1 Introduction↩︎

Let \((M,g)\) be an n-dimensional compact Riemannian manifold, the following classical problem, called the isometric embedding problem is studied in differential geometry. Does there exist an embedding \(u:M\longrightarrow \mathbb{R}^N\) for some \(N\) such that \[u^*g_{\mathrm{can}}=g,\] where \(g_{\mathrm{can}}\) is the Euclidean metric in \(\mathbb{R}^N\)? In 1956, J. Nash famously proved in [3] that there exists a \(C^r\)-class isometric embedding for \(g\in C^r\), with \(r\geq 3\) or \(r=\infty\). Furthermore, for any compact \(n\)-dimensional Riemannian manifold, the optimal value of \(N\) he found was \(N=\frac{3}{2}n(n+1)+4n\).

In [3], Nash developed an iteration nowadays known as the Nash-Moser theorem to address the problem of losing differentiability when taking the usual Newton iteration. Decades later, M.Günther (1989, [4]) significantly simplified Nash’s proof by applying an elliptic operator to develop a different iteration, which avoids the loss of differentiability (see Section 3.2). This allows one to simply use the usual Banach fixed point theorem to conclude the proof. His approach is also exposed in the proceedings [5] of ICM 1990 Kyoto.

Nash and Günther’s construction of the isometric embedding is highly flexible. By employing this method, any \(C^{r\geq 3}\) embedding \(u: M \longrightarrow \mathbb{R}^N\) such that the induced metric is less than or equal to \(g\) can serve as a start to produce an isometric embedding. This great flexibility, on the other hand, often results in the isometric embeddings being noncanonical.

Contrastingly, in 1994, Bérard, Besson, and Gallot [1] constructed an ‘asymptotically isometric’ embedding using the heat kernel of the manifold. This embedding, referred to as the (normalized) heat kernel embedding throughout, maps a compact Riemannian manifold \(M\) into \(\ell^2\), the space of square summable series, and is constructed as follows: \[\Psi_t: x\mapsto \sqrt{2}(4\pi )^{\frac{n}{4}} t^{\frac{n+2}{4}}\cdot \big\{ e^{-\lambda_j t/2}\phi_j(x) \big\}_{j\geq 1}, \text{ for } t>0,\] where \(\lambda_j\) is the \(j\)th eigenvalue of the Laplacian \(\Delta=\mathrm{tr}_g \nabla^2\) of \((M,g)\), here \(\nabla\) is the Levi-Civita connection, and \(\{\phi_j\}_{j\geq 0}\) is an \(L^2\) orthonormal eigenbasis of \(\Delta\). It is worth noting that the embedding \(\Psi_t\) is canonical due to the fact that it is constructed by the heat kernel and therefore the spectral geometry of \((M,g)\) uniquely determines it. A more precise formula in [1] that justifies the above statement is the following, indicating that \(\Psi_t\) tends to an isometry in the following sense: \[\Psi^*_t g_{\mathrm{can}}= g+\frac{t}{3} (\frac{1}{2}\mathrm{Scal}_g\cdot g-\mathrm{Ric}_g)+O(t^2),\] where the \(g_\mathrm{can}\) is the standard Euclidean metric in \(\ell^2\), \(\mathrm{Scal}_g\) is the scalar curvature of \((M,g)\), \(\mathrm{Ric}_g\) is the Ricci curvature of \((M,g)\), and the convergence is in the \(C^r\) sense for any \(r>0\).

In light of the facts that Nash and Günther’s isometric methods ([3],[4]) being flexible but far from being canonical, and Bérard, Besson, and Gallot’s heat kernel embedding ([1]) being canonical but not yet exactly isometric, Wang and Zhu (2015, [2]) embarked on a study aimed at finding a canonical isometric embedding of a compact Riemannian manifold into \(\mathbb{R}^q\) for \(q\gg 1\) by using the heat kernel of this manifold. Their approach begins by first modifying the heat kernel embedding \(\Psi_t\) in [1] to a better approximation with an error term of \(O(t^l)\) for any \(l\geq 2\), and continues by perturbing such an ‘almost isometric’ embedding to an isometric one. Namely for any \(l\geq 1\), by using the \(\Psi_t\), they find a canonical family of ‘almost’ isometric embeddings \(\tilde{\Psi}_t: M \longrightarrow \ell^2\) such that \[\tilde{\Psi}_t^* g_{\mathrm{can}}=g+O(t^l)\] in the \(C^r\) sense for \(r>0\). Subsequently, they find a unique \(C^{r,\alpha}\) isometric embedding \(I_t:M\longrightarrow\mathbb{R}^{q(t)}\) such that \[\norm{ I_t-\tilde{\Psi}_t}_{C^{r,\alpha}(M)}= O(t^{l+\frac{1}{2}-\frac{r+\alpha}{2}}),\] where \(q(t)\geq t^{-\frac{n}{2}-\rho}\), \(\rho>0\), \(r+\alpha<l+\frac{1}{2}\), \(0<\alpha<1\), \(r\geq 2\), \(t\in (0,t_0)\) for some \(t_0>0\) depending on \(r,\alpha, l,\) and \(g\). Additionally, we note Bérard, Besson, and Gallot’s heat kernel embeddings can be used in many other ways. For further references, historical contexts, and other uses of this heat kernel embedding, see [6] and [7].

