Gunther's proof of Nash's isometric embedding theorem

On the mean curvature of Nash isometric embeddings

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, imposing restrictions on the mean curvature vector of the embedding.

紧致流形的全局 Nash-Kuiper 定理

Global Nash-Kuiper theorem for compact manifolds

本文推广 Nash-Kuiper 定理至紧致流形，证明凸曲面等距嵌入三维空间时非刚性在 Hölder 指数小于 1/5 时成立，扩展了二维圆盘及高维结果。

关于Whitney强嵌入定理的等距版本

On the isometric version of Whitney's strong embedding theorem

本文在Nash-Kuiper h-原理框架下，证明了任意n维光滑紧致流形可等距嵌入2n维欧氏空间，且嵌入具有特定Hölder正则性，方法为凸积分与粘合技术。

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

In this paper we study the embedding of Riemannian manifolds in low codimension. The well-known result of Nash and Kuiper 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 $α>2/3$. On the other hand he announced in 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 …

A Nash-Kuiper theorem for $C^{1,\frac{1}{5}-δ}$ immersions of surfaces in $3$ dimensions

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 $C^{1,α}$ isometric immersions for any $α< \frac{1}{5}$. This statement improves previous results by Yu.F. Borisov and of a joint paper of the first and third author with S. Conti.

A note on closed isometric embeddings

This note is about a little extension of Nash's embedding theorem in the case of complete manifolds.

A gap theorem on complete shrinking gradient Ricci solitons

In this short note, using Günther's volume comparison theorem and Yokota's gap theorem on complete shrinking gradient Ricci solitons, we prove that for any complete shrinking gradient Ricci soliton $(M^{n},g,f)$ with sectional curvature $K(g)<A$ and ${\rm Vol}_{f}(M)\geq v$ for some uniform constant $A,v$, there exists a small uniform constant $ε_{n,A,v}>0$ depends only on $n, A$ and $v$, if the scalar curvature $R\leq ε_{n,A,v}$, then $(M,g,f)$ is isometric to the Gaussian soliton $(\mathbb{R}^{n}, g_{E}, \frac{|x|^{2}}{4})$.

关于黎曼等距嵌入的几何学

On the geometry of Riemannian isometric embeddings

针对与$\RR^n$微分同胚且允许Bieberbach群$Γ$等距作用的黎曼流形$M$，研究其到欧几里得空间有界子集及$Γ$-等变等距嵌入问题，通过新用已知技巧得到嵌入维数结果，并证明n维光滑流形到$\RR^{2N}$有界子集的嵌入性。

Isometric Embeddings in Imaging and Vision: Facts and Fiction

We explore the practicability of Nash's Embedding Theorem in vision and imaging sciences. In particular, we investigate the relevance of a result of Burago and Zalgaller regarding the existence of isometric embeddings of polyhedral surfaces in $\mathbb{R}^3$ and we show that their proof does not extended directly to higher dimensions.

等距嵌入共形紧致流形到双曲空间

Isometric Embeddings of Conformally Compact Manifolds into Hyperbolic Spaces

本文证明了共形紧致流形可以等距嵌入到高维双曲空间中，解决了在无穷远处具有负曲率下界流形的嵌入问题。

Approximating continuous maps by isometries

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 result is now known to be a consequence of Gromov's more general $h$-principle. There have been some recent extensions of the Nash-Kuiper Theorem to Euclidean polyhedra, which in some sense provide a very specialized discretization of the $h$-principle. In this paper we will discuss these recent results and provide generalizations to the setting of isometric emb…

On the complexity of isometric immersions of hyperbolic spaces in any codimension

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 Lipschitz map $F:M^m\to\mathbb R^n$, where $M^m$ is a Hadamard manifold whose curvature lies between negative constants. The main result of this paper is that $F$ must perform a substantial compression: For every $r>0$ and integer $k\geq 2$ there exist $k$ geodesic balls of radius $r$ in $M^m$ that are arbitrarily far from each other, but whose images under $F$ a…

Lipschitz and path isometric embeddings of metric spaces

We prove that each sub-Riemannian manifold can be embedded in some Euclidean space preserving the length of all the curves in the manifold. The result is an extension of Nash $C^1$ Embedding Theorem. For more general metric spaces the same result is false, e.g., for Finsler non-Riemannian manifolds. However, we also show that any metric space of finite packing dimension can be embedded in some Euclidean space via a Lipschitz map.

Rigidity for local holomorphic isometric embeddings from ${\BB}^n$ into ${\BB}^{N_1}\times... \times{\BB}^{N_m}$ up to conformal factors

In this article, we study local holomorphic isometric embeddings from ${\BB}^n$ into ${\BB}^{N_1}\times... \times{\BB}^{N_m}$ with respect to the normalized Bergman metrics up to conformal factors. Assume that each conformal factor is smooth Nash algebraic. Then each component of the map is a multi-valued holomorphic map between complex Euclidean spaces by the algebraic extension theorem derived along the lines of Mok and Mok-Ng. Applying holomorphic continuation and analyzing real analytic subvarieties carefully, we show that each component is either a constant map or a proper holomorphic map…

Einstein 可解流形作为对称空间的子流形

Einstein solvmanifolds as submanifolds of symmetric spaces

本文结合 Ado 定理与 Nash 嵌入定理，为包含 Einstein 和 Ricci 孤立子可解流形的一类可解李群构造同时为等距嵌入的忠实表示，进而证明所有 Einstein 可解流形均可实现为对称空间的子流形。

曲面上光滑度量等距嵌入的共形不变量

Conformal invariants of isometric embeddings of the smooth metrics on a surface

本文通过 Nash 等距嵌入定义曲面度量空间上的 Willmore 泛函，证明其在共形类中的下确界为共形不变量，并利用 Simons 间隙定理和 Lawson 极小曲面分别对亏格为零、正亏格及不可定向情形给出最优下界与取等条件的共形刻画。

浸入的外在特征刻画

Extrinsic Characterisations of Immersions

本文回顾空间形变与曲率不变量概念的历史，围绕 Nash 等距浸入定理，探讨如何利用内蕴量与外在量之间的关系刻画浸入，并反思主要以外在量刻画子流形的可能性。