Abstract
We give a brief survey with emphasis on recent advances on regularity of solutions to the special Lagrangian equation and the Hamiltonian stationary Lagrangian equation.
Similar content being viewed by others
References
Alvarez, O., Lasry, J.-M., Lions, P.-L.: Convex viscosity solutions and state constraints. J. Math. Pures Appl. (9) 76(3), 265–288 (1997)
Brendle, S., Warren, M.: A boundary value problem for minimal Lagrangian graphs. J. Differ. Geom. 84(2), 267–287 (2010)
Bhattacharya, A., Chen, J.Y., Warren, M.: Regularity of Hamiltonian stationary equations in symplectic manifolds. ar** for 4th-order Nonlinear Elliptic Equations. Int. Math. Res. Not. 6, 4324–4348 (2019)
Caffarelli, L.A., Cabré, X.: Fully Nonlinear Elliptic Equations. American Mathematical Society Colloquium Publications, vol. 43. American Mathematical Society, Providence, RI (1995)
Caffarelli, L.A., Cabré, X.: Interior \(C^{2,\alpha }\) regularity for a class of nonconvex fully nonlinear elliptic equations. J. Math. Pures Appl. 82, 573–612 (2003)
Chau, A., Chen, J.Y., Yuan, Y.: Lagrangian mean curvature flow for entire Lipschitz graphs II. Math. Ann. 357(1), 165–183 (2013)
Choi, H.-I., Schoen, R.: The space of minimal embeddings of a surface into a three-dimensional manifold of positive Ricci curvature. Invent. Math. 81(3), 387–394 (1985)
Chen, J.Y., Warren, M.: On the regularity of Hamiltonian stationary Lagrangian submanifolds. Adv. Math. 343, 316–352 (2019)
Chen, J.Y., Warren, M.: Compactification of the space of Hamiltonian stationary Lagrangian submanifolds with bounded total extrinsic curvature and volume. J. Differ. Geom. (to appear)
Chen, J.Y., Warren, M.: Radial solutions of a fourth order Hamiltonian stationary equation. J. Differ. Equ. 265(4), 1576–1595 (2018)
Chen, J.Y., Ma, J. M.-S.: On the compactness of Hamiltonian stationary Lagrangian surfaces in Kähler surfaces. Calc. Var. Part. Differ. Equ. 60(2) (2021)
Chen, J.Y., Ma, J. M.-S.: (in preparation)
Chen, J.Y., Warren, M., Yuan, Y.: A priori estimate for convex solutions to special Lagrangian equations and its application. Commun. Pure Appl. Math. 62(4), 583–595 (2009)
Chen, J.Y., Shankar, R., Yuan, Y.: Regularity for convex viscosity solutions of special Lagrangian equation. Commun. Pure Appl. Math. (to appear)
Dazord, P.: Sur la géometie des sous-fibrés et des feuilletages lagrangiense, Ann. Sci. Éc. Norm. Super., IV, Ser.13 (1981), 465-480
Heinz, E.: On elliptic Monge-Ampère equations and Weyl’s embedding problem. J. Anal. Math. 7, 1–52 (1959)
Harvey, R., Lawson, H.B.: Calibrated geometries. Acta Math. 148, 47–157 (1982)
Joyce, D., Lee, Y.-I., Schoen, R.: On the existence of Hamiltonian stationary Lagrangian submanifolds in symplectic manifolds. Am. J. Math. 133(4), 1067–1092 (2011)
Morrey, C.B.: On the analyticity of the solutions of analytic non-linear elliptic systems of partial differential equations. I. Analyticity in the interior. Am. J. Math. 80198-218 (1958)
Nadirashvili, N., Vlăduţ, S.: Singular solution to special Lagrangian equations. Ann. Inst. H. Poincaré Anal. Non Linéaire 27(5), 1179–1188 (2010)
Oh, Y.-G.: Volume minimization of Lagrangian submanifolds under Hamiltonian deformations. Math. Z. 212, 175–192 (1992)
Schoen, R., Wolfson, R.: Minimizing area among Lagrangian surfaces: the map** problem. J. Differ. Geom. 58, 1–86 (2001)
Serrin, J.: Removable singularities of solutions of elliptic equations. Arch. Ratl. Mech. Anal. 17, 67–78 (1964)
Simon, L.: Lectures on geometric measure theory In: Proc. of the Centre for Math. Analysis, vol. 3. Australian National University (1983)
Thomas, R.P., Yau, S.-T.: Special Lagrangians, stable bundles and mean curvature flow. Commun. Anal. Geom. 10, 1075–1113 (2002)
Wang, D., Yuan, Y.: Hessian estimates for special Lagrangian equations with critical and supercritical phases in general dimensions. Am. J. Math. 136, 481–499 (2014)
Warren, M.: A Liouville property for gradient graphs and a Bernstein problem for Hamiltonian stationary equations. Manuscr. Math. 150(1–2), 151–157 (2016)
Warren, M., Yuan, Y.: Hessian estimates for the sigma-2 equation in dimension 3. Commun. Pure Appl. Math. 62(3), 305–321 (2009)
Warren, M., Yuan, Y.: Explicit gradient estimates for minimal Lagrangian surfaces of dimension two. Math. Z. 262(4), 867–879 (2009)
Warren, M., Yuan, Y.: Hessian and gradient estimates for three dimensional special Lagrangian equations with large phase. Am. J. Math. 132(3), 751–770 (2010)
Yuan, Y.: A priori estimates for solutions of fully nonlinear special Lagrangian equations. Ann. Inst. Henri Poincaré Anal. Non Linéaire 18(2), 261–270 (2001)
Yuan, Y.: A Bernstein problem for special Lagrangian equations. Invent. Math. 150(1), 117–125 (2002)
Yuan, Y.: Global solutions to special Lagrangian equations. Proc. Am. Math. Soc. 134(5), 1355–1358 (2006)
Author information
Authors and Affiliations
Corresponding author
Additional information
To Professor Peter Li in the occasion of his 70th birthday.
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
The author is partially supported by NSERC Discovery Grant (22R80062)
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Chen, J. Regularity of Critical Points of the Volume Functional for Lagrangian Submanifolds. J Geom Anal 32, 289 (2022). https://doi.org/10.1007/s12220-022-01041-8
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s12220-022-01041-8
Keywords
- Viscosity solutions
- Special Lagrangian equation
- Hamiltonian stationary Lagrangian equation
- Symplectic manifold