From the view of Kähler geometry and complex geometry, one may seek more embeddings of this canonical type. In analogy to the Kodaira embedding (see, e.g. [8]) in Kähler geometry that preserves the holomorphic structure, in the current paper we find a family of canonical embeddings of compact Riemannian manifolds that preserve the conformal structure. Indeed, an isometric one is already a conformal one, but starting with the heat kernel ‘almost’ isometric embedding and looking into Günther’s method, we have shown that by requesting the result map to be conformal and keeping each step done canonically, one can find a family of canonical conformal embeddings of \((M,g)\) in Euclidean space, with the isometric embedding constructed in [2] as one special case among them. Note that the present paper partially fulfills the proposal of Wang and Zhu, specifically the part concerning the construction of canonical conformal embeddings, see [2].

Throughout, conformal embeddings are referred to embeddings that are conformal maps, see Definition 8. In the following, we present the main theorems, fixing the constant \(\rho>0\), and \(0<\alpha<1\), and using Einstein summation notation throughout.

Note here that \(\Psi_{t,g(t),\eta_i}\) is an almost conformal embedding since the error term \(O(t^l)\) is small when \(t\rightarrow0_+\). And it is a canonical embedding in the sense that it is determined by the geometry of \((M,g)\).

As we will see in Proposition 10, for given \(\eta_i\in C^\infty (M,\mathbb{R})\) and each \(1\leq i\leq l-1\), we will uniquely determine an \(h_i\in \Gamma(\mathrm{Sym}^{\otimes 2}(T^*M))\) depending only on \((M,g)\) and \(\eta_i\), such that 5 holds, and especially \(\frac{\mathrm{tr}_g h_i}{n}=\eta_i\). Then for the metric \(g(t):=g+\sum\limits_{i=1}^{l-1}h_it^i\), the map \(\Psi_{t,g(t),\eta_i}\) is defined as the heat kernel embedding (in the sense of [1]) of \((M,g(t))\).

Given this \(\Psi_{t,g(t),\eta_i}\), as quoted verbatim from [2], we have the following definition:

The following is the second part of our main theorem.

The main techniques in this article can be described as a process of ‘recovering the trace’, and can be outlined as follows. Notice that a map \(u:M \longrightarrow \mathbb{R}^N\) being free (see Definition 17) is a strong condition, which will ensure the existence and uniqueness of the solution to the equation (see Lemma 19): \[\label{0465} P(u)\cdot v=\begin{bmatrix} \xi& f \end{bmatrix}^T,\tag{1}\] where the definition of \(P(u)\) is in Definition 17. However, to address our conformal question, we need to look into the equation \[\label{0466} P_c(u)\cdot v=\begin{bmatrix} \xi & f-\frac{\mathrm{tr}_g f}{n}g \end{bmatrix}^T,\tag{2}\] where \(P_c(u)\) is obtained from \(P(u)\) by subtracting its own trace of the second derivative part (see Definition 17). The challenge lies in the fact that \(P_c(u)\) is not of full rank. To overcome this difficulty, we point out that the solution of 1 is one special solution of 2 . And precisely describing the kernel of \(P_c(u)\), which corresponds to the trace that to be recovered, allows us to obtain all the solutions in the following manner: ‘solutions of eq. ¿eq:0466?’= ‘one special solution’ +‘kernel of \(P_c(u)\)’(see Remark 21). In this context the ‘one special solution’ corresponds to the isometric embeddings attained by Wang and Zhu in [2], hence we will closely follow their construction. This coincides with the general perspective that isometric embeddings are special cases of conformal ones.

