Harmonic Coordinates and the Regularity of Metrics

6B.7 Schauder Theory and the Continuity Method–6B.9 De Giorgi–Nash–Moser built the regularity theory of linear elliptic equations: Schauder estimates when the coefficients are Hölder, $L^p$ estimates when they are continuous, De Giorgi–Nash–Moser when they are merely measurable. This chapter turns that theory on the most important unknown of geometric analysis, a Riemannian metric, and finds that it applies only after a choice has been made. A metric written in coordinates is a matrix-valued function $g_{ij}(x)$, and its smoothness depends on the coordinates: the flat metric of the plane, written in a badly chosen chart, can have coefficients that are only once differentiable. The Ricci tensor, the quantity that appears in the Ricci flow and in Einstein's equations, involves second derivatives of $g_{ij}$, but in arbitrary coordinates it is not an elliptic operator on the metric, for the same reason that the Ricci flow is not parabolic (6A.1 What a PDE Is): it is unchanged by changes of coordinates, so it cannot control them.

The remedy is to choose coordinates adapted to the metric. Harmonic coordinates, coordinate functions that solve the Laplace equation of the metric, are the best choice: in them the Ricci tensor becomes $-\frac12g^{kl}\partial_k\partial_lg_{ij}$ plus terms of lower order, a diagonal elliptic system for the components of the metric, and all of 6B.7 Schauder Theory and the Continuity Method–6B.9 De Giorgi–Nash–Moser applies. The consequences are among the basic facts of Riemannian geometry: a metric is as smooth as its Ricci curvature allows and no smoother (DeTurck–Kazdan), Einstein metrics are real analytic, and a bound on the Ricci curvature controls the metric in $C^{1,\alpha}$ on balls of a definite size (the harmonic radius), which is how compactness theorems for Riemannian manifolds, and Hamilton's compactness theorem for Ricci flows (11B.3 Compactness of Ricci Flows), are proved. In Lorentzian signature the same coordinates, now solving a wave equation, turn Einstein's equations into a hyperbolic system: this is how Yvonne Choquet-Bruhat proved in 1952 that general relativity has a well-posed initial-value problem, and how numerical relativists in 2005 first simulated the merger of two black holes. DeTurck's trick for the Ricci flow (11A.4 Short-Time Existence and Uniqueness) is the parabolic member of the same family.

The chapter is written for a reader who has not yet met manifolds. Everything happens on an open set of $\mathbb{R}^n$ carrying a positive definite matrix field $g_{ij}(x)$; Book 9A (9A.1 Riemannian Metrics and Model Spaces–9A.6 The Laplacian and the Bochner Formula) will interpret it geometrically.

By the end of this chapter you will be able to:

• write the Laplace–Beltrami operator of a metric in coordinates, in divergence and non-divergence form, and identify the contracted Christoffel symbols $\Gamma^k$ with $-\Delta_gx^k$;
• explain why the regularity of a metric is coordinate dependent, and give examples;
• prove that harmonic coordinates exist near every point, with $C^{1,\alpha}$ control;
• derive the formula $R_{ij} = -\frac12g^{kl}\partial_k\partial_lg_{ij} + \frac12(\partial_i\Gamma_j + \partial_j\Gamma_i) + Q_{ij}(g, \partial g)$ and conclude that the Ricci tensor is elliptic in harmonic coordinates;
• prove the DeTurck–Kazdan regularity theorem by Schauder and $L^p$ bootstrapping, and explain why Einstein metrics are analytic;
• define the harmonic radius and state the Jost–Karcher and Anderson estimates and the compactness theorems they give;
• describe Einstein's equations in wave gauge, linearised gravity, and generalised harmonic coordinates in numerical relativity;
• relate harmonic coordinates to harmonic maps and to DeTurck's trick.

