For every integer n≥ 2, let F colon B^n→ B^2n⊂C^n be a smoothly immersed Lagrangian n-ball whose Maslov form is conformal and whose boundary is Legendrian and capillary. Then the image F(B^n) is either an equatorial Lagrangian disk or is contained in a Whitney sphere centered at the origin.

Li–Wang–Weng conjecture on Lagrangian submanifolds

Li, Wang, and Weng proposed a rigidity statement for minimal Lagrangian disks in dimension two, together with an associated annulus classification. The tracked problem extends the disk classification to immersed Lagrangian balls in dimensions $n \ge 2$ .

Known results

Li–Wang–Weng established the two-dimensional minimal Lagrangian disk theorem.
Luo–Sun proved two-dimensional free-boundary rigidity and the annulus-type classification without assuming embeddedness.

September 2026 classification

Dong Gao, Yong Luo, Hui Ma, and Jiabin Yin claim that their paper classifies the relevant immersed Lagrangian balls for all $n \ge 2$ , thereby resolving the cited conjecture in dimension two. The claim is unverified; the paper does not assert a classification of all related Lagrangian objects.

Current status (as of September 2026): The original two-dimensional rigidity conjecture is claimed solved by the September 2026 paper, while the broader classification remains outside its stated scope and the claim awaits verification.

Li–Wang–Weng conjecture on Lagrangian submanifolds

Primary source

Additional references

Progress summary

Solutions 0