Gunther’s proof of Nash’s isometric embedding theorem

Deane Yang Department of Mathematics
Polytechnic University
Six Metrotech Center
Brooklyn NY 11201
yang@math.poly.edu

1. Preface

Around 1987 a German mathematician named Matthias Gunther found a new way of obtaining the existence of isometric embeddings of a Riemannian manifold. His proof appeared in [1, 2]. His approach avoids the so-called Nash-Moser iteration scheme and, therefore, the need to prove smooth tame or Moser-type estimates for the inverse of the linearized operator. This simplifies the proof of Nash’s isometric embedding theorem [3]considerably.

This is an informal expository note describing his proof. It was originally written, because when I first learned Gunther’s proof, it had not appeared either in preprint or published form, and I felt that everyone should know about it. Moreover, since he is at Leipzig, which at the time was part of East Germany, very few mathematicians in the U.S. knew about him or his proof.

Since many still seem to be unaware of Gunther’s proof, even after he gave a talk at the International Congress of Mathematicians at Kyoto in 1990 and published his proof in the proceedings [2], I have updated this note and continue to distribute it. I do, however, encourage you to seek out Gunther’s own presentations of his proof.

2. Introduction

Let M𝑀M be a smooth n𝑛n dimensional manifold. Given an embedding u:M𝐑N:𝑢𝑀superscript𝐑𝑁u:M\to{\mathbf{R}}^{N} , the standard inner product on 𝐑Nsuperscript𝐑𝑁{\mathbf{R}}^{N} induces a Riemannian metric on M𝑀M . We shall denote this metric by dudu𝑑𝑢𝑑𝑢du\cdot du . In particular, given a Riemannian metric g𝑔g on M𝑀M , we say that the embedding u𝑢u is isometric , if

dudu=g𝑑𝑢𝑑𝑢𝑔du\cdot du=g

Let N12n(n+1)𝑁12𝑛𝑛1N\geq\frac{1}{2}n(n+1) . A C2superscript𝐶2C^{2} immersion u:M𝐑N:𝑢𝑀superscript𝐑𝑁u:M\rightarrow{\mathbf{R}}^{N} is free if for every xM𝑥𝑀x\in M ,

iu(x),iju(x), 1i,jn,formulae-sequencesubscript𝑖𝑢𝑥subscript𝑖subscript𝑗𝑢𝑥 1𝑖𝑗𝑛\partial_{i}u(x),\partial_{i}\partial_{j}u(x),\ 1\leq i,j\leq n,

span a min(N,n+12n(n+1))𝑁𝑛12𝑛𝑛1\min(N,n+\frac{1}{2}n(n+1)) dimensional linear subspace of 𝐑Nsuperscript𝐑𝑁{\mathbf{R}}^{N} .

The only place where Gunther’s proof differs from earlier proofs of existence lies in showing that given a smooth, free embedding u0:M𝐑N:subscript𝑢0𝑀superscript𝐑𝑁u_{0}:M\rightarrow{\mathbf{R}}^{N} and a smooth Riemannian metric g𝑔g sufficiently close (in a sense to be made precise later) to du0du0𝑑subscript𝑢0𝑑subscript𝑢0du_{0}\cdot du_{0} , there exists a smooth embedding u:M𝐑N:𝑢𝑀superscript𝐑𝑁u:M\rightarrow{\mathbf{R}}^{N} close to u0subscript𝑢0u_{0} such that

(1) dudu=g.𝑑𝑢𝑑𝑢𝑔du\cdot du=g.

Although it is not necessary, we shall simplify the exposition by assuming the existence of “global” coordinates on M𝑀M . If M𝑀M is compact, this is obtained by embedding M𝑀M smoothly into a torus of larger dimension and extending smoothly the embedding u0subscript𝑢0u_{0} and the metric g𝑔g to the torus so that g𝑔g remains close to du0du0𝑑subscript𝑢0𝑑subscript𝑢0du_{0}\cdot du_{0} . Otherwise, if all we are trying to prove is a local existence theorem, we can assume that M𝑀M is diffeomorphic to an open set in 𝐑nsuperscript𝐑𝑛{\mathbf{R}}^{n} . In the discussion below, x1,,xnsuperscript𝑥1superscript𝑥𝑛x^{1},\ldots,x^{n} are assumed to be global coordinates on M𝑀M . (If M𝑀M does not have global coordinates, then all the calculations below should be done using a fixed smooth background metric g^^𝑔\hat{g} , instead of the flat metric implied by the global coordinates, and its LeviCivita connection. Extra terms involving the curvature of g^^𝑔\hat{g} and the covariant derivative of curvature appear, but they are all of lower order and do not affect the proof at all.)