10.1A metric is a matrix of functions

Let $U \subseteq \mathbb{R}^n$ be open. A Riemannian metric on $U$ is a matrix-valued function $g = (g_{ij}(x))$, symmetric and positive definite at each point. It measures the length of a vector $v$ at $x$ by $|v|_g^2 = g_{ij}(x)v^iv^j$ (summation over repeated indices, as throughout), and the length of a curve by integrating that. The flat metric is $g_{ij} = \delta_{ij}$. We write $g^{ij}$ for the inverse matrix and $|g| = \det(g_{ij})$.

Changing coordinates. If $x = \phi(y)$ is a diffeomorphism, the same metric in the coordinates $y$ has components

so that lengths of curves are the same whichever coordinates compute them. Two observations follow immediately. First, $\tilde g$ involves the first derivatives of $\phi$, so if $\phi$ is only $C^{k+1,\alpha}$ then $\tilde g$ is only $C^{k,\alpha}$, however smooth $g$ was. Pulling the flat metric back by a $C^{2,\alpha}$ diffeomorphism that is not $C^3$ produces a metric with $C^{1,\alpha}$ coefficients that is nonetheless flat: every geometric quantity is as nice as possible, and the coefficients are not. The regularity of the coefficients of a metric is not a geometric property. Second, there is a large group acting (all diffeomorphisms), and any equation for $g$ that is geometric, such as $\operatorname{Ric}(g) = 0$, is invariant under it, so its solutions come in infinite-dimensional families of coordinate representations. An elliptic equation, whose solutions with given boundary values are unique (6A.5 Maximum Principles), cannot have that property. The equation must be supplemented by a choice of coordinates before elliptic theory applies, exactly as the Ricci flow must be supplemented by DeTurck's gauge before parabolic theory applies.

The Laplacian of a metric. The Laplace–Beltrami operator of $g$ is

It is characterised by the integration by parts $\int_U(\Delta_gu)v\,dV_g = -\int_Ug^{ij}\partial_iu\,\partial_jv\,dV_g$ for $v$ with compact support, where $dV_g = \sqrt{|g|}\,dx$ is the volume of the metric, and that characterisation shows that $\Delta_g$ does not depend on the coordinates (the right side is built from lengths and volumes). Two features matter here. In divergence form, $\Delta_g$ has coefficients $\sqrt{|g|}g^{ij}$ that involve no derivatives of $g$: if $g$ is merely bounded and measurable with $\lambda\delta \leq g \leq \Lambda\delta$, the equation $\Delta_gu = 0$ is a uniformly elliptic divergence-form equation, and De Giorgi–Nash–Moser (6B.9 De Giorgi–Nash–Moser) applies to it. In non-divergence form, expanding the derivative,

where $\Gamma^k_{ij} = \frac12g^{kl}(\partial_ig_{jl} + \partial_jg_{il} - \partial_lg_{ij})$ are the Christoffel symbols of $g$ and $\Gamma^k$ their contraction (Exercise 10.5). Applying this to a coordinate function $u = x^k$ gives the identity on which the chapter rests:

So the contracted Christoffel symbols measure how far the coordinate functions are from being harmonic for the metric. For polar coordinates on the plane, $\Delta r = \frac1r$ and $\Delta\theta = 0$: the angle is harmonic, the radius is not (Exercise 10.6).

10.2Harmonic coordinates exist

A coordinate system $(u^1, \dots, u^n)$ on an open set is harmonic for $g$ if each $u^k$ satisfies $\Delta_gu^k = 0$. In such coordinates $\Gamma^k = 0$ for every $k$, by the identity above, read in the new coordinates. Harmonic coordinates are not unique (composing with an affine map preserves harmonicity, and so do many other changes), but they exist near every point, with estimates.

The proposition already gives the first half of the regularity theory: if a metric is $C^{k,\alpha}$ in some coordinates, it is $C^{k,\alpha}$ in harmonic coordinates. Harmonic coordinates never lose regularity. The second half, which is the point, is that they can gain it, and they gain it because of what the Ricci tensor looks like in them.

