Effective Results on Uniformization and Intrinsic GCM Spheres in Perturbations of Kerr

Abstract

This is a follow-up of our paper (Klainerman and Szeftel in Construction of GCM spheres in perturbations of Kerr, Accepted for publication in Annals of PDE) on the construction of general covariant modulated (GCM) spheres in perturbations of Kerr, which we expect to play a central role in establishing their nonlinear stability. We reformulate the main results of that paper using a canonical definition of $\ell =1$ modes on a 2-sphere embedded in a $1+3$ vacuum manifold. This is based on a new, effective, version of the classical uniformization theorem which allows us to define such modes and prove their stability for spheres with comparable metrics. The reformulation allows us to prove a second, intrinsic, existence theorem for GCM spheres, expressed purely in terms of geometric quantities defined on it. A natural definition of angular momentum for such GCM spheres is also introduced, which we expect to play a key role in determining the final angular momentum for general perturbations of Kerr.

Appendix by Camillo De Lellis

We denote by e the Euclidean metric on ${\mathbb {R}}^n$.

Theorem A.1

For every $n\ge 2$ and every $\alpha \in (0,1]$ there is a constant $C = C(n,\alpha )$ with the following property. Let $u\in H^1 (B_2, {\mathbb {R}}^n)$ be such that

$$\begin{aligned} \Vert u^\sharp e - e\Vert _{L^\infty (B_1)} \le \frac{1}{2} \end{aligned}$$

(A.1)

and

$$\begin{aligned} \det Du \ge 0 \qquad \text{ a.e. } \end{aligned}$$

(A.2)

Then there is $A\in SO (n)$ such that

$$\begin{aligned} \Vert Du - A\Vert _{C^\alpha (B_1)} \le C \Vert u^\sharp e - e\Vert _{C^\alpha (B_2)}\, . \end{aligned}$$

(A.3)

Remark A.2

From now on, |A| will denote the Hilbert-Schmidt norm of the matrix $A\in {\mathbb {R}}^{n\times n}$, namely $|A|= \sqrt{\mathrm{tr}\, A^\top A}$ induced by the Hilbert-Schmidt scalar product $\langle A, B\rangle := \mathrm{tr}\, A^\top B$. In particular, using standard coordinates in $B_2$ and identifying $u^\sharp e$ with the corresponding $n\times n$ matrix, we have $|Du|^2 = \mathrm{tr}\, u^\sharp e$. Under the assumption (A.1), we thus conclude immediately that $\Vert Du\Vert _{L^\infty (B_2)} \le C (n)$, namely that the map u is Lipschitz.

Remark A.3

Note that an assumption like (A.2) is needed because it is easy to give examples of Lipschitz maps whose derivative belong to O(n) almost everywhere but which are not affine. In fact such maps are “abundant” in an appropriate sense: in particular they form a residual set in the space $X:=\{u\in \mathrm{Lip}\, (B_2, {\mathbb {R}}^n): u^\sharp e \le e\}$ endowed with the the $L^\infty$ distance, which makes X a compact metric space (cf. [15] for the latter and more subtle results).

The two authors of the paper asked me while preparing their manuscript whether I could provide a reference or a proof for Theorem A.1. While I felt that this should be a well-known “classical fact”, I was unable to find a reference for it. I therefore suggested a simple argument which reduces (A.3) to an important work of [10] which essentially handles a corresponding “$L^2$-estimate”. The reduction is given in this appendix. It uses some elementary facts from Linear Algebra (which are well known and I just include for the reader’s convenience) and a Morrey-type decay. In what follows we denote by $\mathrm{Id}$ the identity matrix in ${\mathbb {R}}^{n\times n}$.

Lemma A.4

We have

$$\begin{aligned} \mathrm{dist}\,(A, SO (n)) = \mathrm{dist}\,(A, O(n)) \qquad \text{ for } \text{ all } A\in {\mathbb {R}}^{n\times n} \text{ with } \det A \ge 0 \end{aligned}$$

(A.4)

and

$$\begin{aligned} \mathrm{dist}\,(A, O(n)) \le |A^\top A - I\,d| \qquad \forall A\in {\mathbb {R}}^{n\times n}\, . \end{aligned}$$

(A.5)

Proof

In order to show (A.4) fix first an arbitrary matrix A with $\det A \ge 0$. Recalling the polar decomposition of matrices there is a symmetric S and a $O_1\in SO (n)$ such that $A = O_1 S$. Next, recalling that every symmetric matrix is diagonalizable, there is $O_2\in SO(n)$ such that $A = O_1 O_2^\top D O_2$ for some diagonal matrix D. Next recall that if O is a diagonal matrix with an even number of entries equal to $-1$ and the remaining equal to 1, then $O\in SO(n)$. If one of the diagonal enties of D is zero, we can then assume without loss of generality that all enties of A are nonnegative. Otherwise, if no diagonal entry is 0, we can assume without loss of generality that they are all positive but at most 1. Since $\det A>0$, we can exclude that one diagonal entry of D is negative and the others are all positive. Summarizing the arguments in the two cases, we can assume that all diagonal entries of D are nonnegative. Since $\mathrm{dist}\,(A, O(n)) = \mathrm{dist}\,(OA, O(n))=\mathrm{dist}\,(AO, O(n))$ and $\mathrm{dist}\,(A, SO(n))= \mathrm{dist}\,(OA, SO(n)) = \mathrm{dist}\,(AO, SO(n))$ for every $O\in SO(n)$, we conclude that it suffices to prove (A.4) for a diagonal matrix A which has all nonnegative entries. Denote them by $\lambda _1, \ldots , \lambda _n$. For any $O\in O(n)$ we can then compute explicitely

