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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.

[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.

[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.

[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.

[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.

[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.

[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.

[8]

H. Lewy, On the existence of a closed convex surface realizing a given Riemannian metric, Proc. Nat. Acad. Sci., 24 (1938), 104-106.

[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.

[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.

[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.

[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.

[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.

[14]

H. Weyl, über die Bestimmung einer geschlossen konvexen Fläche durch ihr Linienelement, Vierteljahrsschrift Naturforsch. Gesellschaft, 61 (1916), 40-72.

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.

[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.

[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.

[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.

[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.

[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.

[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.

[8]

H. Lewy, On the existence of a closed convex surface realizing a given Riemannian metric, Proc. Nat. Acad. Sci., 24 (1938), 104-106.

[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.

[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.

[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.

[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.

[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.

[14]

H. Weyl, über die Bestimmung einer geschlossen konvexen Fläche durch ihr Linienelement, Vierteljahrsschrift Naturforsch. Gesellschaft, 61 (1916), 40-72.

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