10.3The Ricci tensor is elliptic in harmonic coordinates

The Ricci tensor of $g$ is defined, in coordinates, by

which Book 9A (9A.4 Curvature and What It Means) derives from the curvature of the connection and interprets geometrically (for the round sphere of radius $1$ in dimension $n$, $R_{ij} = (n - 1)g_{ij}$). For this chapter all that matters is its structure: it is a sum of terms with two derivatives of $g$, linear in those derivatives with coefficients depending on $g$, and terms quadratic in first derivatives of $g$. Write $Q_{ij}(g, \partial g)$ for any expression of the second kind: a polynomial in $\partial g$, quadratic, with coefficients smooth functions of $g$.

Read the first formula as a statement about symbols (6A.1 What a PDE Is, 6A.4 Fourier Methods and Scale Space). The operator $g \mapsto -\frac12g^{kl}\partial_k\partial_lg_{ij}$ acts on each component by the same scalar elliptic operator, so it is as elliptic as an operator on a system can be. The term $\frac12(\partial_i\Gamma_j + \partial_j\Gamma_i)$ is a symmetrised derivative of a vector field built from first derivatives of $g$: it is the second-order part of a Lie derivative, the infinitesimal change of coordinates. It is responsible for the gauge directions of the symbol computed in 6A.4 Fourier Methods and Scale Space, where the linearised Ricci operator vanished on matrices $\xi v^T + v\xi^T$. Harmonic coordinates delete it (Exercise 10.9).

Regularity. In harmonic coordinates, then, a metric with prescribed Ricci tensor solves a quasilinear elliptic system with diagonal principal part,

and bootstrapping runs as in 6A.6 Weak Solutions and Elliptic Regularity.

This is DeTurck and Kazdan's Theorem 4.5(b); the formula of Proposition 10.2 that it rests on is their Lemma 4.1, which they attribute to Lanczos. With more work the hypothesis $g \in C^2$ can be weakened (to $g \in C^{1,\alpha}$ or $W^{2,p}$ with the Ricci tensor understood weakly), which is the form needed for limits of manifolds below.

Together with Proposition 10.1 this gives the striking corollary: if the Ricci curvature of a metric is smooth in some coordinates (it is a tensor, so its regularity transforms like that of $g$), then in harmonic coordinates the metric is two derivatives smoother than its curvature, and that is optimal. Dennis DeTurck and Jerry Kazdan (1981) also compared harmonic coordinates with the geodesic normal coordinates of 9A.3 Geodesics and the Exponential Map, which are natural geometrically: a metric of class $C^{k,\alpha}$ in some chart is $C^{k,\alpha}$ in harmonic coordinates, but in general only $C^{k-2,\alpha}$ in normal coordinates, which can lose two derivatives (their argument uses the theorem of Calabi and Hartman, 1970, that isometries between $C^{k,\alpha}$ metrics are $C^{k+1,\alpha}$; Myers and Steenrod had proved in 1939 that isometries of smooth metrics are smooth). Einstein metrics are real analytic in harmonic coordinates, which is the regularity fact behind their unique continuation: on a connected, simply connected manifold, two Einstein metrics that agree on an open set agree everywhere up to a diffeomorphism (DeTurck and Kazdan's Theorem 5.4, via a theorem of Myers on analytic metrics), so local rigidity spreads.

Dimension two. In two dimensions there are better coordinates still. Every metric $g$ on a surface is, near each point, conformally flat: there are isothermal coordinates in which $g = e^{2\varphi}(dx^2 + dy^2)$ (Gauss for analytic metrics; Korn in 1914 and Lichtenstein in 1916 for Hölder ones; Chern's short proof in 1955). Isothermal coordinates are harmonic, because $\Delta_g = e^{-2\varphi}(\partial_x^2 + \partial_y^2)$ in them, and the Gauss curvature is $K = -e^{-2\varphi}(\partial_x^2 + \partial_y^2)\varphi$. In dimension $n \geq 3$ conformally flat coordinates are not harmonic (Exercise 10.7), and most metrics have no conformally flat coordinates at all. The two-dimensional picture is the starting point of the uniformisation theorem (5A.5 Uniformization and the Two-Dimensional Ricci Flow) and of the Ricci flow on surfaces, which in isothermal coordinates is a single scalar equation $\partial_t\varphi = e^{-2\varphi}\Delta\varphi$ (11A.12 Ricci Flow on Surfaces).

