Abstract
In this paper we study the embedding of Riemannian manifolds in low codimension. The well-known result of Nash and Kuiper (Nash in Ann. Math. 60:383–396, 1954; Kuiper in Proc. Kon. Acad. Wet. Amsterdam A 58:545–556, 1955; Kuiper in Proc. Kon. Acad. Wet. Amsterdam A 58:683–689, 1955) says that any short embedding in codimension one can be uniformly approximated by C 1 isometric embeddings. This statement clearly cannot be true for C 2 embeddings in general, due to the classical rigidity in the Weyl problem. In fact Borisov extended the latter to embeddings of class C 1,α with α>2/3 in (Borisov in Vestn. Leningr. Univ. 14(13):20–26, 1959; Borisov in Vestn. Leningr. Univ. 15(19):127–129, 1960). On the other hand he announced in (Borisov in Doklady 163:869–871, 1965) that the Nash–Kuiper statement can be extended to local C 1,α embeddings with α<(1+n+n 2)−1, where n is the dimension of the manifold, provided the metric is analytic. Subsequently a proof of the 2-dimensional case appeared in (Borisov in Sib. Mat. Zh. 45(1):25–61, 2004). In this paper we provide analytic proofs of all these statements, for general dimension and general metric.
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Borisov, J.F.: The parallel translation on a smooth surface. I. Vestn. Leningr. Univ. 13(7), 160–171 (1958)
Borisov, J.F.: The parallel translation on a smooth surface. II. Vestn. Leningr. Univ. 13(19), 45–54 (1958)
Borisov, J.F.: On the connection between the spatial form of smooth surfaces and their intrinsic geometry. Vestn. Leningr. Univ. 14(13), 20–26 (1959)
Borisov, J.F.: The parallel translation on a smooth surface. III. Vestn. Leningr. Univ. 14(1), 34–50 (1959)
Borisov, J.F.: On the question of parallel displacement on a smooth surface and the connection of space forms of smooth surfaces with their intrinsic geometries. Vestn. Leningr. Univ. 15(19), 127–129 (1960)
Borisov, J.F.: C 1,α-isometric immersions of Riemannian spaces. Doklady 163, 869–871 (1965)
Borisov, Y.F.: Irregular C 1,β-surfaces with analytic metric. Sib. Mat. Zh. 45(1), 25–61 (2004)
Cohn-Vossen, S.: Zwei Sätze über die Starrheit der Eiflächen. Nachr. Gött. 1927, 125–137 (1927)
Constantin, P., E, W., Titi, E.S.: Onsager’s conjecture on the energy conservation for solutions of Euler’s equation. Commun. Math. Phys. 165(1), 207–209 (1994)
De Lellis, C., Székelyhidi, L.J.: The Euler equations as a differential inclusion. Ann. of Math. (2) 170(3), 1417–1436 (2009) (English summary)
De Lellis, C., Székelyhidi, L.J.: On admissibility criteria for weak solutions of the Euler equations. Arch. Ration. Mech. Anal. 195(1), 225–260 (2010) (English summary)
Eliashberg, Y., Mishachev, N.: Introduction to the h-Principle. Graduate Studies in Mathematics, vol. 48. Am. Math. Soc., Providence (2002)
Eyink, G.L.: Energy dissipation without viscosity in ideal hydrodynamics. I. Fourier analysis and local energy transfer. Physica D 78(3–4), 222–240 (1994)
Frisch, U.: Turbulence. The Legacy of A.N. Kolmogorov, p. 296. Cambridge University Press, Cambridge (1995). ISBN:0-521-45103-5
Gromov, M.: Convex integration of differential relations. Izv. Akad. Nauk USSR 37, 329–343 (1973)
Gromov, M.: Partial Differential Relations. Springer, Berlin (1986)
Herglotz, G.: Über die Starrheit der Eiflächen. Abh. Math. Semin. Hansische Univ. 15, 127–129 (1943)
Jacobowitz, H.: Implicit function theorems and isometric embeddings. Ann. of Math. (2) 95, 191–225 (1972)
Källén, A.: Isometric embedding of a smooth compact manifold with a metric of low regularity. Ark. Mat. 16(1), 29–50 (1978)
Kuiper, N.: On C 1 isometric imbeddings I. Proc. Kon. Acad. Wet. Amsterdam A 58, 545–556 (1955)
Kuiper, N.: On C 1 isometric imbeddings II. Proc. Kon. Acad. Wet. Amsterdam A 58, 683–689 (1955)
Nash, J.: C 1 isometric imbeddings. Ann. Math. 60, 383–396 (1954)
Nash, J.: The imbedding problem for Riemannian manifolds. Ann. Math. 63, 20–63 (1956)
Onsager, L.: Statistical hydrodynamics. Nuovo Cimento (9) 6(Supplemento, 2(Convegno Internazionale di Meccanica Statistica)), 279–287 (1949)
Pogorelov, A.V.: The rigidity of general convex surfaces. Dokl. Akad. Nauk SSSR 79, 739–742 (1951)
Pogorelov, A.V.: Extrinsic Geometry of Convex Surfaces. Translations of Mathematical Monographs, vol. 35. Am. Math. Soc., Providence (1973)
Sabitov, I.H.: Regularity of convex domains with a metric that is regular on Hölder classes. Sib. Mat. Zh. 17(4), 907–915 (1976)
Scheffer, V.: An inviscid flow with compact support in space-time. J. Geom. Anal. 3(4), 343–401 (1993)
Shnirelman, A.: On the nonuniqueness of weak solution of the Euler equation. Commun. Pure Appl. Math. 50(12), 1261–1286 (1997)
Shnirelman, A.: Weak solutions with decreasing energy of incompressible Euler equations. Commun. Math. Phys. 210(3), 541–603 (2000)
Spivak, M.: A Comprehensive Introduction to Differential Geometry, vol. V, 2nd edn. Publish or Perish, Berkeley (1979)
Spring, D.: On the regularity of solutions in convex integration theory. Invent. Math. 104(1), 165–178 (1991). doi:10.1007/BF01245070
Spring, D.: Convex Integration Theory. Birkhäuser, Basel (1998)
Yau, S.-T.: Open problems in geometry. In: Differential Geometry: Partial Differential Equations on Manifolds, Los Angeles, CA, 1990. Proc. Sympos. Pure Math., vol. 54, pp. 1–28. Am. Math. Soc., Providence (1993)
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Camillo De Lellis has been supported by the SFB grant TR 71.
