Prescribing geodesics and a variational problem for Riemannian metrics

Abstract

Given a prescription of unparametrised paths on a manifold \(M\), one path for each tangent direction, we may ask whether these paths agree with the geodesics of a Riemannian metric on \(M\). Generically, this is not the case. Motivated by this fact, we introduce a non-negative functional \(\mathcal{E}\) on the space of Riemannian metrics on \(M\) so that \(\mathcal{E}(g)=0\) if and only if the geodesics of the metric \(g\) agree with the prescribed paths. We compute the variational equations for \(\mathcal{E}\) and show that the conformal variational equation is, perhaps surprisingly, of Yamabe type. This allows us to obtain existence results for conformally critical points of \(\mathcal{E}\). In particular, in the surface case, every conformal class contains a conformally critical metric, unique up to homothety. As a by-product, we establish that the Blaschke metric of a properly convex projective surface is a critical point for \(\mathcal{E}\).

1 Introduction

A prescription of unparametrised paths on a manifold \(M\), one path for each tangent direction, specifies the geometric shape of a family of curves but not the speed at which one travels along them. It is natural to require these paths to be the geodesics of a torsion-free connection on \(TM\), since the geodesics of a Riemannian metric are precisely those of its Levi-Civita connection. Such a prescription is encoded by a projective structure: an equivalence class \(\mathfrak{p}\) of torsion-free connections on \(TM\), where two such connections are equivalent if they have the same geodesics up to parametrisation. Here by a geodesic for \(\nabla \in \mathfrak{p}\) we mean an immersed curve \(\gamma : I \to M\) so that \(\nabla_{\dot{\gamma}}\dot{\gamma}\) vanishes identically. A projective structure \(\mathfrak{p}\) is called metrisable if there exists a Riemannian metric \(g\) on \(M\) whose Levi-Civita connection \({}^g\nabla\) is an element of \(\mathfrak{p}\). A generic projective structure is not metrisable, and necessary and sufficient conditions for metrisability of a two-dimensional projective structure were obtained by Bryant, Dunajski and Eastwood [MR2581355].

A manifold \(M\) equipped with a projective structure will be called a projective manifold. For a closed oriented projective manifold \((M,\mathfrak{p})\) of dimension \(n\geqslant 2\), we introduce a functional \(\mathcal{E}\) on the space \(\mathrm{Riem}(M)\) of Riemannian metrics on \(M\) as follows. Given a representative \(\nabla \in \mathfrak{p}\), the difference \(\nabla-{}^g\nabla\) is a \(1\)-form on \(M\) with values in the endomorphisms of \(TM\); we write \((\cdot)_0\) for the part of such a \(1\)-form taking values in the trace-free endomorphisms of \(TM\) and define \[\mathcal{E} : \mathrm{Riem}(M) \to \mathbb{R}, \qquad g \mapsto \operatorname{Vol}_{M}(g)^{(2-n)/n}\int_{M} |(\nabla-{}^g\nabla)_0|^2_g d\mu_g,\] where \(d\mu_g\) denotes the volume form of \(g\), \(\operatorname{Vol}_M(g)\) the volume of \(M\) with respect to \(d\mu_g\) and \(|\,\cdot\,|_g\) the point-wise norm induced by \(g\) on the fibres of \(S^2(T^*M)\otimes TM\). It turns out that the non-negative functional \(\mathcal{E}\) depends only on the projective equivalence class of \(\nabla\) and, moreover, that \(\mathcal{E}(g)=0\) for \(g \in \mathrm{Riem}(M)\) if and only if \(\mathfrak{p}\) is metrisable by the metric \(g\). The presence of the factor \(\operatorname{Vol}_{M}(g)^{(2-n)/n}\) makes \(\mathcal{E}\) invariant under homothety, that is, \(\mathcal{E}(cg)=\mathcal{E}(g)\) for all positive constants \(c\) and all metrics \(g \in \mathrm{Riem}(M)\).

Setting \(\Phi_g:=\left(\nabla-{}^g\nabla\right)_0\), the variational equations for \(\mathcal{E}\) take the form (see Proposition 3.1 for details) \[T_g(\Phi_g)-2\ell^*_g(\Phi_g)=kg,\] where \(k\) is a constant and \(T_g(\Phi_g) \in \Gamma(S^2(T^*M))\) is a purely algebraic term in \(\Phi_g\). The map \(\ell^*_g\) is the formal adjoint of a first-order differential operator \(\ell_g\) obtained from the linearisation of the map \(g\mapsto{}^g\nabla\), which assigns to a metric its Levi-Civita connection.

An important class of (in general) non-metrisable projective surfaces carrying a distinguished metric – the so-called Blaschke metric – is given by the properly convex projective surfaces. They are of particular interest due to their connection to higher Teichmüller theory. For background we refer to the landmark paper by Hitchin [MR1174252]; see also [MR3966798] for a recent survey. Given a Riemann surface structure on \(M\), every holomorphic cubic differential \(C\) determines a Riemannian metric \(g\) – the Blaschke metric – via Hitchin’s self-duality equations [MR887284], and the pair \((g,C)\) encodes a properly convex projective structure on \(M\) [MR1828223, MR2402597]. Alternatively, the Blaschke metric arises via a hyperbolic affine sphere over the universal cover of \(M\), obtained from a Monge–Ampère equation solved by Cheng–Yau [MR0437805, MR859275]; see [MR2743442] for a survey. Our first main result complements these pictures by relating the Blaschke metric to a variational principle intrinsic to the projective structure:

mainthmmaintheoremone Let \((M,\mathfrak{p})\) be a closed oriented properly convex projective surface of negative Euler characteristic. Then the Blaschke metric of \(\mathfrak{p}\) is a critical point for \(\mathcal{E}\).

We now turn to conformal variations of \(\mathcal{E}\). In two dimensions, solving the variational equations for conformal variations is equivalent to solving a Poisson equation, and this allows us to prove:

mainthmmaintheoremtwo Let \((M,\mathfrak{p})\) be a closed oriented projective surface. Then every conformal structure on \(M\) contains a conformally critical metric, unique up to homothety.

A surprising aspect of \(\mathcal{E}\) is its close connection to the Yamabe equation [MR125546] and to the Einstein–Hilbert functional, which assigns to a metric \(g\) its volume-normalised total scalar curvature \(\operatorname{Vol}_{M}(g)^{(2-n)/n}\int_{M}S_g d\mu_g\). To explain this connection, we use [MR4201191], which shows that the choice of a conformal structure \([g]\) on a projective manifold \((M,\mathfrak{p})\) distinguishes a unique torsion-free connection \({}^{[g]}\nabla\) preserving \([g]\) – a so-called Weyl connection – and a \(1\)-form \(A_{[g]}\) on \(M\) with values in the endomorphisms of \(TM\) so that \({}^{[g]}\nabla+A_{[g]}\) lies in \(\mathfrak{p}\).

For a metric \(g\) on \((M,\mathfrak{p})\), we define the conformal defect as \[V_g=\tfrac{(n+1)(n-2)}{(n+2)}|A_{[g]}|^2_g-\operatorname{tr}_g\operatorname{Ric}\left({}^{[g]}\nabla\right),\] where \(\operatorname{Ric}\left({}^{[g]}\nabla\right)\) denotes the Ricci curvature of \({}^{[g]}\nabla\). The motivation for introducing \(V_g\) is its relevance in finding conformally critical metrics for \(\mathcal{E}\). Recall that, for \(n\geqslant 3\) the conformal Laplacian of \(g\) is defined as \[L_g:=4\left(\frac{n-1}{n-2}\right)\Delta_g + S_g.\] Using \(V_g\), we define the projective-conformal Laplacian \(\mathcal{L}_g=L_g+V_g\) and show that finding a metric in \([g]\) that is conformally critical for \(\mathcal{E}\) amounts to solving an equation of Yamabe type \[\mathcal{L}_g u=C u^{(n+2)/(n-2)}\] for some constant \(C\) and smooth positive function \(u\) on \(M\).