2 Heat kernel embeddings and modifications to almost conformal maps↩︎

Let \((M,g)\) be an n-dimensional compact Riemannian manifold with smooth metric \(g\). Denote the eigenvalues of the Laplacian of \((M,g)\) as \(0=\lambda_0<\lambda_1<\lambda_2<\cdots\), and let \(\{\phi_j\}_{j\geq 0}\subset C^\infty (M)\) be a corresponding \(L^2\)-orthonormal basis of the real eigenfunctions. In other words, this means \(\Delta_g \phi_j=-\lambda_j \phi_j\), and \(\int_M \phi_i\phi_j d\mathrm{vol}_g=\delta_i^j\), for \(i,j\geq 0\). The heat kernel of \((M,g)\) is: \[H(x,y,t)=\sum\limits_{j=1}^\infty e^{-\lambda_j t} \phi_j(x)\phi_j(y),\] where \(x,y\in M\), and \(t>0\). Recall the definition in [1] regarding almost isometric heat kernel embeddings into \(\ell^2\):

One main theorem in [1] can be phrased as the following:

Similar to Proposition 5 of [2], we can perturb the \(t\)-dependent map to a family of \(t\)-dependent maps, each of which is in the form of an almost conformal map. The idea, compared to the isometric case, is to require it to be isometric to some conformal metric \(\lambda^2 g\), instead of the metric \(g\) itself. Here is the definition of conformal map upon which we base our understanding:

With a slight abuse of language, we also refer to a smooth map or a smooth immersion \(f\) as a conformal map as long as it satisfies the equation (3 ). Much attention will be focused on seeking immersions that satisfy (3 ), and ultimately showing the immersions are embeddings. Therefore, such an abuse won’t affect our results.

Next, to obtain the embedding into \(\mathbb{R}^q\), we truncate off the terms beyond the first \(q\) ones, in the following sense.

The following Proposition estimates the truncated tail approaches to \(0\) exponentially, which is due to [2].

The following corollary applies the former discussion to our conformal case.

The truncated almost conformal embedding \(\Psi^{q(t)}_{t,\eta_i}\) will be later perturbed to a family of honest conformal embeddings in Section 6.

3 Günther’s iteration and the modification to conformal case↩︎

We may start by stating some conventions. Assume \(u=(u_1,\cdots,u_N)\in C^\infty (M, \mathbb{R}^N)\) is a smooth embedding, and let the metric \(g \in C^{2,\alpha}(M,\mathrm{Sym}^{ \otimes 2}T^*M)\). Then \(\nabla u=(\nabla u_1,\cdots,\nabla u_N)\in C^\infty (M,T^*M\otimes \mathbb{R}^N)\) as a section, and we further denote \(\nabla u\cdot \nabla u:=\nabla u_1\otimes \nabla u_1+\cdots +\nabla u_N\otimes \nabla u_N\in C^\infty (M,T^*M\otimes T^*M )\) as a \((0,2)\)-tensor field. In local coordinates, \(\nabla u\) may be identified with an \(n\times N\) matrix, where \(n=\dim M\), and \(\nabla u\cdot \nabla u\) can be computed as \(\nabla u \nabla u^T\). Eventually, as we will see in the later sections, \(u\) is meant to represent \(\Psi_t\) and \(\Psi_{t,\eta_i}^{q(t)}\).

Let’s begin with the assumption that \(u\) is given readily as an almost conformal map, which satisfies \[\label{almost32conformal32equation} \nabla u\cdot \nabla u-\frac{\mathrm{tr}_g(\nabla u\cdot \nabla u)}{n}g=-f+\frac{\mathrm{tr}_g f}{n}g,\tag{8}\] where \(f\) is a ‘small’ symmetric 2-tensor, and \(\nabla\) is the Levi-Civita connection of \((M,g)\). To attain our goal of this paper, it will suffice to find a map \(u+v :M \longrightarrow \mathbb{R}^N\) that solves the equation \[\label{conformal32want} \nabla(u+v)\cdot \nabla(u+v)-\frac{\mathrm{tr}_g(\nabla(u+v)\cdot \nabla(u+v))}{n}g=0.\tag{9}\]

Then, after subtract 9 by 8 , our goal becomes finding a \(v\in C^{r,\alpha}(M,\mathbb{R}^N)\), \(r\geq 2\), satisfying the conformal32embedding32equation: \[\label{conformal32embedding32equation} \nabla u\cdot \nabla v-\frac{\mathrm{tr}_g(\nabla u\cdot \nabla v)}{n}g+\nabla v\cdot \nabla u-\frac{\mathrm{tr}_g(\nabla v\cdot \nabla u)}{n}g+\nabla v\cdot \nabla v-\frac{\mathrm{tr}_g(\nabla v\cdot \nabla v)}{n}g=f-\frac{\mathrm{tr}_g f}{n}g.\tag{10}\]

3.1 Free mappings.↩︎

In this subsection, we would like to state the facts about free mapping and apply it to our conformal case. Throughout this paper, when we mention \(C^{r,\alpha}\), \(r\geq 2\), we will fix \(0<\alpha<1\). Also, since finding the optimistic \(N\) is not the goal of this paper, we could take \(N\) to always be greater than or equal to \(n+\frac{1}{2}n(n+1)\).