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Appendix
Appendix
Proof of Corollary 4
First of all, since the theorem is local, without loss of generality we can assume that:
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1.
Ω=B r (0), \(u\in C^{1, \alpha} (\overline{B}_{r} (x))\), \(g\in C^{2,\beta} (\overline{B}_{r} (x))\) and u is an embedding;
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2.
u(Ω) has bounded extrinsic curvature.
Step 1. Density of Regular Points
For any point \(z\in\mathbb{S}^{2}\) we let n(z) be the cardinality of N −1(z). It is easy to see that, for a surface of bounded extrinsic curvature, \(\int_{\mathbb{S}^{2}}n <\infty\) (cf. with Theorem 3 of p. 590 in [26]). Therefore, the set E:={n=∞} has measure zero. Let \(\varOmega_{r}:= N^{-1} (\mathbb{S}^{2}\setminus E)\). Observe that
Otherwise there is a nontrivial smooth open set V such that N(V)⊂E. But then, deg(⋅,V,N)=0 for every \(y\notin N (\overline{V})\), and since |N(V)|=|N(∂V)|=0, it follows that deg(⋅,V,N)=0 a.e. By Corollary 5, ∫ V κ=0, which contradicts κ>0.
Step 2. Convexity Around Regular Points
Note next that, for every x∈Ω r there is a neighborhood U of x such that N(y)≠N(x) for all y∈U∖{x}, i.e. x is regular in the sense of [26] p. 582. Recalling (98), deg(⋅,V,N)≥1 V∖∂V for every V: therefore the index of the map N at every point x∈Ω r is at least 1. So, by the Lemma of page 594 in [26], any point x∈Ω r is an elliptic point relative to the mapping N (that is, there is a neighborhood U of x such that the tangent plane π to u(Ω) in x intersects U∩u(Ω) only in u(x); cf. with page 593 of [26]).
By the discussion of page 650 in [26], u(Ω) has nonnegative extrinsic curvature as defined in IX.5 of [26]. Then, Lemma 2 of page 612 shows that, for every elliptic point y∈u(Ω) there is a neighborhood where u(Ω) is convex. This conclusion applies, therefore, to any y∈Ω r . We next claim the existence of a constant C with the following property. Set ρ(y):=C −1min{1,dist (u(y),u(∂Ω))}. Then
Recall that u is an embedding and hence dist (u(y),u(∂Ω))>0 for every y∈Ω. By (100), (101) gives for any y∈Ω there is a neighborhood where u(Ω) is convex. This would complete the proof.
Step 3. Proof of (101)
First of all, since u is an embedding and \(\|u\|_{C^{1,\alpha}}\) is finite, there is a constant c 0 such that, for any point x, \(B_{c_{0}} (x)\cap u(\varOmega)\) is the graph of a C 1,α function with \(\|\cdot\|_{C^{1,\alpha}}\) norm smaller than 1. In order to prove (101) we assume, without loss of generality, that y=0 and that the tangent plane to u(Ω) at y is {x 3=0}. Denote by π the projection on {x 3=0}. By [27] there is a constant λ>0 (depending only on \(\|g\|_{C^{2,\beta}}\), \(\|\kappa\|_{C^{0}}\) and \(\|\kappa^{-1}\|_{C^{0}}\)) with the following property.
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(Est)
Let U be an open convex set such that U∩u(∂Ω)=∅, diam (U)≤c 0 and U∩u(Ω) is locally convex. Then U∩u(Ω) is the graph of a function f:π(u(Ω)∩U)→ℝ with \(\|f\|_{C^{2, 1/2}}\leq\lambda^{-1}\) and D 2 f≥λId.
We now look for sets U as in (Est) with the additional property that U=V×]−a,a[ and f| ∂V =a (see Fig. 2). Let U m be the maximal set of this form for which the assumptions of (Est) hold. We claim that, either ∂U m ∩u(∂Ω)≠∅, or diam (U m )=c 0. By (Est), this claim easily implies (101). To prove the claim, assume by contradiction that it is wrong and let U m =W m ×]−a m ,a m [ be the maximal set. Let γ=∂U m ∩u(Ω). By the choice of c 0, γ is necessarily the curve ∂W m ×{a}. On the other hand, by the estimates of (Est), it follows that every tangent plane to u(Ω) at a point of γ is transversal to {x 3=0}. So, for a sufficiently small ε>0, the intersection {x 3=a m +ε}∩u(Ω) contains a curve γ′ bounding a connected region D⊂u(Ω) which contains u(Ω)∩U m . By Theorem 8 of page 650 in [26], D is a convex set. This easily shows that U m was not maximal. □
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Conti, S., De Lellis, C., Székelyhidi, L. (2012). h-Principle and Rigidity for C 1,α Isometric Embeddings. In: Holden, H., Karlsen, K. (eds) Nonlinear Partial Differential Equations. Abel Symposia, vol 7. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-25361-4_5
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