10.4The harmonic radius and compactness

Harmonic coordinates exist on small balls; the question for geometry is how small. Define, at a point $p$, the harmonic radius $r_h(p)$ (for fixed $\alpha \in (0, 1)$ and a fixed tolerance $Q > 1$) to be the largest $r$ such that there are harmonic coordinates on the ball of radius $r$ about $p$ in which

The definition is scale invariant: replacing $g$ by $\lambda^2g$ multiplies $r_h$ by $\lambda$ (Exercise 10.10). A lower bound on the harmonic radius is a statement that, on balls of that size, the metric is uniformly close to Euclidean in $C^{1,\alpha}$, which is exactly what is needed to run the elliptic theory of this book on the geometry.

This is due to Jürgen Jost and Hermann Karcher (1982) under a bound on the full curvature tensor, and to Michael Anderson (1990) under a bound on the Ricci curvature alone; the proof is by contradiction and compactness, rescaling a sequence with $r_h \to 0$ so that $r_h = 1$ and showing the limit is flat Euclidean space, which has infinite harmonic radius. The injectivity radius is the size of the largest ball on which geodesics from the centre do not meet again (9A.3 Geodesics and the Exponential Map); without a lower bound on it, a manifold can collapse (a thin flat torus has bounded curvature and tiny harmonic radius), and the theorem fails. With higher derivative bounds on the curvature, Theorem 10.3 upgrades the $C^{1,\alpha}$ control to $C^{k+1,\alpha}$ control.

From the harmonic radius bound follows the Cheeger–Gromov compactness theorem in its harmonic-coordinate form (9B.4 Convergence of Manifolds): a sequence of complete pointed manifolds with $|\operatorname{Ric}| \leq \Lambda$ and injectivity radius at least $i_0$ has a subsequence converging in $C^{1,\beta}$, $\beta < \alpha$, to a limit with a $C^{1,\alpha}$ metric; with bounds on all derivatives of the curvature, the convergence is $C^\infty$. The charts of the limit are limits of harmonic coordinates, glued by transition maps controlled by Arzelà–Ascoli (2B.5 Uniform Convergence and Arzelà–Ascoli). Robert Greene and Hung-Hsi Wu (1988) first used harmonic coordinates to improve Gromov's Lipschitz convergence to $C^{1,\alpha}$ convergence in this way.

The Ricci flow. Hamilton's compactness theorem for Ricci flows (11B.3 Compactness of Ricci Flows) is the parabolic version: a sequence of flows with uniformly bounded curvature and a uniform lower bound on the injectivity radius at one point at one time has a subsequence converging smoothly to a limit flow. Shi's estimates (11A.5 Shi's Derivative Estimates) give bounds on all derivatives of curvature, harmonic coordinates turn curvature bounds into bounds on the metric, and the flow equation controls time derivatives. The hypothesis that does not come for free is the injectivity radius bound, and it is exactly what Perelman's no local collapsing theorem supplies (12A.4 κ-Noncollapsing): a curvature bound on a ball of radius $r$ and $\kappa$-noncollapsing give an injectivity radius lower bound of the form $c(\kappa)\,r$ (Cheeger–Gromov–Taylor; 9B.3 Collapsing and Noncollapsing). That is why Morgan and Tian devote their chapter 5 to geometric limits before they reach the blow-up analysis: every blow-up limit in the proof of the Poincaré conjecture is produced by this machinery, and harmonic coordinates are in its foundations.