Under a conformal change \(g \mapsto \tilde{g}=\exp(2f)g\), the conformal defect changes as \(V_{\tilde{g}}=\exp(-2f)V_g\). As a consequence of this, it follows that the projective scalar curvature \[\mathcal{S}_g:=S_g+V_g\] transforms exactly as the scalar curvature under a conformal change \(g\mapsto \tilde{g}\). Furthermore, we have the identity \[\mathcal{E}(g)=\tfrac{n+2}{(n+1)(n-2)}\operatorname{Vol}_{M}(g)^{(2-n)/n}\int_{M}\mathcal{S}_g d\mu_g,\] showing that our functional closely resembles the Einstein–Hilbert functional. We define the conformal locus of \([g]\) as \(\Sigma_{[g]}:=\left\{p \in M\,|\, V_g(p)=0\right\}\). Note that \(\Sigma_{[g]}\) is a conformal invariant of \([g]\). Building on the solution [MR240748, MR0433500, MR788292] of the Yamabe problem [MR125546] and the theory of elliptic equations of Yamabe type (see in particular [MR2148873]), we obtain:

mainthmmaintheoremthree Let \((M,\mathfrak{p})\) be a closed oriented projective manifold of dimension \(n\geqslant 3\) and \([g]\) a conformal structure on \(M\). Suppose that either \(\Sigma_{[g]}=M\), or that \(n\geqslant 4\) and there exists \(p_0\in M\) with \(V_{\tilde{g}}(p_0)<0\) for some (and hence any) \(\tilde{g}\in [g]\). Then \([g]\) contains a conformally critical metric for \(\mathcal{E}\).

Drawing further on the analogy with the Yamabe equation, it is tempting to speculate that on a closed oriented projective manifold \((M,\mathfrak{p})\), every conformal equivalence class \([g]\) contains a conformally critical metric for \(\mathcal{E}\). Some of the cases left open by [thm:maintheoremthree] appear to be within reach of the localisation theory for elliptic equations of Yamabe type developed by Druet and Hebey [MR2148873]. Others – most notably the borderline regime in which \(V_g\geqslant 0\) with \(\Sigma_{[g]}\neq\emptyset\) – parallel the situation in which the classical Yamabe problem was settled by Schoen [MR788292] via the positive mass theorem of Schoen and Yau [MR526976, MR612249]. We hope to return to these unresolved cases in future work.

Acknowledgements

The author is grateful to Rod Gover, Luca Giudici and Tobias Weth for helpful conversations.

2 Setup

2.1 Projective structures

Let \(M\) be a closed oriented smooth \(n\)-manifold with \(n\geqslant 2\). Recall that the difference of two torsion-free connections \(\nabla\) and \(\nabla^{\prime}\) on \(TM\) is a section of the vector bundle \(E=S^2(T^*M)\otimes TM\). For a section \(\Phi=(\Phi^i_{jk})\) of \(E\) we define \(\mathop{\mathrm{Con}}(\Phi) \in \Omega^1\) by the pointwise tensor contraction \[[\mathop{\mathrm{Con}}(\Phi)]_i=\Phi^k_{ik}.\] For \(\alpha=(\alpha_i) \in\Omega^1\) we obtain a section \(\operatorname{Sym}(\alpha)\) of \(E\) by the rule \[\left[\operatorname{Sym}(\alpha)\right]^i_{jk}=\delta^i_j\alpha_k+\delta^i_k\alpha_j.\] Observe that \(\mathop{\mathrm{Con}}(\operatorname{Sym}(\alpha))=(n+1)\alpha\).

For a section \(\Phi\) of \(E\) we denote by \(\Phi_0\) its part in the kernel of the above contraction, that is, \[ \Phi_0:=\Phi-\tfrac{1}{n+1}\operatorname{Sym}\left(\mathop{\mathrm{Con}}(\Phi)\right).\](2.1) In particular, we have \[\left(\operatorname{Sym}(\alpha)\right)_0=\operatorname{Sym}(\alpha)-\tfrac{1}{(n+1)}\operatorname{Sym}(\mathop{\mathrm{Con}}(\operatorname{Sym}(\alpha)))=\operatorname{Sym}(\alpha)-\operatorname{Sym}(\alpha)=0\] for all \(\alpha \in\Omega^1\), so that \[E\simeq E_0\oplus T^*M,\] where \(E_0\) denotes the contraction-free part of \(E\).

Definition 2.1. Two torsion-free connections \(\nabla\) and \(\nabla^{\prime}\) on \(TM\) are called projectively equivalent if every geodesic of \(\nabla\) can be reparametrised to become a geodesic of \(\nabla^{\prime}\). A projective structure on \(M\) is an equivalence class \(\mathfrak{p}\) of projectively equivalent torsion-free connections on \(TM\). A manifold \(M\) equipped with a projective structure \(\mathfrak{p}\) will be called a projective manifold.

Remark 2.2. By a geodesic of \(\nabla\) we mean an immersed curve \(\gamma : I \to M\) which satisfies \(\nabla_{\dot\gamma}\dot\gamma=0\).

We have – see [MR532831] for a modern reference:

Lemma 2.3 (Cartan, Eisenhart, Weyl). Two torsion-free connections \(\nabla\) and \(\nabla^{\prime}\) on \(TM\) are projectively equivalent if and only if \((\nabla-\nabla^{\prime})_0=0\).

2.2 A choice of metric

Besides a projective structure \(\mathfrak{p}\), we now also fix a Riemannian metric \(g\) on \(M\). The metric \(g=(g_{ij})\) equips each real tensor bundle \(E \to M\) with a bundle metric. Abusing notation, we denote this bundle metric on all the various tensor bundles by \(\langle\cdot,\cdot\rangle_g\). For instance, if \(\Phi=(\Phi^i_{jk})\) and \(\Psi=(\Psi^i_{jk})\) are sections of \(S^2(T^*M)\otimes TM\), then \[\langle\Phi,\Psi\rangle_g=\Phi^i_{jk}g^{jb}g^{kc}g_{ia}\Psi^a_{bc},\] where we write \(g^{\sharp}=(g^{ij})\) for the dual metric. Correspondingly, we obtain an inner product on the set \(\Gamma(E)\) of smooth sections of \(E\) by defining \[\langle\!\langle s_1,s_2\rangle\!\rangle_g=\int_M\langle s_1,s_2\rangle_g d\mu_g,\] where \(d\mu_g\) denotes the volume form of \(g\) and \(s_1,s_2\in \Gamma(E)\). As usual, we write \(|s|_g:=\sqrt{\langle s,s\rangle_g}\) and likewise \(\Vert s\Vert_g:=\sqrt{\langle\!\langle s,s\rangle\!\rangle_g}\) for all \(s \in \Gamma(E)\). For what follows, we record the identities \[ |\Phi|^2_{\exp(2f)g}=\mathrm{e}^{-2f}|\Phi|^2_g\](2.2) and \[ d\mu_{\exp(2f)g}=\mathrm{e}^{nf}d\mu_g,\](2.3) where \(f \in C^{\infty}\) and \(\Phi \in \Gamma(S^2(T^*M)\otimes TM)\).