To clarify the definition, note that although for any point \(x\in M\), \(P\) could be thought of as a map \(C^{\infty}(M)\longrightarrow T^*_x M \oplus\mathrm{Sym}^{\otimes 2} T_x^*M\), in our discussion, \(P\) is always applied to a fixed free mapping \(u\), which makes it a linear operator. The definition of \(P(u)\) can be viewed as follows: \[\begin{array}{ccccc} C^{\infty}(M,\mathbb{R}^N)&\overset{P|_x}{\xrightarrow{}} &\mathbb{R}^N\otimes(T^*_x M \oplus\mathrm{Sym}^{\otimes 2} T_x^*M) & \overset{\text{in normal coordinates}}{\xrightarrow{}}& \mathcal{L}(\mathbb{R}^N,\mathbb{R}^{n+\frac{n(n+1)}{2}})\\ & & & &\\ u& \xmapsto{} & \begin{bmatrix} \nabla u\\ \nabla \nabla u \end{bmatrix}& \xrightarrow{}& \begin{bmatrix} \nabla_i u_m\\ \nabla_j \nabla_k u_m \end{bmatrix}_{\begin{array}{c} \scriptstyle 1\leq i,j,k \leq n, \\ \scriptstyle 1\leq m\leq N \end{array}} \end{array}.\]

3.2 Günther’s lemma in conformal case.↩︎

In this subsection, we would follow Günther [4] to perform a detailed computation, summarizing the results as a lemma at the end. Again, the well-known Einstein summation notation is employed throughout.

Let \(\nabla\) be the Levi-Civita connection of \((M,g)\). The Laplacian in use is the connection Laplacian, \(\Delta:=\mathrm{tr}\nabla^2\), applicable to all the functions and tensors of at least \(C^2\) smooth. The Ricci curvature is defined as \(R_{ik}:=R_{lik}{}^l\) in our convention.

In our setting, the eigenvalues are the \(\lambda_i\)’s satisfying \(\Delta f +\lambda f=0\), hence \(0<\lambda_1\leq \lambda_2\leq \cdots\). Thus for some positive constant number \(\epsilon\), the \(\Delta-\epsilon\) is an isomorphism between \(C^{r,\alpha}\) and \(C^{r-2,\alpha}\) of functions and tensors of various sizes on compact \(M\), for \(r\geq 2\). Additionally, it has a unique inverse denoted as \((\Delta-\epsilon)^{-1}\). For our application, we shall take \(\epsilon=1\) throughout.

Let \(u\in C^\infty (M, \mathbb{R}^N)\) be a free embedding approximating the given smooth metric \(g\) as in 8 . Let \(r\geq 2\), our goal is to find \(v\in C^{r,\alpha}(M,\mathbb{R}^N)\) satisfying 10 .

We first consider the term with trace: \(\nabla u\cdot \nabla v+\nabla v\cdot \nabla u +\nabla v\cdot \nabla v\), apply \(\Delta-1\) to it and set it equal to \(f\): \[(\Delta-1)(\nabla u\cdot \nabla v)+(\Delta-1)(\nabla v\cdot \nabla u)+(\Delta-1)(\nabla v\cdot \nabla v)=(\Delta-1)f,\] where the third derivatives of \(v\) are to be considered as distributions if \(r<3\), and this won’t affect the following computation.

