h-Principle and Rigidity for C ^1,α Isometric Embeddings

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.

Appendix

Proof of Corollary 4

First of all, since the theorem is local, without loss of generality we can assume that:

• 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;

• 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

$$ \varOmega_r\quad \mbox{is dense in}\ \varOmega.$$

(100)

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

$$ u(\varOmega)\cap B_{\rho(y)} (y)\quad \mbox{is convex for all}\ y\in\varOmega_r.$$

(101)

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.

• (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. □

Fig. 2

The convex sets of type V×]−a,a[ among which we choose the maximal one U _m