The bundle \(E_0\) with \(E=S^2(T^*M)\otimes TM\) is irreducible as a \(\mathrm{GL}^+(n,\mathbb{R})\)-bundle, it is however not irreducible as a \(\mathrm{SO}(n)\)-bundle. Consider the metric trace \[\operatorname{tr}_g : \Gamma(E_0) \to \Gamma(TM), \quad \Phi^i_{jk} \mapsto g^{jk}\Phi^i_{jk}\] and the inclusion \[\iota_g : \Gamma(TM) \to \Gamma(E_0),\quad X \mapsto (g\otimes X)_0\] which satisfy \[\operatorname{tr}_g \circ \iota_g = \tfrac{(n+2)(n-1)}{(n+1)}\mathrm{Id}_{\Gamma(TM)}.\] Consequently, we can decompose \(\Phi \in \Gamma(E_0)\) as \[ X=\frac{1}{m} \operatorname{tr}_g \Phi\quad \text{and} \quad A=\Phi-(g\otimes X)_0,\](2.4) where \(m=\tfrac{(n+2)(n-1)}{(n+1)}\). By construction, \(\operatorname{tr}_gA\) vanishes identically. Furthermore, a simple computation shows that this decomposition is orthogonal. That is, if \(Z\) is a vector field on \(M\) and \(\Phi\) a section of \(E_0\) which satisfies \(\operatorname{tr}_g\Phi\equiv 0\), then \(\langle (g\otimes Z)_0,\Phi\rangle_g\) vanishes identically as well. Moreover, for the decomposition (2.4), we have \[ |\Phi|^2_g=|A|^2_g+m|X|^2_g,\](2.5) since \(|(g\otimes Y)_0|^2_g=m|Y|^2_g\) for every vector field \(Y\) on \(M\).

Remark 2.4. Also observe that for \(\Psi \in \Gamma(E_0)\) and a \(1\)-form \(\alpha\) on \(M\) we have \[ \langle \Psi,\operatorname{Sym}(\alpha)\rangle_g=0,%\Phi^i_{jk}g^{jb}g^{kc}g_{ia}(\delta^a_b\alpha_c+\delta^a_c\alpha_b)=\Phi^i_{jk}(\delta^j_ig^{kc}\alpha_c+\delta^k_ig^{jb}\alpha_b)=0.\](2.6) as can be verified with a simple computation.

2.3 A projectively invariant functional

Let \(\mathrm{Riem}(M)\) denote the set of Riemannian metrics on \(M\). For a projective structure \(\mathfrak{p}\) and orientation on \(M\) we consider the following projectively invariant non-negative functional \[\mathcal{E} : \mathrm{Riem}(M) \to \mathbb{R}, \qquad g \mapsto \operatorname{Vol}_{M}(g)^{(2-n)/n}\int_{M} |(\nabla-{}^g\nabla)_0|^2_g d\mu_g,\] where \(\nabla \in \mathfrak{p}\) and \({}^g\nabla\) denotes the Levi-Civita connection of \(g\). Note that the functional attains the value \(0\) if and only if there exists a Riemannian metric \(g\) whose Levi-Civita connection belongs to \(\mathfrak{p}\). Also observe that if \(\hat{\mathcal{F}} : \mathrm{Riem}(M) \to \mathbb{R}\) is a functional on the set of Riemannian metrics for which there exists a constant \(k\) such that \(\hat{\mathcal{F}}(\mathrm{e}^{2c}g)=\mathrm{e}^{kc}\hat{\mathcal{F}}(g)\) for all constants \(c\) and metrics \(g\), then the functional \(\mathcal{F}\) defined by \(\mathcal{F}(g)=\operatorname{Vol}_{M}(g)^{-k/n}\hat{\mathcal{F}}(g)\) is invariant under rescaling a metric by a positive constant. Since rescaling a metric by a positive constant does not change its Levi-Civita connection, (2.2) and (2.3) imply that \[\mathcal{E}(cg)=\mathcal{E}(g)\] for all positive constants \(c\) and all metrics \(g \in \mathrm{Riem}(M)\).

Remark 2.5 (Notation). The functional \(\mathcal{E}\) depends on a choice of projective structure \(\mathfrak{p}\) on \(M\), as do a number of other objects we shall define. To keep the notation uncluttered, we leave this dependency implicit throughout.

3 Variational equations

Let \(V\) be a Fréchet space and \(U\subset V\) an open subset. The partial derivative of a function \(\mathcal{F} : U \to \mathbb{R}\) at \(g \in U\) in the direction of \(h \in V\) is defined by \[\mathcal{F}^{\prime}_g(h):=\lim_{t\to 0}\frac{1}{t}(\mathcal{F}(g+th)-\mathcal{F}(g)),\] provided the limit exists. The set \(\Gamma(S^2(T^*M))\) is a Fréchet space and the set of Riemannian metrics on \(M\) is an open subset thereof, hence \(\mathrm{Riem}(M)\) is a Fréchet manifold whose tangent space at \(g\) is canonically isomorphic to \(\Gamma(S^2(T^*M))\).

We say a Riemannian metric \(g\) on \(M\) is critical for \(\mathcal{E}\) provided \(\mathcal{E}^{\prime}_g(h)=0\) for all \(h \in T_g\mathrm{Riem}(M)\simeq \Gamma(S^2(T^*M))\). Likewise, we say a metric \(g\) on \(M\) is conformally critical for \(\mathcal{E}\), provided \(\mathcal{E}^{\prime}_g(h)=0\) for all \(h\) of the form \(h=fg\) with \(f \in C^{\infty}\).

For a Riemannian metric \(g\) on \(M\), \(h \in \Gamma(S^2(T^*M))\) and \(t \in \mathbb{R}\), we write \(g_t=g+th\) as well as \[\Phi_{g_t}:=\left(\nabla-{}^{g_t}\nabla\right)_0 \qquad \text{and} \qquad \Phi_g:=\Phi_{g_0}.\] Let \(\Gamma\) denote the map which assigns to a Riemannian metric \(g\) its Levi-Civita connection and let \(\Gamma^{\prime}_g : \Gamma(S^2(T^*M)) \to \Gamma(E)\) denote its linearisation at \(g \in \mathrm{Riem}(M)\) \[\Gamma^{\prime}_g(h)=\lim_{t\to 0}\frac{1}{t}\left({}^{g_t}\nabla-{}^g\nabla\right).\] Using \(\Gamma^{\prime}_g\) we obtain a linear first order differential operator \[\ell_g : \Gamma(S^2(T^*M)) \to \Gamma(E_0), \quad h \mapsto \left(\Gamma^{\prime}_g(h)\right)_0.\] We let \[\ell^*_g : \Gamma(E_0) \to \Gamma(S^2(T^*M))\] denote the formal adjoint satisfying \[ \int_M\langle \Psi,\ell_g(h)\rangle_g d\mu_g=\int_M\langle \ell^*_g(\Psi),h\rangle_g d\mu_g\](3.1) for all \(h \in \Gamma(S^2(T^*M))\) and all \(\Psi \in \Gamma(E_0)\).