10.5Einstein's equations in wave coordinates

In general relativity the metric $g_{\mu\nu}$ on four-dimensional spacetime has signature $(-, +, +, +)$, and Einstein's vacuum equations are $R_{\mu\nu} = 0$. The algebra of Proposition 10.2 does not see the signature, so in coordinates satisfying $\square_gx^\mu = 0$, where $\square_g$ is the wave operator of the Lorentzian metric (the same formula as $\Delta_g$), the equations become

a quasilinear system of wave equations, one for each component, coupled only through lower-order terms (6B.14 Hyperbolic Equations and Systems). Théophile de Donder (1921) and Cornelius Lanczos (1922) used such coordinates; Yvonne Choquet-Bruhat (then Fourès-Bruhat) proved in 1952 that the Cauchy problem for Einstein's equations is well posed in them, provided the initial data satisfy the constraint equations, and that the wave-coordinate condition, once imposed initially, propagates. This was the first proof that general relativity is a deterministic theory in the sense of the PDE theory of this book.

Gravitational waves. Linearising about the Minkowski metric, $g = \eta + h$, in the linearised wave gauge (the Lorenz or de Donder gauge, $\partial^\mu\bar h_{\mu\nu} = 0$ for the trace-reversed perturbation $\bar h_{\mu\nu} = h_{\mu\nu} - \frac12\eta_{\mu\nu}\eta^{\alpha\beta}h_{\alpha\beta}$), Einstein's equations with matter become $\square\bar h_{\mu\nu} = -16\pi G\,T_{\mu\nu}/c^4$: the flat wave equation, whose solutions travel at the speed of light. Einstein derived gravitational waves this way in 1916–18, with the quadrupole formula for the energy they carry. The orbital decay of the binary pulsar discovered by Russell Hulse and Joseph Taylor in 1974 matched the quadrupole formula (Nobel prize 1993), and the LIGO detectors observed gravitational waves directly from a black-hole merger on 14 September 2015 (Nobel prize 2017).

10.6Harmonic maps and DeTurck's trick

A coordinate system is a map $u : (U, g) \to (\mathbb{R}^n, \delta)$, and $\Delta_gu^k = 0$ says that this map is harmonic (6C.7 Harmonic Maps and Their Heat Flow): a critical point of the energy $\frac12\int g^{ij}\partial_iu^k\partial_ju^k\,dV_g$. For a map into a curved target $(N, h)$, harmonicity is $\tau(u) = 0$, with the tension field

where $\tilde\Gamma$ are the Christoffel symbols of $h$. For the identity map from $(M, g)$ to $(M, \tilde g)$ this is

the vector field of DeTurck's trick (11A.4 Short-Time Existence and Uniqueness). So "the identity is harmonic from $g$ to a fixed background $\tilde g$" is a coordinate-free version of "the coordinates are harmonic", with $\tilde g$ playing the role of the flat metric, and $W$ is the coordinate-free $\Gamma^k$.

The Ricci–DeTurck flow $\partial_tg = -2\operatorname{Ric}(g) + \mathcal L_Wg$ subtracts precisely the gauge term of Proposition 10.2: in coordinates, $\mathcal L_Wg_{ij} = \partial_iW_j + \partial_jW_i + Q(g, \partial g)$, and $W$ differs from $\Gamma$ by terms without derivatives of $g$, so

a strictly parabolic system with diagonal principal part. Its solutions are pulled back to Ricci flows by the diffeomorphisms generated by $W$, and those diffeomorphisms solve the harmonic map heat flow from $(M, g(t))$ to $(M, \tilde g)$. Uniqueness of the Ricci flow is proved by running this backwards: given two Ricci flows with the same initial metric, solve the harmonic map heat flow for each, which converts both into Ricci–DeTurck flows with the same initial data, and these are equal by uniqueness for strictly parabolic systems (Hamilton for compact manifolds; Bing-Long Chen and Xi-Ping Zhu in 2006 for complete manifolds with bounded curvature). Morgan and Tian cite Chen and Zhu's theorem for the uniqueness of the standard solution used in surgery, and give instead a more elementary argument that exploits its rotational symmetry (11B.6 Rotationally Symmetric Flows and the Standard Solution).

So one idea, coordinates that solve the Laplace or heat equation of the metric they describe, underlies the regularity of Einstein metrics, the compactness of spaces of metrics, the well-posedness of general relativity, and the existence and uniqueness of the Ricci flow.