Let v=u-u0𝑣𝑢subscript𝑢0v=u-u_{0} and h=g-du0du0𝑔𝑑subscript𝑢0𝑑subscript𝑢0h=g-du_{0}\cdot du_{0} . For convenience we shall denote

ui=u0xi,uij=2u0xixj.formulae-sequencesubscript𝑢𝑖subscript𝑢0superscript𝑥𝑖subscript𝑢𝑖𝑗superscript2subscript𝑢0superscript𝑥𝑖superscript𝑥𝑗u_{i}=\frac{\partial u_{0}}{\partial x^{i}},\ u_{ij}=\frac{\partial^{2}u_{0}}{% \partial x^{i}\partial x^{j}}.

Then ( 1) is equivalent to

(2) uijv+ujiv+ivjv=hij, 1i,jn.formulae-sequencesubscript𝑢𝑖subscript𝑗𝑣subscript𝑢𝑗subscript𝑖𝑣subscript𝑖𝑣subscript𝑗𝑣subscript𝑖𝑗formulae-sequence 1𝑖𝑗𝑛u_{i}\cdot\partial_{j}v+u_{j}\cdot\partial_{i}v+\partial_{i}v\cdot\partial_{j}% v=h_{ij},\ 1\leq i,j\leq n.

Applying the standard “integration by parts” trick, ( 2) can be rewritten as

(3) i(ujv)+j(uiv)-2uijv+ivjv=hij.subscript𝑖subscript𝑢𝑗𝑣subscript𝑗subscript𝑢𝑖𝑣2subscript𝑢𝑖𝑗𝑣subscript𝑖𝑣subscript𝑗𝑣subscript𝑖𝑗\partial_{i}(u_{j}\cdot v)+\partial_{j}(u_{i}\cdot v)-2u_{ij}\cdot v+\partial_% {i}v\cdot\partial_{j}v=h_{ij}.

This can be written abstractly in the following form:

L0v+Q(v,v)=h,subscript𝐿0𝑣𝑄𝑣𝑣L_{0}v+Q(v,v)=h,

where L0subscript𝐿0L_{0} is a linear operator and Q𝑄Q is bilinear. Nash’s trick, when N12n(n+1)+n𝑁12𝑛𝑛1𝑛N\geq\frac{1}{2}n(n+1)+n , was to observe that the linear differential operator L0subscript𝐿0L_{0} could be inverted by a zeroth order differential operator M0subscript𝑀0M_{0} . More recently, M. Gromov and Bryant-Griffiths-Yang have found cases where N<12n(n+1)+n𝑁12𝑛𝑛1𝑛N<\frac{1}{2}n(n+1)+n and L0subscript𝐿0L_{0} admits a right inverse M0subscript𝑀0M_{0} which “loses” a fixed number of derivatives. In all cases there is a loss in regularity, so that standard implicit function theorems or contraction map arguments do not seem to apply. Instead, the socalled NashMoser iteration scheme must be used.

Gunther’s ingenious trick can be decribed as follows: He finds new nonlocal bilinear operators Q1subscript𝑄1Q_{1} and Q2subscript𝑄2Q_{2} such that

(4) Q=L0Q1+Q2,𝑄subscript𝐿0subscript𝑄1subscript𝑄2Q=L_{0}Q_{1}+Q_{2},

where Q1subscript𝑄1Q_{1} is zeroth order and Q2subscript𝑄2Q_{2} is of any given negative order, i.e. it is a bilinear smoothing operator. Actually, in the specific situation here, the operator Q2subscript𝑄2Q_{2} will be identically zero. Then the contraction mapping argument can be applied to the equation

v=M0(h-Q1(v,v))-Q2(v,v).𝑣subscript𝑀0subscript𝑄1𝑣𝑣subscript𝑄2𝑣𝑣v=M_{0}(h-Q_{1}(v,v))-Q_{2}(v,v).