Furthermore, for a section \(\Psi\) of \(E_0\) we let \(T_g(\Psi)\) denote the unique symmetric covariant \(2\)-tensor field so that \[\left.\tfrac{\partial}{\partial t}\right|_{t=0}|\Psi|^2_{g_t}d\mu_{g_t}=\langle T_g(\Psi),h\rangle_g d\mu_g\] for all \(h \in \Gamma(S^2(T^*M))\). In analogy with the physics literature, we call \(T_g(\Psi)\) the stress-energy tensor of \(\Psi\). The integrand \(|\Phi_{g_t}|^2_{g_t}d\mu_{g_t}\) depends on \(t\) through two distinct quantities: the section \(\Phi_{g_t}\) and the metric pairing together with the volume form \(|\cdot|^2_{g_t}d\mu_{g_t}\). The chain rule therefore gives \[ \left.\tfrac{\partial}{\partial t}\right|_{t=0}|\Phi_{g_t}|^2_{g_t}d\mu_{g_t}=\left.\tfrac{\partial}{\partial t}\right|_{t=0}|\Phi_g|^2_{g_t}d\mu_{g_t}+\left.\tfrac{\partial}{\partial t}\right|_{t=0}|\Phi_{g_t}|^2_g d\mu_g,\](3.2) where in the first term on the right the section is frozen at \(\Phi_g\), and in the second the pairing and volume are frozen at \(g\). By the definition of the stress-energy tensor, the first term equals \(\langle T_g(\Phi_g),h\rangle_g d\mu_g\). By the Leibniz rule applied to the inner product, the second term equals \(2\langle \Phi_g,\left.\tfrac{\partial}{\partial t}\right|_{t=0}\Phi_{g_t}\rangle_g d\mu_g\). We thus obtain \[ \left.\tfrac{\partial}{\partial t}\right|_{t=0}|\Phi_{g_t}|^2_{g_t}d\mu_{g_t}=\langle T_g(\Phi_g),h\rangle_g d\mu_g+2\langle \Phi_g,\left.\tfrac{\partial}{\partial t}\right|_{t=0}\Phi_{g_t}\rangle_g d\mu_g.\](3.3) Writing \[\Phi_{g_t}=(\nabla-{}^g\nabla-({}^{g_t}\nabla-{}^g\nabla))_0,\] we observe that the difference \(\nabla-{}^g\nabla\) is independent of \(t\). Setting \(H:=\Gamma^{\prime}_g(h)\), the definition of \(\Gamma^{\prime}_g\) gives \[\left.\tfrac{\partial}{\partial t}\right|_{t=0}({}^{g_t}\nabla-{}^g\nabla)=H,\] and since \(\ell_g(h)=H_0\) by definition of \(\ell_g\), we conclude \[\left.\tfrac{\partial}{\partial t}\right|_{t=0}\Phi_{g_t}=-H_0=-\ell_g(h).\] Substituting into (3.3) thus yields \[ \left.\tfrac{\partial}{\partial t}\right|_{t=0}|\Phi_{g_t}|^2_{g_t}d\mu_{g_t}=\langle T_g(\Phi_g),h\rangle_gd\mu_g-2\langle \Phi_g,\ell_g(h)\rangle_g d\mu_g.\](3.4)

Now consider \[\hat{\mathcal{E}}(g):=\int_M|(\nabla-{}^g\nabla)_0|^2_gd\mu_g=\int_M|\Phi_g|^2_gd\mu_g=\Vert\Phi_g\Vert^2_g\] so that, interchanging \(\left.\tfrac{\partial}{\partial t}\right|_{t=0}\) with \(\int_M\) and integrating (3.4) over \(M\), the first two equalities below follow; the third uses the defining property (3.1) of \(\ell^*_g\) to push \(\ell_g\) across the inner product: \[\begin{aligned} \hat{\mathcal{E}}^{\prime}_g(h)=\left.\tfrac{\partial}{\partial t}\right|_{t=0}\int_M|\Phi_{g_t}|^2_{g_t}d\mu_{g_t}&=\int_M\langle T_g(\Phi_g),h\rangle_g-2\langle \Phi_g,\ell_g(h)\rangle_g d\mu_g\\ &=\int_M\langle T_g(\Phi_g)-2\ell^*_g(\Phi_g),h\rangle_g d\mu_g. \end{aligned}\]

Thinking of the volume of \(M\) with respect to a metric as a functional \(\operatorname{Vol}_{M} : \mathrm{Riem}(M) \to \mathbb{R}\), we have the standard identity \[(\operatorname{Vol}_{M})^{\prime}_g(h)=\frac{1}{2}\int_M(\operatorname{tr}_g\! h)d\mu_g=\frac{1}{2}\int_M\langle g,h\rangle_gd\mu_g.\] Since \(\mathcal{E}(g)=\operatorname{Vol}_{M}(g)^{(2-n)/n}\hat{\mathcal{E}}(g)\), the product rule gives \[\mathcal{E}^{\prime}_g(h)=\operatorname{Vol}_{M}(g)^{(2-n)/n}\hat{\mathcal{E}}^{\prime}_g(h)+\tfrac{2-n}{n}\operatorname{Vol}_{M}(g)^{(2-n)/n-1}\hat{\mathcal{E}}(g)\,(\operatorname{Vol}_M)^{\prime}_g(h).\] Substituting the formula for \((\operatorname{Vol}_M)^{\prime}_g(h)\) above and using \(\hat{\mathcal{E}}(g)=\Vert\Phi_g\Vert^2_g\), we obtain \[\mathcal{E}^{\prime}_g(h)=\operatorname{Vol}_{M}(g)^{(2-n)/n}\left[\hat{\mathcal{E}}^{\prime}_g(h)-\left(\frac{n-2}{2n}\right)\frac{\Vert\Phi_g\Vert^2_g}{\operatorname{Vol}_{M}(g)}\int_M \langle g,h\rangle_g d\mu_g\right].\] Combining this with the expression for \(\hat{\mathcal{E}}^{\prime}_g(h)\) derived above, and setting \(k:=\left(\frac{n-2}{2n}\right)\frac{\Vert\Phi_g\Vert^2_g}{\operatorname{Vol}_{M}(g)}\), we conclude \[ \mathcal{E}^{\prime}_g(h)=\operatorname{Vol}_{M}(g)^{(2-n)/n}\int_M\langle T_g(\Phi_g)-2\ell^*_g(\Phi_g)-kg,\,h\rangle_g d\mu_g.\](3.5) By the fundamental lemma of the calculus of variations, \(\mathcal{E}^{\prime}_g(h)=0\) for all \(h \in \Gamma(S^2(T^*M))\) if and only if the symmetric tensor \(T_g(\Phi_g)-2\ell^*_g(\Phi_g)-kg\) vanishes identically. We thus have:

Proposition 3.1. A Riemannian metric \(g\) is critical for \(\mathcal{E}\) if and only if \[T_g(\Phi_g)-2\ell^*_g(\Phi_g)=kg,\] where the constant \(k\) is given by \(k=\left(\frac{n-2}{2n}\right)\frac{\Vert\Phi_g\Vert^2_g}{\operatorname{Vol}_{M}(g)}\).