10.7History

Gauss constructed isothermal coordinates for analytic surfaces in the 1820s; Korn (1914) and Lichtenstein (1916) proved their existence for Hölder metrics, and Chern gave a short proof in 1955. Einstein derived gravitational waves in 1916–18, de Donder introduced harmonic coordinates in 1921 and Lanczos used them in 1922. Myers and Steenrod proved that isometries are smooth in 1939. Choquet-Bruhat's local existence theorem for Einstein's equations appeared in 1952, and Morrey's analyticity theorem for nonlinear elliptic systems in 1958. Calabi and Hartman studied the regularity of isometries in 1970. DeTurck's trick appeared in 1983, two years after DeTurck and Kazdan's regularity paper (1981). Jost and Karcher's harmonic radius estimate dates from 1982, Greene and Wu's $C^{1,\alpha}$ compactness from 1988, and Anderson's Ricci-bounded version from 1990. Hulse and Taylor discovered the binary pulsar in 1974. Pretorius's binary black hole simulation appeared in 2005 and the first direct detection of gravitational waves in 2015.

10.8Exercises

Solution

(a) $\partial_i\log\det g = \operatorname{tr}(g^{-1}\partial_ig) = g^{kl}\partial_ig_{kl}$, and $\Gamma^k_{ik} = \frac12g^{kl}(\partial_ig_{kl} + \partial_kg_{il} - \partial_lg_{ik}) = \frac12g^{kl}\partial_ig_{kl}$, since the last two terms cancel under the symmetric $g^{kl}$. (b) $\frac{1}{\sqrt{|g|}}\partial_i(\sqrt{|g|}g^{ik}) = \partial_ig^{ik} + g^{ik}\Gamma^l_{il} = -g^{ip}g^{kq}\partial_ig_{pq} + \frac12g^{ik}g^{pq}\partial_ig_{pq}$. Meanwhile $\Gamma^k = g^{pq}\Gamma^k_{pq} = g^{pq}g^{kl}(\partial_pg_{ql} - \frac12\partial_lg_{pq})$, and relabelling shows the two agree with opposite signs. (c) $\Delta_gu = \frac{1}{\sqrt{|g|}}\partial_i(\sqrt{|g|}g^{ij})\partial_ju + g^{ij}\partial_i\partial_ju = -\Gamma^j\partial_ju + g^{ij}\partial_i\partial_ju$. For $u = x^k$ the second derivatives vanish and $\partial_jx^k = \delta^k_j$.

Solution

The nonzero Christoffel symbols are $\Gamma^r_{\theta\theta} = -r$ and $\Gamma^\theta_{r\theta} = \Gamma^\theta_{\theta r} = \frac1r$. With $g^{rr} = 1$, $g^{\theta\theta} = r^{-2}$: $\Gamma^r = g^{\theta\theta}\Gamma^r_{\theta\theta} = -\frac1r$ and $\Gamma^\theta = 2g^{r\theta}\Gamma^\theta_{r\theta} = 0$. So $\Delta r = \frac1r$ and $\Delta\theta = 0$, as the formula $\Delta = \partial_r^2 + \frac1r\partial_r + \frac1{r^2}\partial_\theta^2$ confirms. $(r, \theta)$ is not harmonic. $\Delta\log r = -\frac{1}{r^2} + \frac{1}{r^2} = 0$, so $(\log r, \theta)$ is harmonic; in it the metric is $e^{2s}(ds^2 + d\theta^2)$ with $s = \log r$, isothermal coordinates (the complex logarithm).