The splitting is obtained as follows: Let

Δ=i=1ni2.Δsuperscriptsubscript𝑖1𝑛superscriptsubscript𝑖2\Delta=\sum_{i=1}^{n}\partial_{i}^{2}.

Then Δ-1Δ1\Delta-1 is an invertible elliptic operator on M𝑀M . Apply it to both sides of ( 3). Rearranging the terms and then applying (Δ-1)-1superscriptΔ11(\Delta-1)^{-1} to the resulting equation, we obtain;

i(ujv+Qj(v,v))+j(uiv+Qi(v,v))-2uijv+Qij(v,v)=hij,subscript𝑖subscript𝑢𝑗𝑣subscript𝑄𝑗𝑣𝑣subscript𝑗subscript𝑢𝑖𝑣subscript𝑄𝑖𝑣𝑣2subscript𝑢𝑖𝑗𝑣subscript𝑄𝑖𝑗𝑣𝑣subscript𝑖𝑗\partial_{i}(u_{j}\cdot v+Q_{j}(v,v))+\partial_{j}(u_{i}\cdot v+Q_{i}(v,v))-2u% _{ij}\cdot v+Q_{ij}(v,v)=h_{ij},

where

Qi(v,v)subscript𝑄𝑖𝑣𝑣\displaystyle Q_{i}(v,v) =\displaystyle= (Δ-1)-1(Δ-1)vivsuperscriptΔ11Δ1𝑣subscript𝑖𝑣\displaystyle(\Delta-1)^{-1}(\Delta-1)v\cdot\partial_{i}v
Qij(v,v)subscript𝑄𝑖𝑗𝑣𝑣\displaystyle Q_{ij}(v,v) =\displaystyle= (Δ-1)-1(2k=1nikvjkv+ivjv-2(Δ-1)vijv).superscriptΔ112superscriptsubscript𝑘1𝑛subscript𝑖subscript𝑘𝑣subscript𝑗subscript𝑘𝑣subscript𝑖𝑣subscript𝑗𝑣2Δ1𝑣subscript𝑖subscript𝑗𝑣\displaystyle(\Delta-1)^{-1}(2\sum_{k=1}^{n}\partial_{i}\partial_{k}v\cdot% \partial_{j}\partial_{k}v+\partial_{i}v\cdot\partial_{j}v-2(\Delta-1)v\cdot% \partial_{i}\partial_{j}v).

Since u0subscript𝑢0u_{0} is free, there exists a unique 𝐑Nsuperscript𝐑𝑁{\mathbf{R}}^{N} -valued bilinear operator Q0subscript𝑄0Q_{0} such that uiQ0=Qisubscript𝑢𝑖subscript𝑄0subscript𝑄𝑖u_{i}\cdot Q_{0}=Q_{i} and uijQ0=Qijsubscript𝑢𝑖𝑗subscript𝑄0subscript𝑄𝑖𝑗u_{ij}\cdot Q_{0}=Q_{ij} . The isometric embedding equation now becomes

L0(v-Q0(v,v))=h,subscript𝐿0𝑣subscript𝑄0𝑣𝑣L_{0}(v-Q_{0}(v,v))=h,

where

(L0v)ij=i(ujv)+j(uiv)-2uijv.subscriptsubscript𝐿0𝑣𝑖𝑗subscript𝑖subscript𝑢𝑗𝑣subscript𝑗subscript𝑢𝑖𝑣2subscript𝑢𝑖𝑗𝑣(L_{0}v)_{ij}=\partial_{i}(u_{j}\cdot v)+\partial_{j}(u_{i}\cdot v)-2u_{ij}% \cdot v.