We also need:

Lemma 3.2. For \(\Psi\in \Gamma(E_0)\), the stress-energy tensor \(T_g(\Psi) \in \Gamma(S^2(T^*M))\) is given by \[T_g(\Psi)=\frac{1}{2}|\Psi|^2_gg+\Psi\circledast_g\!\Psi,\] where \[(\Psi\circledast_g\!\Psi)_{ij}=\Psi^k_{uv}\Psi^a_{bc}g_{ia}g_{jk}g^{ub}g^{vc}-2\Psi^k_{jc}\Psi^a_{ib}g_{ak}g^{bc}.\] In particular \[ \operatorname{tr}_g\left(T_g(\Psi)\right)=\left(\tfrac{n}{2}-1\right)|\Psi|^2_g.\](3.6)

Proof. Recall the standard identities \[\left.\tfrac{\partial}{\partial t}\right|_{t=0}d\mu_{g_t}=\frac{1}{2}(\operatorname{tr}_g\!h) d\mu_g\qquad \text{and}\qquad \left.\tfrac{\partial}{\partial t}\right|_{t=0}\left(g_t^{\sharp}\right)^{ij}=-h^{ij}\] where we raise indices of \(h\) with the metric \(g\). Then \[\begin{gathered} |\Psi|^2_{g_t}d \mu_{g_t}=\Psi^k_{ij}\Psi^a_{bc}\left(g_{ak}+th_{ak}\right) \left(g^{ib}-th^{ib}+\mathcal{O}(t^2)\right)\\\left(g^{jc}-th^{jc}+\mathcal{O}(t^2)\right)\left(1+\frac{1}{2}tg_{pq}h^{pq}+\mathcal{O}(t^2)\right)d\mu_{g}. \end{gathered}\] From this we calculate \[\begin{gathered} \left.\tfrac{\partial}{\partial t}\right|_{t=0}|\Psi|^2_{g_t}d\mu_{g_t}=\Big[\Psi^k_{uv}\Psi^a_{bc}g_{ia}g_{jk}g^{ub}g^{vc}-2\Psi^k_{jc}\Psi^a_{ib}g_{ak}g^{bc}\Big.\\\Big.+\tfrac{1}{2}|\Psi|^2_gg_{ij}\Big]h^{ij}d\mu_g, \end{gathered}\] so that \[\left.\tfrac{\partial}{\partial t}\right|_{t=0}|\Psi|^2_{g_t}d\mu_{g_t}=\langle T_g(\Psi),h\rangle_g d\mu_g,\] as claimed. The identity (3.6) follows from an elementary computation which we omit. ◻

Lemma 3.3. For \(g \in \mathrm{Riem}(M)\) the operator \[\ell^*_g : \Gamma(E_0) \to \Gamma(S^2(T^*M))\] is given by \[\Psi^i_{jk} \mapsto \tfrac{1}{2}\left({}^g\nabla_k\Psi^k_{ij}-g^{uv}g_{ik}{}^g\nabla_u\Psi^k_{jv}-g^{uv}g_{jk}{}^g\nabla_u\Psi^k_{iv}\right).\]

Proof. We need to show that for \(g \in \mathrm{Riem}(M)\), \(h \in \Gamma(S^2(T^*M))\) and \(\Psi \in \Gamma(E_0)\), we have \[\int_M\langle \Psi,\ell_g(h)\rangle_g d\mu_g=\int_M\langle \ell^*_g(\Psi),h\rangle_g d\mu_g.\] With our notation \(H=\Gamma^{\prime}_g(h)\), an elementary computation gives \[H^k_{ij}=\frac{1}{2}g^{kl}\left({}^g\nabla_ih_{jl}+{}^g\nabla_j h_{il}-{}^g\nabla_lh_{ij}\right).\] Recall from (2.1) that \[H=H_0+\tfrac{1}{(n+1)}\operatorname{Sym}(\mathop{\mathrm{Con}}(H)).\] Since \(\Psi \in \Gamma(E_0)\) and \(\ell_g(h)=H_0\), we have \[\langle \Psi,H\rangle_g=\langle \Psi,H_0\rangle_g+\tfrac{1}{(n+1)}\langle \Psi,\operatorname{Sym}(\mathop{\mathrm{Con}}(H))\rangle_g=\langle \Psi,H_0\rangle_g=\langle \Psi,\ell_g(h)\rangle_g,\] where we use (2.6). Contracting \(g_{ka}g^{al}=\delta^l_k\) gives \[\langle \Psi,H\rangle_g=\frac{1}{2}\Psi^k_{ij}g^{ib}g^{jc}\left({}^g\nabla_b h_{ck}+{}^g\nabla_c h_{bk}-{}^g\nabla_k h_{bc}\right).\] The first two terms contribute equally (swap \(i\leftrightarrow j\), \(b\leftrightarrow c\) and use \(\Psi^k_{ij}=\Psi^k_{ji}\)), so that \[\langle \Psi,H\rangle_g=\Psi^k_{ij}g^{ib}g^{jc}{}^g\nabla_b h_{ck}-\frac{1}{2}\Psi^k_{ij}g^{ib}g^{jc}{}^g\nabla_k h_{bc}.\] Consider the vector field \(Y=(Y^{l})\) on \(M\) with \[Y^{l}=g^{il}g^{jc}\Psi^k_{ij}h_{ck}.\] Since \(\int_M \operatorname{div}_g\!Y\,d\mu_g=0\) by Stokes’ theorem, we obtain \[\int_M g^{ib}g^{jc}\nabla_b(\Psi^k_{ij}h_{ck})d \mu_g=0\] and hence \[\int_Mg^{ib}g^{jc}\Psi^k_{ij}{}^g\nabla_b h_{ck}\,d\mu_g=-\int_M g^{ib}g^{jc}({}^g\nabla_b\Psi^k_{ij})\,h_{ck}\,d\mu_g.\] Like-wise we obtain \[-\frac{1}{2}\int_M\Psi^k_{ij}g^{ib}g^{jc}{}^g\nabla_k h_{bc}\,d\mu_g=\frac{1}{2}\int_M g^{ib}g^{jc}({}^g\nabla_k\Psi^k_{ij})\,h_{bc}\,d\mu_g.\] In the second integral, substituting \(h_{bc}=g_{bp}g_{cq}h^{pq}\) gives \(\frac{1}{2}({}^g\nabla_k\Psi^k_{pq})\,h^{pq}\). In the first, substituting \(h_{ck}=g_{cp}g_{kq}h^{pq}\) gives \(-g^{ub}g_{kq}({}^g\nabla_b\Psi^k_{up})\,h^{pq}\). Symmetrising in \(p,q\) and relabelling summation indices (using \(\Psi^k_{up}=\Psi^k_{pu}\)) yields \[[\ell^*_g(\Psi)]_{pq}=\frac{1}{2}\left({}^g\nabla_k\Psi^k_{pq}-g^{uv}g_{pk}{}^g\nabla_u\Psi^k_{qv}-g^{uv}g_{qk}{}^g\nabla_u\Psi^k_{pv}\right). \qedhere\] ◻

3.1 Decomposing \(\boldsymbol{\Phi_g}\)

Decomposing \(\Phi_g\) as in (2.4) \[X_g:=\frac{1}{m}\operatorname{tr}_g \Phi_g\qquad \text{and} \qquad A_{g}:=\Phi_g-(g\otimes X_g)_0\] it turns out that \(A_{g}\) depends only on the conformal equivalence class \([g]\) of \(g\). To this end let \(f \in C^{\infty}\) and consider \(\tilde{g}=\exp(2f)g\). Using the identity (see [MR867684]) \[ {}^{\tilde{g}}\nabla={}^g\nabla-g\otimes\operatorname{grad}_g\! f+\operatorname{Sym}(\mathrm{d}f),\](3.7) we obtain \[\Phi_{\tilde{g}}=(\nabla-{}^{\tilde{g}}\nabla)_0=\Phi_g+(g\otimes \operatorname{grad}_g f)_0=\Phi_g+g\otimes \operatorname{grad}_g f-\frac{1}{(n+1)}\operatorname{Sym}(\mathrm{d}f).\] where we use that \(\mathop{\mathrm{Con}}(g\otimes \operatorname{grad}_g f)=\mathrm{d}f\). Hence we compute \[ \begin{aligned} X_{\tilde{g}}&=\frac{1}{m}\operatorname{tr}_{\tilde{g}}\!\Phi_{\tilde{g}}=\mathrm{e}^{-2f}\left(X_g+\frac{n}{m}\operatorname{grad}_g f-\frac{2}{(n+1)m}\operatorname{grad}_g f\right)\\ &=\mathrm{e}^{-2f}\left(X_g+\operatorname{grad}_g f\right). \end{aligned}\](3.8) This also gives \[\begin{aligned} A_{\tilde{g}}&=\Phi_{\tilde{g}}-(\tilde{g}\otimes X_{\tilde{g}})_0\\ &=\Phi_g+g\otimes \operatorname{grad}_g f-\tfrac{1}{(n+1)}\operatorname{Sym}(\mathrm{d}f)-\left(g\otimes(X_g+\operatorname{grad}_g f)\right)_0\\ &=A_g+g\otimes \operatorname{grad}_g f-\tfrac{1}{(n+1)}\operatorname{Sym}(\mathrm{d}f)-(g\otimes \operatorname{grad}_g f)_0\\ &=A_g-\tfrac{1}{(n+1)}\operatorname{Sym}(\mathrm{d}f)+\tfrac{1}{(n+1)}\operatorname{Sym}(\mathop{\mathrm{Con}}(g\otimes \operatorname{grad}_g f))=A_{g}. \end{aligned}\] We will henceforth write \(A_{[g]}\) instead of \(A_g\). The conformal invariance of \(A_{[g]}\) was previously observed in [MR4201191]. We included the argument for the convenience of the reader.