$$\begin{aligned} |A - O|^2 = \sum _i \lambda _i^2 + n - 2\sum _i \lambda _i O_{ii}\, . \end{aligned}$$

Observe that $-1\le O_{ii}\le 1$ because O is orthogonal. Since $\lambda _i \ge 0$ for every i we then conclude $|A-O|^2 \ge \sum _i \lambda _i^2 + n - 2 \sum _i \lambda _i = |A-I\,d|^2$. This however shows that $\mathrm{dist}\,(A, O(n))^2 = |A-I\,d|^2 = \mathrm{dist}\,(A, SO(n))^2$.

As for (A.5) fix $A\in {\mathbb {R}}^{n\times n}$ and let $O\in O(n)$ be such that $\mathrm{dist}\,(A, O(n)) = |A-O|$. Since both sides of the inequality take the same value for A and $O^{-1} A$, we can assume that $O = I\,d$. By the minimality condition of $I\,d$ we must have that $A-I\,d$ is orthogonal (in the Hilbert-Schmidt scalar product) to the tangent to O(n) at $I\,d$, which is the space of skew-symmetric matrices. We therefore conclude that A is symmetric and, again applying the O(n) invariance of the inequality, we can assume w.l.o.g. that it is diagonal. Let $\lambda _i= A_{ii}$ be the diagonal entries and observe that none of them can be negative: if $\lambda _k <0$ then the matrix B which has $B_{ij} =0$ for $i\ne j$, $B_{ii}=1$ for $i\ne k$ and $B_{kk}=-1$ satisfies $B\in O(n)$ and $|A-B|< |A-I\,d|$. (A.5) is thus reduced to proving

$$\begin{aligned} \sum _i (\lambda _i-1)^2 \le \sum _i (\lambda _i^2-1)^2\, \end{aligned}$$

under the assumption that $\lambda _i \ge 0$ for every i. This is equivalent to prove $(x-1)^2 \le (x^2-1)^2= (x-1)^2 (x+1)^2$ for $x\ge 0$, which is obvious. $\square$

Proof of Theorem A.1

Fix $x\in B_1$ and let S(x) be the unique positive definite symmetric matrix such that $S(x)^2 = u^\sharp e (x)$. Observe that, by (A.1), we have

$$\begin{aligned} |S(x)|+|S(x)^{-1}|\le C\, . \end{aligned}$$

Let v be the map $v(y) := u (y) (S(x))^{-1}$ and observe further that

$$\begin{aligned} \mathrm{dist}\,(Dv (y), SO(n))&\le \mathrm{dist}\,(Dv (y), O(n)) \le |Dv^\top (y) Dv (y) - I\,d|\\&\le |S(x)^{-1} Du^\top (y) Du (y) S(x)^{-1} - I\,d|\\&= |S(x)^{-1} (Du^\top (y) Du (y) - S(y)^2)S(x)^{-1}|\\&= | S(x)^{-1} (u^\sharp e (y) - u^\sharp e (x)) S(x)^{-1}|\\&\le 4 [u^\sharp e]_{\alpha , B_2} |x-y|^\alpha = 4 [u^\sharp e - e]_{\alpha , B_2} |x-y|^\alpha \end{aligned}$$

(where we use the standard notation $[f]_{\alpha , \Omega }:= \sup \{\frac{|f(x)-f(y)|}{|x-y|^\alpha } : x,y\in \Omega \}$). In particular, for every $r\le 1$ we can apply the Friesecke-James-Müller inequality, namely [10, Theorem 3.1], to find a matrix $A (x,r)\in SO(n)$ with the property that

(A.6)

(note that [10, Theorem 3.1] is stated for a general open set U in place of $B_r (x)$, with a constant depending on U; however an obvious scaling argument shows that the constant is the same for balls of arbitrary radii). Recalling that

we conclude

Morrey’s estimate then implies that $Du\in C^\alpha (B_1)$ and

$$\begin{aligned} {[}Du]_{\alpha , B_1} \le C [u^\sharp e - e]_{\alpha , B_2}\, . \end{aligned}$$

(A.7)

On the other hand, again by the Friesecke-James-Müller estimate, there is $A\in SO (n)$ such that

(A.8)

(A.7) and (A.8) immediately imply (A.3). $\square$