Local flexibility for open partial differential relations. (English) Zbl 1497.35485
Summary: We show that local deformations, near closed subsets, of solutions to open partial differential relations can be extended to global deformations, provided all but the highest derivatives stay constant along the subset. The applicability of this general result is illustrated by a number of examples, dealing with convex embeddings of hypersurfaces, differential forms, and lapse functions in Lorentzian geometry.
The main application is a general approximation result by sections that have very restrictive local properties on open dense subsets. This shows, for instance, that given any \(K \in \mathbb{R}\) every manifold of dimension at least 2 carries a complete \(C^{1, 1}\)-metric which, on a dense open subset, is smooth with constant sectional curvature \(K\). Of course, this is impossible for \(C^2\)-metrics in general.
The main application is a general approximation result by sections that have very restrictive local properties on open dense subsets. This shows, for instance, that given any \(K \in \mathbb{R}\) every manifold of dimension at least 2 carries a complete \(C^{1, 1}\)-metric which, on a dense open subset, is smooth with constant sectional curvature \(K\). Of course, this is impossible for \(C^2\)-metrics in general.
MSC:
| 35R01 | PDEs on manifolds |
| 58A20 | Jets in global analysis |
| 53B25 | Local submanifolds |
| 53B30 | Local differential geometry of Lorentz metrics, indefinite metrics |
| 53A10 | Minimal surfaces in differential geometry, surfaces with prescribed mean curvature |
Keywords:
local and global deformations; convex embeddings of hypersurfaces; differential forms; lapse functions; Lorentzian geometry; constant sectional curvatureReferences:
| [1] | Alvarez‐Gavela, D. Refinements of the holonomic approximation lemma. Algebr. Geom. Topol.18 (2018), no. 4, 2265-2303. doi: https://doi.org/10.2140/agt.2018.18.2265 · Zbl 1393.58002 · doi:10.2140/agt.2018.18.2265 |
| [2] | Alzaareer, H.; Schmeding, A.Differentiable mappings on products with different degrees of differentiability in the two factors. Expo. Math.33 (2015), no. 2, 184-222. doi: https://doi.org/10.1016/j.exmath.2014.07.002 · Zbl 1330.46039 · doi:10.1016/j.exmath.2014.07.002 |
| [3] | Beem, J. K.; Ehrlich, Paul E.; Easley, Kevin L. Global Lorentzian geometry. (English summary) Second edition. Monographs and Textbooks in Pure and Applied Mathematics, 202. Marcel Dekker, New York, 1996. · Zbl 0846.53001 |
| [4] | Bernal, A. N.; Sanchez, M.Smoothness of time functions and the metric splitting of globally hyperbolic spacetimes. Comm. Math. Phys.257 (2005), no. 1, 43-50. doi: https://doi.org/10.1007/s00220‐005‐1346‐1 · Zbl 1081.53059 · doi:10.1007/s00220‐005‐1346‐1 |
| [5] | Bernal, A. N.; Sanchez, M.Further results on the smoothability of Cauchy hypersurfaces and Cauchy time functions. Lett. Math. Phys.77 (2006), no. 2, 183-197. doi: https://doi.org/10.1007/s11005‐006‐0091‐5 · Zbl 1126.53043 · doi:10.1007/s11005‐006‐0091‐5 |
| [6] | Bryant, R. L. Some remarks on G_2‐structures. Proceedings of Gokova Geometry‐Topology Conference2005, 75-109. Gokova Geometry/Topology Conference (GGT), Gokova, 2006. · Zbl 1115.53018 |
| [7] | Burago, D.; Burago, Y.; Ivanov, S.A course in metric geometry. Graduate Studies in Mathematics, 33. American Mathematical Society, Providence, R.I., 2001. doi: https://doi.org/10.1090/gsm/033 · Zbl 0981.51016 · doi:10.1090/gsm/033 |