For what follows we recall the standard identity for a vector field \(Y\) on \(M\) \[\operatorname{div}_{\tilde{g}}Y=\operatorname{div}_g\! Y+nY(f),\] which, together with (3.8), gives \[ \operatorname{div}_{\tilde{g}}X_{\tilde{g}}=\mathrm{e}^{-2f}\left(\operatorname{div}_g\! X_g+(n-2)X_g(f)-\Delta_g f+(n-2)|\mathrm{d}f|^2_g\right),\](3.9) where \(\Delta_g=-\operatorname{div}_g \operatorname{grad}_g\).

3.2 Conformal variations

As an immediate corollary of Proposition 3.1 we have:

Corollary 3.4. A metric \(g\) is conformally critical for \(\mathcal{E}\) if and only if \[ \operatorname{div}_g(\operatorname{tr}_g\!\Phi_g)=\left(\frac{n}{2}-1\right)\left(|\Phi_g|^2_g-\frac{\Vert\Phi_g\Vert_g^2}{\operatorname{Vol}_{M}(g)}\right).\](3.10)

Proof. Observe that we have \[\operatorname{tr}_g\left(\ell^*_g(\Phi_g)\right)=\tfrac{1}{2}\operatorname{div}_g(\operatorname{tr}_g\Phi_g),\] where \[\operatorname{div}_g(\operatorname{tr}_g\Phi_g)={}^g\nabla_ig^{jk}(\Phi_g)^i_{jk}.\] Furthermore, recall (3.6) \[\operatorname{tr}_g(T_g(\Phi_g))=\left(\tfrac{n}{2}-1\right)|\Phi_g|^2_g.\] For \(h=f g\) with \(f \in C^{\infty}\), we obtain \[\begin{aligned} \mathcal{E}^{\prime}_g(fg)&=\operatorname{Vol}_{M}(g)^{(2-n)/n}\int_M f \operatorname{tr}_g\left(T_g(\Phi_g)-2\ell^*_g(\Phi_g)\right)-kn f d\mu_g\\ &=\operatorname{Vol}_{M}(g)^{(2-n)/n}\int_M f\left[\left(\frac{n}{2}-1\right)|\Phi_g|^2_g-\operatorname{div}_g(\operatorname{tr}_g\Phi_g)-kn\right]d\mu_g, \end{aligned}\] and the claim follows. ◻

4 The surface case

4.1 Example: Projective structures from holomorphic cubic differentials

Let \((M,g)\) be an oriented Riemannian \(2\)-manifold and let \(J : TM \to TM\) denote counter clockwise rotation by \(\pi/2\) with respect to the orientation and metric \(g\). We consider a cubic differential \(C\) on the Riemann surface \((M,J)\). Its real part is a totally symmetric \((0,3)\) tensor field \(A=(A_{ijk})\) on \(M\) that is totally trace-free with respect to \(g\), that is, \(g^{ij}A_{ijk}\) vanishes identically. Define \(\alpha=(\alpha^i_{jk}) \in \Gamma(E)\) by \(\alpha^i_{jk}=g^{il}A_{ljk}\) and consider the projective structure \(\mathfrak{p}\) arising from \({}^g\nabla+\alpha\). By construction, \(\alpha \in \Gamma(E_0)\) and \(\Phi_g=\alpha\). Applying Proposition 3.1, the variational equations become \(T_g(\alpha)=2\ell^*_g(\alpha)\). By (3.6), the tensor field \(T_g(\alpha)\) is totally trace-free with respect to \(g\) and hence encodes a quadratic differential on \((M,J)\). The map \(T_g\) thus gives rise to a quadratic form on \(\Gamma(K^3_{M,J})\) with values in \(\Gamma(K^2_{M,J})\), where \(K_{M,J}\) denotes the canonical bundle of \((M,J)\). Unsurprisingly, this quadratic form is trivial (as can be checked with a direct computation), hence \(T_g(\alpha)\) vanishes identically and the variational equations simplify to \(\ell^*_g(\alpha)=0\). A computation gives \(2\ell^*_g(\alpha)=-\operatorname{div}_g \alpha\) with \(\left(\operatorname{div}_g \alpha\right)_{ij}={}^g\nabla_k \alpha^k_{ij}\). This last condition is well known to be equivalent to the holomorphicity of \(C\). From this we obtain:

Proof. By [MR1828223, MR2402597], on a closed oriented surface \(M\) of negative Euler characteristic every properly convex projective structure \(\mathfrak{p}\) arises from a unique pair \((g,C)\) with \(C\) holomorphic via the above construction. That is, the torsion-free connection \({}^g\nabla+\alpha\) on \(TM\) is a representative connection of \(\mathfrak{p}\). Additionally, the pair satisfies \(K_g=-1+2|C|^2_g\), where \(K_g\) denotes the Gauss curvature of \(g\). The metric \(g\) is known as the Blaschke metric of \(\mathfrak{p}\). The above computations then show that \(T_g(\alpha)=0\) and \(\ell^*_g(\alpha)=0\) by the holomorphicity of \(C\), so the Blaschke metric \(g\) is critical for \(\mathcal{E}\). ◻

4.2 Existence and uniqueness of conformally critical metrics

Corollary 3.4 implies that for \(n=2\) a Riemannian metric \(g\) is conformally critical if and only if \(\operatorname{div}_g\!X_g=0\). Using (3.9), looking for a conformally critical metric \(\tilde{g}=\mathrm{e}^{2f}g\) in the conformal equivalence class of \(g\) thus gives the PDE \[\begin{aligned} \operatorname{div}_{\tilde{g}}X_{\tilde{g}}=\mathrm{e}^{-2f}\left(\operatorname{div}_g\! X_g-\Delta_g f\right)=0. \end{aligned}\] Hence we obtain:

Proof. Recall that for \(u \in C^{\infty}\) the Poisson equation \(\Delta_gf=u\) on a closed oriented Riemannian manifold \((M,g)\) has a smooth solution, unique up to adding a constant, provided \(\int_Mud\mu_g=0\). In our case \(u=\operatorname{div}_g\! X_g\), hence the claim is a consequence of Stokes’ theorem. ◻

5 The projective Yamabe equation

5.1 The projective-conformal Laplacian

From now on we assume \(n\geqslant 3\). Consider a closed oriented conformal \(n\)-manifold \((M,[g])\). Recall that the Yamabe problem asks whether there exists a metric \(\tilde{g} \in [g]\) whose scalar curvature \(S_{\tilde{g}}\) is constant. For \(f \in C^{\infty}\) the scalar curvature of the metric \(\tilde{g}=\exp(2f)g\) is \[S_{\tilde{g}}=\mathrm{e}^{-2f}\left(S_g+2(n-1)\Delta_g f-(n-1)(n-2)|\mathrm{d}f|^2_g\right),\] where \(\Delta_g f=-\operatorname{div}_g\operatorname{grad}_g f\). Writing \(\tilde{g}=u^{4/(n-2)}g\) for a positive function \(u\), the previous equation becomes \[4\left(\frac{n-1}{n-2}\right)\Delta_gu+S_gu=u^{(n+2)/(n-2)}S_{\tilde{g}}.\] Finding a metric \(\tilde{g} \in [g]\) with constant scalar curvature \(C\) thus amounts to solving \(L_g u=C u^{(n+2)/(n-2)}\), where \(L_g=4\left(\tfrac{n-1}{n-2}\right)\Delta_g+S_g\) denotes the so-called conformal Laplacian.

