Asian Journal of Mathematics

Volume 22 (2018)

Number 1

Isometric embedding via strongly symmetric positive systems

Pages: 1 – 40

DOI: http://dx.doi.org/10.4310/AJM.2018.v22.n1.a1

Authors

Gui-Qiang Chen (Mathematical Institute, University of Oxford, United Kingdom)

Jeanne Clelland (Department of Mathematics, University of Colorado, Boulder, Col., U.S.A.)

Marshall Slemrod (Department of Mathematics, University of Wisconsin, Madison, Wisc., U.S.A.)

Dehua Wang (Department of Mathematics, University of Pittsburgh, Pennsylvania, U.S.A.)

Deane Yang (Department of Mathematics, New York University, New York, N.Y., U.S.A.)

Abstract

We give a new proof for the local existence of a smooth isometric embedding of a smooth $3$-dimensional Riemannian manifold with nonzero Riemannian curvature tensor into $6$-dimensional Euclidean space. Our proof avoids the sophisticated arguments via microlocal analysis used in earlier proofs.

In Part 1, we introduce a new type of system of partial differential equations (PDE), which is not one of the standard types (elliptic, hyperbolic, parabolic) but satisfies a property called strong symmetric positivity. Such a PDE system is a generalization of and has properties similar to a system of ordinary differential equations with a regular singular point. A local existence theorem is then established by using a novel local-to-global-to-local approach. In Part 2, we apply this theorem to prove the local existence result for isometric embeddings.

2010 Mathematics Subject Classification

Primary 53B20, 53C42. Secondary 35F50.

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The authors gratefully acknowledge the support of a SQuaRE grant from the American Institute of Mathematics, without which this project would not have been possible. We all thank our late friend Thomas H. Otway for helpful discussions. G.-Q. Chen was supported in part by the UK Engineering and Physical Sciences Research Council Award EP/L015811/1. J. Clelland was supported in part by NSF grants DMS-0908456 and DMS-1206272. M. Slemrod was supported in part by Simons Collaborative Research Grant 232531 and a Visiting Senior Research Fellowship at Keble College (Oxford). D. Wang was supported in part by NSF grants DMS-1312800 and DMS-1613213. D. Yang was supported in part by NSF grant DMS-1007347.

Received 9 August 2016

Accepted 5 October 2016

Published 10 May 2018