Solution

$g_{ij} = e^{2\varphi}\delta_{ij}$, $\partial_pg_{ql} = 2e^{2\varphi}\partial_p\varphi\,\delta_{ql}$, and $g^{kl} = e^{-2\varphi}\delta_{kl}$, so $\Gamma^k_{pq} = \frac12e^{-2\varphi}\cdot2e^{2\varphi}(\partial_p\varphi\delta_{qk} + \partial_q\varphi\delta_{pk} - \partial_k\varphi\delta_{pq})$. Contracting with $g^{pq} = e^{-2\varphi}\delta^{pq}$: $\Gamma^k = e^{-2\varphi}(\partial_k\varphi + \partial_k\varphi - n\partial_k\varphi) = (2 - n)e^{-2\varphi}\partial_k\varphi$. This vanishes for all $\varphi$ exactly when $n = 2$; for $n \geq 3$ only when $\varphi$ is constant.

Solution

(a) $g_{ij}$ is a sum of products of first derivatives of $\psi$, which are $C^{1,\alpha}$; its derivatives involve second derivatives of $\psi$, which are only $C^\alpha$ in general, so $g \notin C^2$ unless they happen to be. (b) By the integration-by-parts characterisation, $\Delta_g(f\circ\psi) = (\Delta_\delta f)\circ\psi$ for an isometry $\psi$. With $f = x^k$, $\Delta_\delta x^k = 0$, so $\Delta_g\psi^k = 0$. (c) In the coordinates $y = \psi(x)$, by the transformation law, the metric is $\delta$. This is the coordinate dependence of regularity in its purest form: the roughness of $g$ was entirely in the choice of coordinates, and harmonic coordinates removed it, as Theorem 10.3 predicts ($\operatorname{Ric} = 0$).

Solution

(a) At $g = \delta$, $\Gamma_k = g^{pq}(\partial_pg_{qk} - \frac12\partial_kg_{pq})$ linearises to $\partial_ph_{pk} - \frac12\partial_kh_{pp}$. (b) Replacing $\partial$ by $2\pi i\xi$ and dropping the common factor: $\hat h\xi - \frac12(\operatorname{tr}\hat h)\xi = 0$. (c) Substituting $\hat h\xi = \frac12(\operatorname{tr}\hat h)\xi$: $\xi(\hat h\xi)^T + (\hat h\xi)\xi^T = (\operatorname{tr}\hat h)\xi\xi^T$, which cancels the second term, leaving $|\xi|^2\hat h$. (d) For $\hat h = \xi v^T + v\xi^T$: $\hat h\xi = |\xi|^2v + (v\cdot\xi)\xi$ and $\frac12(\operatorname{tr}\hat h)\xi = (v\cdot\xi)\xi$, so the condition gives $|\xi|^2v = 0$, $v = 0$ for $\xi \neq 0$. The gauge condition is transverse to the gauge directions, so on the remaining matrices the Ricci operator is elliptic.

Solution

$\Delta_{\lambda^2g} = \lambda^{-2}\Delta_g$, so $\lambda u^k$ is harmonic for $\lambda^2g$. In the coordinates $y = \lambda u$, the metric $\lambda^2g$ has components $\lambda^2g_{ij}\frac{\partial u^i}{\partial y^a}\frac{\partial u^j}{\partial y^b} = g_{ab}(u)$ evaluated at $u = y/\lambda$: the same matrix function, rescaled in its argument. So the bounds $Q^{-1}\delta \leq g \leq Q\delta$ hold unchanged, and $\partial_y = \lambda^{-1}\partial_u$ gives $\sup|\partial_yg| = \lambda^{-1}\sup|\partial_ug|$ and $[\partial_yg]_\alpha = \lambda^{-1-\alpha}[\partial_ug]_\alpha$, exactly compensated by the factors $(\lambda r)$ and $(\lambda r)^{1+\alpha}$. Hence $r_h(\lambda^2g) = \lambda r_h(g)$. If $|\operatorname{Ric}(g)| \leq \Lambda$, then $\lambda^2g$ with $\lambda^2 = \Lambda$ has $|\operatorname{Ric}| \leq 1$ (Ricci is scale invariant as a $(0, 2)$-tensor, and $|\cdot|$ is measured with the metric, which divides by $\lambda^2$), so $r_h(\lambda^2g) \geq c$ and $r_h(g) \geq c\Lambda^{-1/2}$.