Careful treatment is needed by using a local coordinates, denoted as \(\{x^i\}_{1\le i \le n}\), which are not necessarily normal coordinates. The first term to compute is the quadratic term \((\Delta-1)(\nabla_iv\cdot \nabla_j v)\) involving \(v\):

\[\begin{align} (\Delta&-1)(\nabla_iv \,dx^i\cdot\nabla_jv \,dx^j)\\ =&\nabla^l(\nabla_l(\nabla_iv \,dx^i\cdot\nabla_jv\,dx^j))-\nabla_iv\cdot\nabla_jv\,dx^i\otimes dx^j\\ =&\Delta(\nabla_iv\,dx^i)\cdot(\nabla_jv\,dx^j)+(\nabla_iv\,dx^i)\cdot\Delta(\nabla_jv\,dx^j)\\ &+2\nabla^l(\nabla_iv\,dx^i)\cdot\nabla_l(\nabla_jv\,dx^j)-\nabla_iv\cdot\nabla_jv\,dx^i\otimes dx^j\\ =&\nabla_i(\Delta v)\,dx^i\cdot\nabla_jv\,dx^j+R_i{}^k\nabla_kv \,dx^i\cdot\nabla_jv\,dx^j+\nabla_iv\,dx^i\cdot\nabla_j(\Delta v)\,dx^j\\ \, &+\nabla_iv \,dx^i\cdot R_j{}^k\nabla_kv\,dx^j+2\nabla^l(\nabla_iv\,dx^i)\cdot\nabla_l(\nabla_jv\,dx^j)-\nabla_iv\cdot\nabla_jv\,dx^i\otimes dx^j\\ =&\nabla_i(\Delta v\cdot\nabla_j v\,dx^j)dx^i-\Delta v\cdot\nabla_i(\nabla_j v \,dx^j) dx^i+ \nabla_j(\nabla_iv\,dx^i\cdot\Delta v)dx^j-\nabla_j(\nabla_iv\,dx^i)dx^j\cdot \Delta v \\ \,&+(R_i{}^k\nabla_jv+R_j{}^k\nabla_iv)\cdot \nabla_kv\,dx^i\otimes dx^j +2\nabla^l(\nabla_iv\,dx^i)\cdot\nabla_l(\nabla_jv\,dx^j)-\nabla_iv\cdot\nabla_jv\,dx^i\otimes dx^j\\ =&2L_{ij}(v,v) dx^i\otimes dx^j+\nabla_i(\Delta v\cdot\nabla_j v\,dx^j)dx^i+\nabla_j(\nabla_iv\,dx^i\cdot\Delta v)dx^j, \end{align}\] here for brevity, we denoted

\[\begin{align} L_{ij}(v,v)\,dx^i\otimes dx^j:=&\nabla^l(\nabla_iv\,dx^i)\cdot\nabla_l(\nabla_jv\,dx^j)-\Delta v\cdot\nabla_i(\nabla_jv\,dx^j)dx^i\notag\\ &+\{-\frac{1}{2}\nabla_iv \cdot\nabla_jv+\frac{1}{2}(R_i{}^k\nabla_jv+R_j{}^k\nabla_iv)\cdot \nabla_kv\}dx^i\otimes dx^j.\label{24616} \end{align}\tag{12}\] For the other terms involving \(u\) and \(v\), we get the following by switching the Laplacian and the covariant derivative: \[\begin{align} (\Delta&-1)(\nabla_iu dx^i\cdot \nabla_jv dx^j)\\ =&(\Delta-1)(\nabla_j(\nabla_iu dx^i\cdot v)dx^j)-(\Delta-1)(\nabla_j(\nabla_iu dx^i)dx^j\cdot v)\\ =&\nabla_j((\Delta-1)(\nabla_iu\,dx^i\cdot v))dx^j+\{2R_j{}^k{}_i{}^n\nabla_k(\nabla_n u\cdot v) +R_{j}{}^k{}_i{}^n(\nabla_m u\cdot v)(-\Gamma_{kn}^m)+\nabla^k({R_{jki}}^n)(\nabla_n u\cdot v)\\ &+g^{kl}{R_{jmi}}^n(\nabla_n u\cdot v) (-\Gamma_{lk}^m)+R_j{}^n\nabla_n(\nabla_i u\cdot v) +R_j{}^n (\nabla_m u\cdot v) (-\Gamma_{ni}^m)\}dx^i\otimes dx^j \\&-(\Delta-1)(\nabla_j\nabla_iu\cdot v\,dx^i\otimes dx^j) -(\Delta-1)(\nabla_nu\,\Gamma_{ji}^n\cdot v\,dx^i\otimes dx^j). \end{align}\] Similar computation for the other one, \[\begin{align} (\Delta&-1)(\nabla_ju\,dx^j\cdot \nabla_iv\,dx^i)\\=&\nabla_i((\Delta-1)(\nabla_ju\,dx^j\cdot v))dx^i+\{2R_i{}^k{}_j{}^n\nabla_k(\nabla_n u\cdot v) +R_{i}{}^k{}_j{}^n(\nabla_m u\cdot v)(-\Gamma_{kn}^m)+\nabla^k({R_{ikj}}^n)(\nabla_n u\cdot v)\\ &+g^{kl}{R_{imj}}^n(\nabla_n u\cdot v) (-\Gamma_{lk}^m)+R_i{}^n\nabla_n(\nabla_j u\cdot v) +R_i{}^n (\nabla_m u\cdot v) (-\Gamma_{nj}^m)\}dx^i\otimes dx^j\\ &-(\Delta-1)(\nabla_i\nabla_ju\cdot v\, dx^i\otimes dx^j)-(\Delta-1)(\nabla_n u\, \Gamma_{ij}^n\cdot v\, dx^i\otimes dx^j). \end{align}\] We could denote the following notion of \(r_{ij}^n\), for \(w=w_ndx^n\in C^{r,\alpha}(M,T_x^*M)\): \[\label{24619} \begin{align} r_{ij}^nw_ndx^i\otimes dx^j:=&\{ 2R_i{}^k{}_j{}^n\nabla_kw_n +R_{i}{}^k{}_j{}^mw_n(-\Gamma_{km}^n)+\nabla^k({R_{ikj}}^n)w_n \\ &+g^{kl}{R_{imj}}^nw_n (-\Gamma_{lk}^m)+R_i{}^m w_n (-\Gamma_{mj}^n)\}dx^i\otimes dx^j. \end{align}\tag{13}\]