We now return to the problem of finding a conformally critical metric on a closed oriented projective manifold of dimension \(n\geqslant 3\). Writing \[ \mathcal{S}_g:=\frac{2(n+1)}{(n+2)}\left[\left(\frac{n}{2}-1\right)|\Phi_g|^2_g-\operatorname{div}_g(\operatorname{tr}_g\Phi_g)\right]\](5.1) the equation (3.10) is equivalent to the statement that a metric \(g\) is conformally critical for \(\mathcal{E}\) if and only if \(\mathcal{S}_g\) is constant.

Lemma 5.1. Let \(f \in C^{\infty}\). Under a conformal change \(g \mapsto \tilde{g}=\mathrm{e}^{2f}g\), the function \(\mathcal{S}_g\) transforms as the scalar curvature, that is, \[\mathcal{S}_{\tilde{g}}=\mathrm{e}^{-2f}(\mathcal{S}_g+2(n-1)\Delta_g f-(n-1)(n-2)|\mathrm{d}f|^2_g),\] where \(\Delta_g f=-\operatorname{div}_g\!\operatorname{grad}_g\! f\).

Proof. From the formula \[\Phi_{\tilde{g}}=\Phi_g+g\otimes \operatorname{grad}_g f-\frac{1}{(n+1)}\operatorname{Sym}(\mathrm{d}f)\] we compute \[ \begin{aligned} |\Phi_{\tilde{g}}|^2_{\tilde{g}}&=\mathrm{e}^{-2f}|\Phi_g+(g\otimes\operatorname{grad}_g\! f)_0|^2_g\\ &=\mathrm{e}^{-2f}\left(|\Phi_g|^2_g+ 2m X_g(f)+|(g\otimes \operatorname{grad}_g\! f)_0|^2_g\right)\\ &=\mathrm{e}^{-2f}\left(|\Phi_g|^2_g+2mX_g(f)+m|\mathrm{d}f|^2_g\right), \end{aligned}\](5.2) where we have used the identities \[|\operatorname{Sym}(\mathrm{d}f)|^2_g=2(n+1)|\mathrm{d}f|^2_g\quad \text{and}\quad |g\otimes \operatorname{grad}_g\! f|^2_g=n|\mathrm{d}f|^2_g\] as well as \[\langle g\otimes \operatorname{grad}_g\! f,\operatorname{Sym}(\mathrm{d}f)\rangle_g=2|\mathrm{d}f|^2_g.\] Now using (3.9) and (5.2) we obtain \[\begin{aligned} \mathcal{S}_{\tilde{g}}&=\frac{(n-1)(n-2)}{m}|\Phi_{\tilde{g}}|^2_{\tilde{g}}-2(n-1)\operatorname{div}_{\tilde{g}}X_{\tilde{g}}\\ &=\mathrm{e}^{-2f}\left[\mathcal{S}_g+2(n-1)(n-2)X_g(f)+(n-1)(n-2)|\mathrm{d}f|^2_g\right.\\ &\phantom{=}\;\left.-2(n-1)(n-2)X_g(f)+2(n-1)\Delta_g f-2(n-1)(n-2)|\mathrm{d}f|^2_g\right]\\ &=\mathrm{e}^{-2f}\left(\mathcal{S}_g+2(n-1)\Delta_g f-(n-1)(n-2)|\mathrm{d}f|^2_g\right), \end{aligned}\] as claimed. ◻

Finding a metric \(\tilde{g}\) in a conformal equivalence class \([g]\) which is conformally critical for \(\mathcal{E}\) is thus equivalent to solving an equation of Yamabe type \(\mathcal{L}_g u=C u^{(n+2)/(n-2)}\), where \[\mathcal{L}_g:=4\left(\frac{n-1}{n-2}\right)\Delta_g + \mathcal{S}_g.\]

Remark 5.2. We will refer to \(\mathcal{L}_g\) as the projective-conformal Laplacian.

5.2 The projective scalar curvature

As an immediate consequence of Stokes’ theorem and definition (5.1) we have

Proposition 5.3. Let \(g\) be a Riemannian metric on the closed oriented projective manifold \((M,\mathfrak{p})\) of dimension \(n \geqslant 3\). Then \[ \int_M\mathcal{S}_g\,d\mu_g = \hat{m} \,\Vert\Phi_g\Vert^2_g \geqslant 0,\](5.3) where \(\hat{m}=\tfrac{(n+1)(n-2)}{(n+2)}\), with equality if and only if \(\mathfrak{p}\) contains the Levi-Civita connection of \(g\).

Denoting by \(\sigma(\mathcal{L}_g)\) the spectrum of \(\mathcal{L}_g\), we also conclude:

Proposition 5.4. Let \((M,\mathfrak{p})\) be a closed oriented projective manifold of dimension \(n\geqslant 3\) and \(g\) a Riemannian metric on \(M\). Then \(\sigma(\mathcal{L}_g)\) is non-negative and moreover, contains \(0\) if and only if \(\mathfrak{p}\) is metrisable by a metric \(\tilde{g}\) in the conformal equivalence class of \(g\).

Proof. For every positive function \(u\) on \(M\), writing \(\tilde{g}=u^{4/(n-2)}g\) and using \(\mathcal{S}_{\tilde{g}}=u^{-(n+2)/(n-2)}\mathcal{L}_gu\) as well as \(d\mu_{\tilde{g}}=u^{2n/(n-2)}d\mu_g\), we obtain \[\int_Mu\,\mathcal{L}_gu\,d\mu_g=\int_Mu^{2n/(n-2)}\mathcal{S}_{\tilde{g}}\,d\mu_g=\int_M\mathcal{S}_{\tilde{g}}\,d\mu_{\tilde{g}}=\hat{m}\,\Vert\Phi_{\tilde{g}}\Vert^2_{\tilde{g}}\geqslant 0.\] Since, by standard elliptic PDE theory, the first eigenfunction of the self-adjoint elliptic operator \(\mathcal{L}_g\) can be chosen positive, it follows that the first eigenvalue \(\lambda_1\) of \(\mathcal{L}_g\) satisfies \(\lambda_1\geqslant 0\) and hence the full spectrum is non-negative. If \(\lambda_1=0\), the corresponding positive eigenfunction \(u\) satisfies \(\mathcal{L}_gu=0\), so that \(\Vert\Phi_{\tilde{g}}\Vert^2_{\tilde{g}}=0\) for \(\tilde{g}=u^{4/(n-2)}g\), meaning \(\mathfrak{p}\) contains the Levi-Civita connection of \(\tilde{g}\in [g]\). Conversely, if \(\tilde{g}=u^{4/(n-2)}g\) metrises \(\mathfrak{p}\), then \(\Phi_{\tilde{g}}=0\) and hence \(\mathcal{L}_gu=u^{(n+2)/(n-2)}\mathcal{S}_{\tilde{g}}=0\), so that \(0\) is in the spectrum of \(\mathcal{L}_g\). ◻