Given h=hijdxidxjsubscript𝑖𝑗𝑑superscript𝑥𝑖𝑑superscript𝑥𝑗h=h_{ij}dx^{i}dx^{j} , define M0h=vsubscript𝑀0𝑣M_{0}h=v , where for every xM𝑥𝑀x\in M , v(x)𝑣𝑥v(x) is the unique vector lying in the span of ui(x),uij(x)subscript𝑢𝑖𝑥subscript𝑢𝑖𝑗𝑥u_{i}(x),u_{ij}(x) , 1i,jnformulae-sequence1𝑖𝑗𝑛1\leq i,j\leq n , satisfying the following equations

uivsubscript𝑢𝑖𝑣\displaystyle u_{i}\cdot v =\displaystyle= 00\displaystyle 0
-2uijv2subscript𝑢𝑖𝑗𝑣\displaystyle-2u_{ij}\cdot v =\displaystyle= hijsubscript𝑖𝑗\displaystyle h_{ij}

Clearly, M0subscript𝑀0M_{0} is a right inverse for L0subscript𝐿0L_{0} . Therefore, to solve ( 3), it suffices to solve the following:

v=M0h+Q0(v,v).𝑣subscript𝑀0subscript𝑄0𝑣𝑣v=M_{0}h+Q_{0}(v,v).

Define Φ(v)=M0h+Q0(v,v)Φ𝑣subscript𝑀0subscript𝑄0𝑣𝑣\Phi(v)=M_{0}h+Q_{0}(v,v) . If v2,αsubscriptnorm𝑣2𝛼\|v\|_{2,\alpha} , 0<α<10𝛼10<\alpha<1 , is sufficiently small, then ΦΦ\Phi is a contraction mapping on a neighborhood of 0C2,α(M,𝐑N)0superscript𝐶2𝛼𝑀superscript𝐑𝑁0\in C^{2,\alpha}(M,{\mathbf{R}}^{N}) . Moreover, the linear operator I-Q0(v,)𝐼subscript𝑄0𝑣I-Q_{0}(v,\cdot) is an elliptic zeroth order operator and therefore if hh is Ck,αsuperscript𝐶𝑘𝛼C^{k,\alpha} , k2𝑘2k\geq 2 , then so is v𝑣v . In particular, if hh is smooth, so is v𝑣v .

We have therefore obtained the following:

Theorem 1 (Nash, Gunther [3, 1, 2]).

Let M𝑀M be an n𝑛n -dimensional torus and u0:MRNnormal-:subscript𝑢0normal-→𝑀superscriptnormal-R𝑁u_{0}:M\rightarrow{\mathbf{R}}^{N} , N12n(n+1)+n𝑁12𝑛𝑛1𝑛N\geq\frac{1}{2}n(n+1)+n , a smooth, free immersion. Then given 0<α<10𝛼10<\alpha<1 , there exists ϵ>0italic-ϵ0\epsilon>0 (depending on u0subscript𝑢0u_{0} and α𝛼\alpha ) such that given any C2,αsuperscript𝐶2𝛼C^{2,\alpha} Riemannian metric g𝑔g , g-du0du02,α<ϵsubscriptnorm𝑔normal-⋅𝑑subscript𝑢0𝑑subscript𝑢02𝛼italic-ϵ\|g-du_{0}\cdot du_{0}\|_{2,\alpha}<\epsilon , there exists a C2,αsuperscript𝐶2𝛼C^{2,\alpha} immersion u𝑢u close to u0subscript𝑢0u_{0} such that dudu=gnormal-⋅𝑑𝑢𝑑𝑢𝑔du\cdot du=g . Moreover, if g𝑔g is Ck,αsuperscript𝐶𝑘𝛼C^{k,\alpha} , 2k2𝑘2\leq k\leq\infty , the immersion u𝑢u is Ck,αsuperscript𝐶𝑘𝛼C^{k,\alpha} .

References

  • 1 Matthias Gunther, On the perturbation problem associated to isometric embeddings of Riemannian manifolds, Annals of Global Analysis and Geometry 7 (1989), 69–77.
  • 2 by same author, Isometric embeddings of Riemannian manifolds, Proceedings of the International Congress of Mathematicians (Kyoto, 1990), Mathematical Society of Japan, 1991, pp. 1137–1143.
  • 3 John Nash, The imbedding problem for Riemannian manifolds, Annals of Mathematics 63 (1956), 20–63.