The regularity of Special Legendrian Integral Cycles

Bellettini, Costante; Rivière, Tristan

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 11 (2012) no. 1, pp. 61-142.

Special Legendrian Integral Cycles in $S^5$ are the links of the tangent cones to Special Lagrangian integer multiplicity rectifiable currents in Calabi-Yau 3-folds. We show that Special Legendrian Cycles are smooth except possibly at isolated points.

Classification : 49Q15, 32Q25

[1] Jr. Almgren and J. Frederick, “Almgren’s big Regularity Paper”, World Scientific Monograph Series in Mathematics, 1, “$Q$-valued Functions Minimizing Dirichlet’s Integral and the Regularity of Area-minimizing Rectifiable Currents up to Codimension 2”, with a preface by Jean E. Taylor and Vladimir Scheffer, World Scientific Publishing Co. Inc., River Edge, NJ, 2000, xvi+955. | MR | Zbl

[2] N. Aronszajn, A unique continuation theorem for solutions of elliptic partial differential equations or inequalities of second order, J. Math. Pures Appl. (9) 36 (1957), 235–249. | MR | Zbl

[3] S. X.-D. Chang, Two-dimensional area minimizing integral currents are classical minimal surfaces, J. Amer. Math. Soc. 1 (1988), 699–778. | MR | Zbl

[4] C. De Lellis and E. Spadaro, “$Q$-Valued Functions Revisited”, Mem. Amer. Math. Soc., 211 (2011), n. 991, vi+79. | MR | Zbl

[5] S. K. Donaldson and R. P. Thomas, Gauge Theory in higher dimensions, In: “The Geometric Universe" (Oxford, 1996), Oxford Univ. Press, 1998, 31-47. | MR | Zbl

[6] L. C. Evans and R. F. Gariepy, “Measure Theory and Fine Properties of Functions”, Studies in Advanced Mathematics, CRC Press, Boca Raton, FL, 1992, viii+268. | MR | Zbl

[7] H. Federer, “Geometric Measure Theory”, Die Grundlehren der mathematischen Wissenschaften, Band 153, Springer-Verlag New York Inc., New York, 1969, xiv+676. | MR | Zbl

[8] M. Giaquinta, G. Modica and J. Souček, “Cartesian Currents in the Calculus of Variations. I”, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], 37, Cartesian currents, Springer-Verlag, Berlin, 1998, xxiv+711. | MR | Zbl

[9] D. Gilbarg and N. S. Trudinger, “Elliptic Partial Differential Equations of Second Order”, Classics in Mathematics, Reprint of the 1998 edition, Springer-Verlag, Berlin, 2001, xiv+517. | MR | Zbl

[10] R. Harvey and H. B. Jr. Lawson, Calibrated geometries, Acta Math. 148 (1982), 47–157. | MR | Zbl

[11] M. Haskins, Special Lagrangian cones, Amer. J. Math. 126 (2004), 845–871. | MR | Zbl

[12] D. D. Joyce, “Riemannian Holonomy Groups and Calibrated Geometry”, Oxford Graduate Texts in Mathematics, 12, Oxford University Press, Oxford, 2007, x+303. | MR | Zbl

[13] M. J. Micallef and B. White, The structure of branch points in minimal surfaces and in pseudoholomorphic curves, Ann. of Math. (2) 141 (1995), 35–85. | MR | Zbl

[14] F. Morgan, “Geometric Measure Theory”, Fourth edition, A beginner’s guide, Elsevier/Academic Press, Amsterdam, 2009, viii+249. | MR | Zbl

[15] C. B. Jr. Morrey, “Multiple Integrals in the Calculus of Variations”, Die Grundlehren der mathematischen Wissenschaften, Band 130, Springer-Verlag New York, Inc., New York, 1966, ix+506. | MR | Zbl

[16] D. Pumberger and T. Rivière, Uniqueness of tangent cones for semi-calibrated 2-cycles, Duke Math. J. 152 (2010), 441–480. | MR | Zbl

[17] T. Rivière and G. Tian, The singular set of $J$-holomorphic maps into projective algebraic varieties, J. Reine Angew. Math. 570 (2004), 47–87. 58J45. | MR | Zbl

[18] T. Rivière and G. Tian, The singular set of 1-1 integral currents, Ann. of Math. (2) 169 (2009), 741–794. | MR | Zbl

[19] L. Simon, “Lectures on Geometric Measure Theory”, Proceedings of the Centre for Mathematical Analysis, Australian National University, 3, Australian National University Centre for Mathematical Analysis, Canberra, 1983, vii+272. | MR | Zbl

[20] J. Simons, Minimal varieties in riemannian manifolds, Ann. of Math. (2) 88 (1968), 62–105. | MR | Zbl

[21] C. H. Taubes, “$\mathrm{SW}\Rightarrow \mathrm{Gr}$: From the Seiberg-Witten Equations to Pseudo-holomorphic Curves". Seiberg Witten and Gromov invariants for symplectic 4-manifolds., 1–102, First Int. Press Lect. Ser., 2, Int. Press, Somerville, MA, 2000. | MR | Zbl

[22] G. Tian, Gauge theory and calibrated geometry. I, Ann. of Math. (2) 151 (2000), 193–268. | EuDML | MR | Zbl

[23] B. White, Tangent cones to two-dimensional area-minimizing integral currents are unique, Duke Math. J. 50 (1983), 143–160. | MR | Zbl