Related papers: h-Principle and Rigidity for $C^{1,\alpha}$ Isomet…

This paper is devoted to investigating the isometric immersion problem of Riemannian manifolds in a high codimension. It has recently been demonstrated that any short immersion from an $n$-dimensional smooth compact manifold into…

The Nash-Kuiper Theorem states that the collection of $C^1$-isometric embeddings from a Riemannian manifold $M^n$ into $\mathbb{E}^N$ is $C^0$-dense within the collection of all smooth 1-Lipschitz embeddings provided that $n < N$. This…

We prove a version of Whitney's strong embedding theorem for isometric embeddings within the general setting of the Nash-Kuiper h-principle. More precisely, we show that any $n$-dimensional smooth compact manifold admits infinitely many…

We prove that, given a $C^2$ Riemannian metric $g$ on the $2$-dimensional disk $D_2$, any short $C^1$ immersion of $(D_2,g)$ into $\mathbb R^3$ can be uniformly approximated with $C^{1,\alpha}$ isometric immersions for any $\alpha <…

We obtain global extensions of the celebrated Nash-Kuiper theorem for $C^{1,\theta}$ isometric immersions of compact manifolds with optimal H\"older exponent. In particular for the Weyl problem of isometrically embedding a convex compact…

We study the problem of constructing $C^{1,\theta}$ isometric immersions of Riemannian metrics on $n$-dimensional domains into $\mathbb{R}^{n+1}$. While the classical Nash--Kuiper theorem established the flexibility of $C^1$ isometries,…

Given any short immersion from an $n$-dimensional bounded and simply connected domain into $\mathbb{R}^{n+1}$ and any H\"older exponent $\alpha<(1+n^2-n)^{-1}$, we construct a $C^{1, \alpha}$ isometric immersion arbitrarily close in the…

In this paper we use the convex integration technique enhanced by an extra iteration originally due to K\"all\'en and revisited by Kr\"oner to provide a local $h$-principle for isometric embeddings in the class $C^{1,1-\epsilon}$ for…

We consider isometric immersions of complete connected Riemannian manifolds into space forms of nonzero constant curvature. We prove that if such an immersion is compact and has semi-definite second fundamental form, then it is an embedding…

In this paper, we obtain the following generalisation of isometric $C^1$-immersion theorem of Nash and Kuiper. Let $M$ be a smooth manifold of dimension $m$ and $H$ a rank $k$ subbundle of the tangent bundle $TM$ with a Riemannian metric…

We study isometric embeddings of $C^2$ Riemannian manifolds in the Euclidean space and we establish that the H\"older space $C^{1,\frac{1}{2}}$ is critical in a suitable sense: in particular we prove that for $\alpha > \frac{1}{2}$ the…

Although the Nash theorem solves the isometric embedding problem, matters are inherently more involved if one is further seeking an embedding that is well-behaved from the standpoint of submanifold geometry. More generally, consider a…

The celebrated Nash Embedding Theorem asserts that every closed Riemannian manifold can be isometrically embedded into a sufficiently high-dimensional Euclidean space. In this paper, we prove an analogous result in the conformally compact…

In this paper we prove weak L^{1,p} (and thus C^{\alpha}) compactness for the class of uniformly mean-convex Riemannian n-manifolds with boundary satisfying bounds on curvature quantities, diameter, and (n-1)-volume of the boundary. We…

J. Nash proved that the geometry of any Riemannian manifold M imposes no restrictions to be embedded isometrically into a (fixed) ball B_{\mathbb{R}^{N}}(1) of the Euclidean space R^N. However, the geometry of M appears, to some extent,…

We prove a priori bounds for the trace of the second fundamental form of a $C^4$ isometric embedding into $R^{n+1}$ of a metric $g$ of non-negative sectional curvature on $S^n$, in terms of the scalar curvature, and the diameter of $g$.…

We consider a priori estimates of Weyl's embedding problem of $(\mathbb{S}^2, g)$ in general $3$-dimensional Riemannian manifold $(N^3, \bar g)$. We establish interior $C^2$ estimate under natural geometric assumption. Together with a…

This paper introduces a probabilistic formulation for the isometric embedding of a Riemannian manifold $(M^n,g)$ into Euclidean space $\mathbb{R}^q$. Given $\alpha \in ]\tfrac{1}{2},1]$, we show that a $C^{1,\alpha}$ embedding $u: M \to…

Let $\Sigma$ be a codimension one submanifold of an $n$-dimensional Riemannian manifold $M$, $n\geqslant 2$. We give a necessary condition for an isometric immersion of $\Sigma$ into $\mathbb R^q$ equipped with the standard Euclidean…

For any compact Riemannian manifold $(V,g)$ and any Lorentzian manifold $(W,h)$, we prove that any spacelike embedding $f: V \rightarrow W$ that is long ($g\leq f^{*}h$) can be $C^{0}$-approximated by a $C^{1}$ isometric embedding $F: (V,g)…