Because of the analogy to the scalar curvature, we will refer to \(\mathcal{S}_g\) as the projective scalar curvature of \(g\). For what follows it is advantageous to express \(\mathcal{S}_g\) in terms of the usual scalar curvature of \(g\), the “scalar curvature” of a certain Weyl connection and \(A_{[g]}\). It follows from (3.7) and (3.8) that the torsion-free connection \[ {}^{[g]}\nabla:={}^g\nabla+g\otimes X_g-\operatorname{Sym}(X_g^{\flat})\](5.4) depends only on the conformal equivalence class of \(g\). Here \(\flat\) is taken with respect to \(g\). The connection \({}^{[g]}\nabla\) on \(TM\) is a Weyl connection for \([g]\), that is, it is torsion-free and its parallel transport maps are angle preserving with respect to \([g]\). In fact, all Weyl connections for \([g]\) are of the form (5.4) for some metric \(g \in [g]\) and some vector field \(X\). For a pair \((g,X)\) we write \[S_{g,X}:=\operatorname{tr}_g \mathrm{Ric}\left({}^{[g]}\nabla\right),\] where \(\mathrm{Ric}\left({}^{[g]}\nabla\right)\) denotes the Ricci curvature of the Weyl connection \({}^{[g]}\nabla\) determined by \((g,X)\). A standard calculation – see Section 7 – gives: \[ S_{g,X} =S_g+2(n-1)\operatorname{div}_g\!X-(n-2)(n-1)|X|^2_g,\](5.5) where \(S_g\) denotes the scalar curvature of \(g\).

Proposition 5.5. For the projective scalar curvature we have \[\mathcal{S}_g=S_g-S_{g,X_g}+\hat{m} |A_{[g]}|^2_g.\]

Proof. Using the definition (5.1), (5.5) and (2.5), we compute \[\begin{aligned} \mathcal{S}_g&=\frac{2(n+1)}{(n+2)}\left[\left(\frac{n}{2}-1\right)|\Phi_g|^2_g-\operatorname{div}_g(\operatorname{tr}_g\Phi_g)\right]\\ &=\hat{m} |\Phi_g|^2_g-2(n-1)\operatorname{div}_g(X_g)\\ &=\hat{m} |A_{[g]}|^2_g+\hat{m} m |X_g|^2_g-2(n-1)\operatorname{div}_g(X_g)\\ &=S_g-\big(S_g+2(n-1)\operatorname{div}_g(X_g)-(n-2)(n-1)|X_g|^2_g\big)+\hat{m} |A_{[g]}|^2_g\\ &=S_g-S_{g,X_g}+\hat{m} |A_{[g]}|^2_g, \end{aligned}\] where we also used that \(\hat{m} m =(n-2)(n-1)\). ◻

Remark 5.6. In terms of \(X_g\) and \(A_{[g]}\) we thus have \[\mathcal{S}_g=\hat{m} |A_{[g]}|^2_g+(n-2)(n-1)|X_g|^2_g-2(n-1)\operatorname{div}_g(X_g).\]

6 Existence of conformally critical metrics

For a Riemannian metric \(g\) on the projective manifold \((M,\mathfrak{p})\) we define:

Definition 6.1. The conformal defect of \(g\) on \((M,\mathfrak{p})\) is the difference of the projective – and standard scalar curvature: \[V_g:=\mathcal{S}_g-S_g=\hat{m}|A_{[g]}|^2_g -S_{g,X_g}.\]

Remark 6.2.

Notice that \(\mathcal{L}_g=L_g+V_g\), so that \(V_g\) measures the defect of the projective-conformal Laplacian to agree with the conformal Laplacian.
Recall that we define the conformal locus of \(g\) as \[\Sigma_{[g]}=\left\{p \in M\,|\, V_g(p)=0\right\}.\] Under a conformal change \(\tilde{g}=\mathrm{e}^{2f}g\), the conformal defect satisfies \(V_{\tilde{g}}=\mathrm{e}^{-2f}V_g\). In particular, the conformal locus does only depend on the conformal equivalence class of \(g\).

6.1 Reduction to the Yamabe problem

In the special case where the conformal locus of \([g]\) is all of \(M\), the projective-conformal Laplacian and the conformal Laplacian agree, hence finding a conformally critical metric for \(\mathcal{E}\) amounts to solving the Yamabe problem. The affirmative solution to the Yamabe problem [MR125546] by Trudinger [MR240748], Aubin [MR0433500], and Schoen [MR788292] thus implies (we also refer to [MR888880] and [MR2148873] for surveys):

Theorem 6.3. Let \((M,\mathfrak{p})\) be a closed oriented projective manifold of dimension \(n\geqslant 3\). Suppose the conformal structure \([g]\) on \(M\) satisfies \(\Sigma_{[g]}=M\). Then \([g]\) contains a conformally critical metric for \(\mathcal{E}\).

Proof. By assumption \(V_g\equiv 0\), so \(\mathcal{L}_g=L_g\) and the criticality equation reduces to the Yamabe equation, which has a positive smooth solution by the work of Yamabe, Trudinger, Aubin and Schoen. ◻

6.2 Negative conformal defect

We treat in this subsection the case where the conformal defect \(V_g\) takes a negative value at some point of \(M\). Since \(V_{\tilde{g}}=\mathrm{e}^{-2f}V_g\) for \(\tilde{g}=\mathrm{e}^{2f}g\), the hypothesis that \(V_g\) is negative at a given point is independent of the choice of representative metric in \([g]\).

Theorem 6.4. Let \((M,\mathfrak{p})\) be a closed oriented projective manifold of dimension \(n\geqslant 4\) and \([g]\) a conformal structure on \(M\). Suppose there exists \(p_0\in M\) such that \(V_{\tilde{g}}(p_0)<0\) for some (and hence any) \(\tilde{g} \in [g]\). Then \([g]\) contains a conformally critical metric for \(\mathcal{E}\).

Proof. By Section 5.1, finding a metric \(\tilde{g}\in[g]\) which is conformally critical for \(\mathcal{E}\) is equivalent to finding a smooth positive function \(u\) on \(M\) that solves \(\mathcal{L}_gu=Cu^{(n+2)/(n-2)}\) for some constant \(C\geqslant 0\). Setting \(h:=\tfrac{n-2}{4(n-1)}\mathcal{S}_g\) and \(\lambda:=\tfrac{n-2}{4(n-1)}C\), this rewrites as the elliptic equation of Yamabe type \[\Delta_gu+h\,u=\lambda\,u^{(n+2)/(n-2)}\] in the sense of [MR2148873]. The hypothesis \(V_g(p_0)<0\) becomes \[h(p_0)<\tfrac{n-2}{4(n-1)}S_g(p_0),\] that is, the linear term of the equation drops below the linear term of the standard Yamabe equation at the point \(p_0\). Existence of a smooth positive solution then follows from [MR2148873] and the discussion thereafter. ◻

Combining Theorem 6.3 and Theorem 6.4 we have:

Remark 6.5. Theorem 6.4 covers, in particular, the empty-locus subcase \(V_g < 0\) on \(M\). The complementary empty-locus subcase \(V_g > 0\) on \(M\) remains open.