Combining everything, we get: \[\label{vanishing32terms} \begin{align} &(\Delta-1)(\nabla_iu\cdot\nabla_jv\,dx^i\otimes dx^j+\nabla_ju\cdot\nabla_iv\,dx^i\otimes dx^j+\nabla_iv\cdot\nabla_jv \, dx^i\otimes dx^j)\\ =& \nabla_j\{(\Delta-1)(\nabla_iu\,dx^i\cdot v)+\nabla_iv\,dx^i\cdot\Delta v\}dx^j+\nabla_i\{(\Delta-1)(\nabla_ju\,dx^j\cdot v)+\Delta v\cdot\nabla_j v\,dx^j\}dx^i\\ &+\{2L_{ij}(v,v)+r^n_{ij}(\nabla_n u\cdot v)+R_i^n\nabla_n(\nabla_ju \cdot v)+r^n_{ji}(\nabla_n u\cdot v)+R_j^n\nabla_n(\nabla_iu \cdot v)\}dx^i\otimes dx^j\\ &-2(\Delta-1)(\nabla_i(\nabla_ju\,dx^j)dx^i\cdot v). \end{align}\tag{14}\] In [4], Günther made the key observation that if \(\nabla_i u \cdot v dx^i\) equal to \(-(\Delta-1)^{-1}\{\Delta v\cdot \nabla_iv\,dx^i\}\), multiple benefits will arise, including the ability to avoid loss of differentiability. In fact, if we force \(\nabla_i u \cdot v dx^i=-(\Delta-1)^{-1}\{\Delta v\cdot \nabla_iv\,dx^i\}\), the equation \[f=\nabla v \cdot\nabla u+\nabla u \cdot\nabla v +\nabla v\cdot \nabla v\] can be solved by the \(v\) that satisfies the following: \[\label{defi32of32u42v} \begin{align} \nabla_i u \,dx^i \cdot v =&-(\Delta-1)^{-1}\{\Delta v\cdot \nabla_iv\,dx^i\},\\ \nabla_i(\nabla_j u \,dx^j) dx^i\cdot v =& \frac{1}{2}(\Delta-1)^{-1}(\{2L_{ij}(v,v)+(r^n_{ij}+r^n_{ji})(\nabla_n u\cdot v )\\ &+R_j{}^n\nabla_n(\nabla_i u\cdot v )+R_i{}^n\nabla_n(\nabla_ju \cdot v)\}dx^i\otimes dx^j)-\frac{1}{2}f_{ij}dx^i\otimes dx^j\\ = & \frac{1}{2}(\Delta-1)^{-1}\Big([2L_{ij}(v,v)+(r_{ij}^n+r_{ji}^n)(-(\Delta-1)^{-1}\{\Delta v\cdot \nabla v\})_n\\ &+R_j{}^n\nabla_n((\Delta-1)^{-1}\{\Delta v\cdot \nabla v\})_i+R_i{}^n\nabla_n((\Delta-1)^{-1}\{\Delta v\cdot \nabla v\})_j]dx^i\otimes dx^j \Big)\\ &-\frac{1}{2}f_{ij}dx^i\otimes dx^j. \end{align}\tag{15}\] Here, the notation \((\quad )_j\) denotes the coefficient of the 1-form inside it with respect to \(dx^j\), i.e., \(w=(w)_jdx^j\) for any 1-form \(w\). Note that, aside from \(f\), the right-hand side of 15 does not involve \(u\), is quadratic about \(v\), and does not lose the differentiability of \(v\), i.e., if \(v\in C^{r,\alpha}(M,\mathbb{R}^N)\) then the terms are in \(C^{r,\alpha}(M,\mathrm{Sym}^{\otimes 2} T_x^*M)\). Then, define \(Q_u(v,v)\in C^{r,\alpha}(M,\mathbb{R}^N)\) as the unique solution (thanks to Lemma 19) of the following: \[\label{defi32of32Q40v44v41} \begin{align} \nabla u\cdot Q_u(v,v) =&-(\Delta-1)^{-1}\{\Delta v\cdot \nabla v\}, \\ \nabla\nabla u\cdot Q_u(v,v) =& M_{ij}(v), \end{align}\tag{16}\] \[\begin{align} \text{where } \quad M_{ij}(v):=&\frac{1}{2}(\Delta-1)^{-1}([2L_{ij}(v,v)+(r_{ij}^n+r_{ji}^n)(-(\Delta-1)^{-1}\{\Delta v\cdot \nabla v\})_n\\ +R_j{}^n&\nabla_n((\Delta-1)^{-1}\{\Delta v\cdot \nabla v\})_i+R_i{}^n\nabla_n((\Delta-1)^{-1}\{\Delta v\cdot \nabla v\})_j]dx^i\otimes dx^j ). \end{align}\]

