32

| issue = 4

| pages = 244–246

| year = 2005

| url = http://www.austms.org.au/Publ/Gazette/2005/Sep05/millennium.pdf

}}

In March 2012, Simon Brendle gave a proof of this conjecture, based on maximum principle techniques.{{cite journal |last=Brendle |first=Simon |title=Embedded minimal tori in S³ and the Lawson conjecture | journal = Acta Mathematica | volume = 211 | pages = 177–190 | year = 2013 |issue=2 | doi=10.1007/s11511-013-0101-2|s2cid=119317563 |doi-access=free |arxiv=1203.6597 }}