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June  2019, 39(6): 3463-3477. doi: 10.3934/dcds.2019143

## Isometric embedding with nonnegative Gauss curvature under the graph setting

 Department of Mathematics, Rutgers University, New Brunswick, NJ 08901, USA

Received  September 2018 Revised  November 2018 Published  February 2019

We study the regularity of the isometric embedding $X:$ $(B(O, r), g)$ $\rightarrow$ $(\mathbb{R}^3, g_{can})$ of a 2-ball with nonnegatively curved $C^4$ metric into $\mathbb{R}^3$. Under the assumption that $X$ can be expressed in the graph form, we show $X \in C^{2,1}$ near $P$, which is optimal by Iaia's example.

Citation: Xumin Jiang. Isometric embedding with nonnegative Gauss curvature under the graph setting. Discrete & Continuous Dynamical Systems - A, 2019, 39 (6) : 3463-3477. doi: 10.3934/dcds.2019143
##### References:
 [1] P. Daskalopoulos and O. Savin, On Monge-Ampère Equations with Homogeneous Right-Hand Sides, Comm. Pure Appl. Math, 62 (2009), 639-676. doi: 10.1002/cpa.20263. Google Scholar [2] P. Guan, Regularity of a class of quasilinear degenerate elliptic equations, Advances in Mathematics, 132 (1997), 24-45. doi: 10.1006/aima.1997.1677. Google Scholar [3] P. Guan and Y. Y. Li, The Weyl problem with nonnegative Gauss curvature, J. Diff. Geom., 39 (1994), 331-342. doi: 10.4310/jdg/1214454874. Google Scholar [4] P. Guan and E. Sawyer, Regularity of subelliptic Monge-Ampère equations in the plane, Trans. Amer. Math. Soc., 361 (2009), 4581-4591. doi: 10.1090/S0002-9947-09-04640-6. Google Scholar [5] J. Hong and C. Zuily, Isometric embedding of the 2-sphere with non negative curvature in $\mathbb{R}^3$, Math. Z., 219 (1995), 323-334. doi: 10.1007/BF02572368. Google Scholar [6] J. Iaia, The Weyl problem for surfaces of nonnegative curvature, Geometric Analysis and Nonlinear Partial Differential Equations (Denton, TX, 1990), 213–220, Lecture Notes in Pure and Appl. Math., 144, Dekker, New York, 1993. Google Scholar [7] J. Iaia, Isometric embeddings of surfaces with nonnegative curvature in $\mathbb{R}^3$, Duke Math. J., 67 (1992), 423-459. doi: 10.1215/S0012-7094-92-06717-2. Google Scholar [8] H. Lewy, On the existence of a closed convex surface realizing a given Riemannian metric, Proc. Nat. Acad. Sci., 24 (1938), 104-106. Google Scholar [9] Y. Y. Li and G. Weinstein, A priori bounds for co-dimension one isometric embeddings, Amer. J. of Math., 121 (1999), 945-965. doi: 10.1353/ajm.1999.0035. Google Scholar [10] C. S. Lin, The local isometric embedding in $\mathbb{R}^3$ of 2-dimensional Riemannian manifolds with nonnegative curvature, J. Diff. Geom., 21 (1985), 213-230. doi: 10.4310/jdg/1214439563. Google Scholar [11] L. Nirenberg, The Weyl and Minkowski problems in differential geometry in the large, Comm. Pure Appl. Math., 6 (1953), 337-394. doi: 10.1002/cpa.3160060303. Google Scholar [12] A. V. Pogorelov, An example of a two-dimensional Riemannian metric admitting no local realization in $E_3$, Dokl. Akad. Nauk SSSR, 198 (1971), 42-43. Google Scholar [13] F. Schulz, Regularity Theory for Quasilinear Elliptic Systems and Monge-Ampère Equations in Two Dimensions, Lect. Note. in Math., 1445, Springer-Verlag, Berlin, 1990. doi: 10.1007/BFb0098277. Google Scholar [14] H. Weyl, über die Bestimmung einer geschlossen konvexen Fläche durch ihr Linienelement, Vierteljahrsschrift Naturforsch. Gesellschaft, 61 (1916), 40-72. Google Scholar

show all references

##### References:
 [1] P. Daskalopoulos and O. Savin, On Monge-Ampère Equations with Homogeneous Right-Hand Sides, Comm. Pure Appl. Math, 62 (2009), 639-676. doi: 10.1002/cpa.20263. Google Scholar [2] P. Guan, Regularity of a class of quasilinear degenerate elliptic equations, Advances in Mathematics, 132 (1997), 24-45. doi: 10.1006/aima.1997.1677. Google Scholar [3] P. Guan and Y. Y. Li, The Weyl problem with nonnegative Gauss curvature, J. Diff. Geom., 39 (1994), 331-342. doi: 10.4310/jdg/1214454874. Google Scholar [4] P. Guan and E. Sawyer, Regularity of subelliptic Monge-Ampère equations in the plane, Trans. Amer. Math. Soc., 361 (2009), 4581-4591. doi: 10.1090/S0002-9947-09-04640-6. Google Scholar [5] J. Hong and C. Zuily, Isometric embedding of the 2-sphere with non negative curvature in $\mathbb{R}^3$, Math. Z., 219 (1995), 323-334. doi: 10.1007/BF02572368. Google Scholar [6] J. Iaia, The Weyl problem for surfaces of nonnegative curvature, Geometric Analysis and Nonlinear Partial Differential Equations (Denton, TX, 1990), 213–220, Lecture Notes in Pure and Appl. Math., 144, Dekker, New York, 1993. Google Scholar [7] J. Iaia, Isometric embeddings of surfaces with nonnegative curvature in $\mathbb{R}^3$, Duke Math. J., 67 (1992), 423-459. doi: 10.1215/S0012-7094-92-06717-2. Google Scholar [8] H. Lewy, On the existence of a closed convex surface realizing a given Riemannian metric, Proc. Nat. Acad. Sci., 24 (1938), 104-106. Google Scholar [9] Y. Y. Li and G. Weinstein, A priori bounds for co-dimension one isometric embeddings, Amer. J. of Math., 121 (1999), 945-965. doi: 10.1353/ajm.1999.0035. Google Scholar [10] C. S. Lin, The local isometric embedding in $\mathbb{R}^3$ of 2-dimensional Riemannian manifolds with nonnegative curvature, J. Diff. Geom., 21 (1985), 213-230. doi: 10.4310/jdg/1214439563. Google Scholar [11] L. Nirenberg, The Weyl and Minkowski problems in differential geometry in the large, Comm. Pure Appl. Math., 6 (1953), 337-394. doi: 10.1002/cpa.3160060303. Google Scholar [12] A. V. Pogorelov, An example of a two-dimensional Riemannian metric admitting no local realization in $E_3$, Dokl. Akad. Nauk SSSR, 198 (1971), 42-43. Google Scholar [13] F. Schulz, Regularity Theory for Quasilinear Elliptic Systems and Monge-Ampère Equations in Two Dimensions, Lect. Note. in Math., 1445, Springer-Verlag, Berlin, 1990. doi: 10.1007/BFb0098277. Google Scholar [14] H. Weyl, über die Bestimmung einer geschlossen konvexen Fläche durch ihr Linienelement, Vierteljahrsschrift Naturforsch. Gesellschaft, 61 (1916), 40-72. Google Scholar
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