By the freeness of \(u\) and by applying Lemma 19, \(Q_u(v,v)\) exists and is unique if we require that \(Q_u(v,v)(x)\perp \mathrm{Ker}P(u)(x)\) for any \(x\in M\). Hence 14 can be rewritten as \[\begin{align} &(\Delta -1)(\nabla u\cdot \nabla v+\nabla v\cdot \nabla u+ \nabla v \cdot \nabla v)\\ =&2\mathrm{Sym}(\nabla\{(\Delta-1)(\nabla u\cdot (v-Q_u(v,v)))\})-2(\Delta-1)\{\nabla\nabla u\cdot (v-Q_u(v,v))\}, \end{align}\] where \(\mathrm{Sym}\) is the symmetrization of a 2-tensor. Using the notation \(\mathrm{tr}^\perp_g\) in Remark 9, the conformal equation (10 ) can be rewritten as \[\mathrm{tr}_g^\perp(f)= \mathrm{tr}_g^\perp\Big(2(\Delta-1)^{-1}\mathrm{Sym}(\nabla\{(\Delta-1)(\nabla u\cdot (v-Q_u(v,v)))\})-2\nabla\nabla u\cdot (v-Q_u(v,v)) \Big).\]

To summarize, we have the conformal version of Günther’s lemma.

4 The properties of \(P_c\), and its right inverses↩︎

4.1 Singularity of \(P_c(\Psi_t)P_c^T(\Psi_t)\).↩︎

In Proposition 20, for any free embedding \(u\in C^\infty (M,\mathbb{R}^N)\), we have seen that \(P_c(u)\) is of rank \(\frac{n(n+3)}{2}-1\) in normal coordinates at a point \(x\in M\). Hence, naturally, \(P_c(u)P_c^T(u)(x)\) is singular and of rank \(\frac{n(n+3)}{2}-1\).

To apply this to our case, we need to show that the functions that of interest are free mappings. It is established in [2] that \(\Psi_t:M\longrightarrow \ell^2\) is a free mapping by expanding the heat kernel. In Corollary 27, we will justify the freeness of the truncated map \(\Psi_{t,\eta_i}^{q(t)}: M\longrightarrow \mathbb{R}^{q(t)}\) as defined in Definition 12.

For additional insight, in this subsection, we will explicitly present the matrix expression of \(P_c(\Psi_t)P_c^T(\Psi_t)\) and compute its rank in proposition 25. Here with respect to normal coordinates in the neighbourhood of \(x\in M\), \(P(\Psi_t)(x)\) and \(P_c(\Psi_t)(x)\) are taken as \(\frac{n(n+3)}{2}\times \infty\) matrices for \(\Psi_t:M\longrightarrow \ell^2\), while \(P(\Psi_{t,\eta_i}^{q(t)})(x)\) and \(P_c(\Psi_{t,\eta_i}^{q(t)})(x)\) are taken as \(\frac{n(n+3)}{2}\times \infty\) matrices.