| [8] | Cannas da Silva, A. Lectures on symplectic geometry. Lecture Notes in Mathematics, 1764. Springer, Berlin, 2001. doi: 10.1007/978‐3‐540‐45330‐7 · Zbl 1016.53001 |
| [9] | Conti, S.; De Lellis, C.; Szekelyhidi, L., Jr.h‐principle and rigidity for C^1, α isometric embeddings. Nonlinear partial differential equations, 83-116. Abel Symposia, Vol. 7. Springer, Heidelberg, 2012. doi: 10.1007/978‐3‐642‐25361‐4_5 · Zbl 1255.53038 |
| [10] | Dovgoshey, O.; Martio, O.; Ryazanov, V.; Vuorinen, M.The Cantor function. Expo. Math.24 (2006), no. 1, 1-37. doi: https://doi.org/10.1016/j.exmath.2005.05.002 · Zbl 1098.26006 · doi:10.1016/j.exmath.2005.05.002 |
| [11] | Eschenburg, J.‐H.Local convexity and nonnegative curvature‐Gromov’s proof of the sphere theorem. Invent. Math.84 (1986), no. 3, 507-522. doi: https://doi.org/10.1007/BF01388744 · Zbl 0594.53034 · doi:10.1007/BF01388744 |
| [12] | Fox, R. H.On topologies for function spaces. Bull. Amer. Math. Soc.51 (1945), 429-432. doi: https://doi.org/10.1090/S0002‐9904‐1945‐08370‐0 · Zbl 0060.41202 · doi:10.1090/S0002‐9904‐1945‐08370‐0 |
| [13] | Gilbarg, D.; Trudinger, N. S.Elliptic partial differential equations of second order. Classics in Mathematics. Springer, Berlin, 2001. · Zbl 1042.35002 |
| [14] | Graf, M.Volume comparison for C^1, 1‐metrics. Ann. Global Anal. Geom.50 (2016), no. 3, 209-235. doi: https://doi.org/10.1007/s10455‐016‐9508‐2 · Zbl 1362.53078 · doi:10.1007/s10455‐016‐9508‐2 |
| [15] | Greene, R. E.Complete metrics of bounded curvature on noncompact manifolds. Arch. Math. (Basel)31 (1978/79), no. 1, 89-95. doi: https://doi.org/10.1007/BF01226419 · Zbl 0373.53018 · doi:10.1007/BF01226419 |
| [16] | Greene, R. E.; Wu, H.C^∞ approximations of convex, subharmonic, and plurisubharmonic functions. Ann. Sci. I’École Norm. Sup. (4)12 (1979), no. 1, 47-84. · Zbl 0415.31001 |
| [17] | Greene, R. E.; Wu, H.Lipschitz convergence of Riemannian manifolds. Pacific J. Math.131 (1988), no. 1, 119-141. · Zbl 0646.53038 |
| [18] | Gromov, M. L. Stable mappings of foliations into manifolds. Izv. Akad. Nauk SSSR Ser. Mat.33 (1969), 707-734. · Zbl 0197.20404 |
| [19] | Gromov, M.Partial differential relations. Ergebnisse der Mathematik und ihrer Grenzgebiete (3), Vol. 9. Springer, Berlin, 1986. doi: https://doi.org/10.1007/978‐3‐662‐02267‐2 · Zbl 0651.53001 · doi:10.1007/978‐3‐662‐02267‐2 |
| [20] | Gromov, M.Metric structures for Riemannian and non‐Riemannian spaces. Progress in Mathematics, 152. BirkhäuserBoston, Boston, 1999. · Zbl 0953.53002 |
| [21] | Gromov, M.Metric inequalities with scalar curvature. Geom. Funct. Anal.28 (2018), no. 3, 645-726. doi: https://doi.org/10.1007/s00039‐018‐0453‐z · Zbl 1396.53068 · doi:10.1007/s00039‐018‐0453‐z |
| [22] | Hirsch, M. W.Differential topology. Graduate Texts in Mathematics, 33. Springer, New York, 1994. |
| [23] | Kuiper, N. H. On C^1‐isometric imbeddings. I, II Nederl. Akad. Wetensch. Proc. Ser. A. 58 = Indag. Math. 17 (1955), 545-556, 683-689. · Zbl 0067.39601 |
| [24] | Nash, J.C^1 isometric imbeddings. Ann. of Math. (2)60 (1954), 383-396. doi: https://doi.org/10.2307/1969840 · Zbl 0058.37703 · doi:10.2307/1969840 |
| [25] | Nikolaev, I. G.Smoothness of the metric of spaces with bilaterally bounded curvature in the sense of A. D. Aleksandrov. Sibirsk. Mat. Zh.24 (1983), no. 2, 114-132. · Zbl 0547.53011 |
| [26] | Peters, S.Convergence of Riemannian manifolds. Compositio Math.62 (1987), no. 1, 3-16. · Zbl 0618.53036 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