We need a proposition on linear algebra first. In the remainder of this paper, \(J_n\) denotes an \(n\times n\) matrix with all entries equal \(1\).

The following proposition is due to [2]. It is interesting to compare it with the case of \(P_c(\Psi_t)P_c^T(\Psi_t)\) described in Proposition 25.

4.2 Right inverses of \(P_c\).↩︎

In the last subsection, we see that the \(P_c(\Psi_{t,\eta_i}^{q(t)})P_c^T(\Psi_{t,\eta_i}^{q(t)})(x)\) is of rank \(\frac{n(n+3)}{2}-1\). This implies that we cannot expect to find a right inverse operator of \(P_c(\Psi_{t,\eta_i}^{q(t)})\) for arbitrary right-hand vectors, but still can find the right inverses of \(P_c(\Psi_{t,\eta_i}^{q(t)})\) for the right-hand ones in the image of \(P_c(\Psi_{t,\eta_i}^{q(t)})\). This illustrates the difference between the local conformal embedding question and the local isometric one.

Recall in Corollary 16, the remainder term \(O(t^l)\) is a symmetric 2-tensor subtracting its own trace, we denote it as \(h\), which corresponds to the small difference term \(f-\frac{\mathrm{tr}_g f}{n}g\) in the conformal embedding equation 10 . Additionally, let \(G\subset \mathrm{Sym}^{\otimes 2}T^*M\) be the subbundle defined by:

\[\label{definition32of32bundle32G} G_x:=\{ s_x\in \mathrm{Sym}^{\otimes 2}T^*M \big|\, \mathrm{tr}_{g(x)}s_x=0\}.\tag{17}\] Thus, \(h\in C^{r,\alpha}(M,G)\), and \(P_c(\Psi_{t,\eta_i}^{q(t)})\) becomes surjective onto \(C^{r,\alpha}(M,G)\), thus has right inverse.

Our goal now is to explicitly construct a family of solutions of the following equation \[P_c(\Psi_{t,\eta_i}^{q(t)})\cdot v=\begin{bmatrix} 0 \\ h \end{bmatrix},\] where we are solving for \(v\in C^{r,\alpha}(M,\mathbb{R}^{q(t)}), \text{ and } v(x)\perp \mathrm{Ker}P(\Psi_{t,\eta_i}^{q(t)})(x)\). Note that it is a version of ?? with \(\Psi_{t,\eta_i}^{q(t)}\) involved.

4.3 Estimates about \(E\).↩︎

In order to apply the implicit function theorem, we need to prepare the estimates of the norms \(\rVert E(\Psi_{t,\eta_i}^{q(t)})\lVert_{C^{r,\alpha}}\) and \(\rVert E(\Psi_{t,\eta_i}^{q(t)})(0,h)\lVert_{C^{r,\alpha}}\), where the latter represents the right inverse operator with \((0,h)\) as the input.

The following analytic preliminaries will be used in latter computations.

5 Günther’s implicit function theorem↩︎

As discussed in Remark 23 and Theorem 28, our goal is to seek the solutions of \(v\in C^{r,\alpha}(M, \mathbb{R}^{q(t)})\), \(r\geq2\), in the following equation: \[\label{iteration32equation} E(\Psi_{t,\eta_i}^{q(t)})(0,-\frac{1}{2}h) +kE(\Psi_{t,\eta_i}^{q(t)})(0, g) =E(\Psi_{t,\eta_i}^{q(t)})(0,-\frac{1}{2}h+k\cdot g)=v-Q_{\Psi_{t,\eta_i}^{q(t)}}(v,v).\tag{19}\] Recall that the definition of \(Q_{\Psi_{t,\eta_i}^{q(t)}}(v,v)\) is given as in 16 with \(u\) substituted by \(\Psi_{t,\eta_i}^{q(t)}\).

The following theorem states the unique existence of the solution of 19 under a control condition of \(E(\Psi_{t,\eta_i}^{q(t)})\).

The proof of this theorem is essentially due to the main theorem of Günther [4]. For the completeness of the article, we will present it here.

6 The main theorem: conformal embeddings↩︎

In this section, the formerly prepared propositions and theorems are employed to prove the main theorem of this paper.

The main theorem can be divided into two propositions: the first claims that we could find a family of conformal immersions \(C_t\) depending on a function \(k_t\in C^{r,\alpha}(M)\) of \(O(t^l)\), hence we denote it as \(C_{t,k_t}\), and the second checks that this \(C_{t,k_t}\) is one to one, hence an embedding.

References↩︎