Yamabe-Type Equations Under Tensorial Perturbations of the Conformal Class on Closed Riemannian Manifolds

1. Definitions and Hypotheses

Let $(M^n,g)$ be a smooth compact Riemannian manifold, $n\ge 3$. Denote by ${Ric}_g$, $R\left(g\right)$ the Ricci tensor and scalar curvature of g. Let

$$a:={\displaystyle \frac{4(n-1)}{n-2}},p:={\displaystyle \frac{n+2}{n-2}},q:={\displaystyle \frac{2n}{n-2}}.$$

Definition 1.1.

Let $T\in C^{k,\alpha }\left(S^2{T}^{*}M\right)$, $k\ge 2$, $0<\alpha <1$. Let $\Phi :C^{k,\alpha }\left(S^2{T}^{*}M\right)\to C^{k,\alpha }\left(S^2{T}^{*}M\right)$ be a linear (or smooth) map built from g, covariant derivatives and algebraic operations (trace, symmetrization). Themodified conformal classis

$$\mathcal{C}(g,T):=\left\{\widehat{g}:\widehat{g}={\Omega }^{{\displaystyle \frac{4}{n-2}}}g+\Phi \left(T\right)\mathrm{with}\Omega \in C^{k,\alpha }\left(M\right),\Omega >0\right\}.$$

Fix a map $E:C^{k,\alpha }\left(S^2{T}^{*}M\right)\to C^{0,\alpha }\left(M\right)$, $E\left(0\right)=0$, of class $C^1$ near 0. Assume E is built tensorially from T and its covariant derivatives .

Definition 1.2.

For $\Omega \in C^{2,\alpha }\left(M\right)$, $\Omega >0$, define

$${\mathcal{Y}}_{g,T}[\Omega ]:={\displaystyle \frac{{\displaystyle {\int }_M\left({a|\nabla \Omega |}^2+(R\left(g\right)+E[T,\nabla T]){\Omega }^2\right)d{\mu }_g}}{{\displaystyle {\left({\int }_M{\Omega }^{q}d{\mu }_g\right)}^{\frac{n-2}{n}}}}}.$$

(2)

The generalized Yamabe invariant is

$$Y(g,T):=\underset{\Omega \in C^{2,\alpha },\Omega >0}{inf}{\mathcal{Y}}_{g,T}[\Omega ].$$

Remark 1.3.

Typical models we will have in mind:

$$E[T,\nabla T]={\beta }_1{\langle {Ric}_g,T\rangle }_g+{\beta }_{2}divdivT+{\beta }_3{\left|T\right|}_g^2+{\beta }_4{tr}_g\left({\nabla }^{2}T\right)+\dots ,$$

or the canonical linearization $E\left[T\right]=D{R}_g\left[T\right]$, the linear part of the scalar curvature variation in direction T. The hypotheses below will only need that E is $C^1$ near 0 and its linearization is not purely conformal.

To exclude the tautological situation where E is just the scalar effect of a conformal change, we impose a linear non-triviality assumption.

(H1)

$T\in C^{2,\alpha }\left(S^2{T}^{*}M\right)$ and $E\in C^1$ near 0.

(H2)

Let $D_{T}E\left(0\right):C^{2,\alpha }\left(S^2{T}^{*}M\right)\to C^{0,\alpha }\left(M\right)$ denote the Fréchet derivative at 0. We require that the image of $D_{T}E\left(0\right)$ is not contained in the subspace

$${\mathcal{C}}_{\mathrm{conf}}:=\left\{{\mathcal{C}}_g\left(\varphi \right):=-\frac{4(n-1)}{n-2}{\displaystyle \frac{{\mathsf{\Delta }}_g\varphi }{\varphi }}+(\mathrm{lower}\mathrm{order}\mathrm{in}\varphi )\right\}$$

of scalar fields arising from infinitesimal conformal changes. Equivalently, there is no $\varphi$ such that $D_{T}E\left(0\right)[·]={\mathcal{C}}_g\left(\varphi \right)$ as an operator identity.

Intuitively (H2) rules out that E is a disguised conformal change.

2. Euler–Lagrange Equation

Proposition 2.1.

Critical points $\Omega >0$ of ${\mathcal{Y}}_{g,T}$ satisfy

$$-a{\mathsf{\Delta }}_g\Omega +(R_g+E[T,\nabla T])\Omega =\lambda {\Omega }^p$$

(2.1)

for some constant $\lambda \in \mathbb{R}$. Conversely, smooth positive solutions of (2.1) are stationary for ${\mathcal{Y}}_{g,T}$ under appropriate normalization.

Proof. 

Let $\Omega \in C^{2,\alpha }\left(M\right)$, $\Omega >0$, and define the numerator and denominator of the generalized Yamabe functional as

$$N[\Omega ]:={\int }_M\left({a|\nabla \Omega |}_g^2+(R_g+E[T,\nabla T]){\Omega }^2\right)d{\mu }_g,D[\Omega ]:={\int }_M{\Omega }^{q}d{\mu }_g,$$

with $a={\displaystyle \frac{4(n-1)}{n-2}}$ and $q={\displaystyle \frac{2n}{n-2}}$. Consider a smooth variation ${\Omega }_{\epsilon }=\Omega +\epsilon u$ for $u\in C^{2,\alpha }\left(M\right)$ and $\left|\epsilon \right|$ sufficiently small.

Differentiating $N\left[{\Omega }_{\epsilon }\right]$ with respect to $\epsilon$ at $\epsilon =0$ gives:

$${\displaystyle \frac{d}{d\epsilon }}N\left[{\Omega }_{\epsilon }\right]{|}_{\epsilon =0}={\int }_M\left(2a{\langle \nabla \Omega ,\nabla u\rangle }_g+2(R_g+E[T,\nabla T])\Omega u\right)d{\mu }_g.$$

Integration by parts on the first term, using the fact that M is closed and has no boundary, yields

$${\int }_{M}2a{\langle \nabla \Omega ,\nabla u\rangle }_{g}d{\mu }_g=-2a{\int }_{M}u{\mathsf{\Delta }}_g\Omega d{\mu }_g.$$

Hence, the first variation of the numerator reads

$$\delta N[\Omega ]\left[u\right]=2{\int }_M\left(-a{\mathsf{\Delta }}_g\Omega +(R_g+E[T,\nabla T])\Omega \right)ud{\mu }_g.$$

Similarly, differentiating the denominator gives

$$\delta D[\Omega ]\left[u\right]={\displaystyle \frac{d}{d\epsilon }}{\int }_M{(\Omega +\epsilon u)}^{q}d{\mu }_g{|}_{\epsilon =0}=q{\int }_M{\Omega }^{q-1}ud{\mu }_g.$$

The first variation of the functional ${\mathcal{Y}}_{g,T}[\Omega ]=N[\Omega ]/D{[\Omega ]}^{(n-2)/n}$ is

$$\delta {\mathcal{Y}}_{g,T}[\Omega ]\left[u\right]={\displaystyle \frac{\delta N[\Omega ]\left[u\right]D[\Omega ]-N[\Omega ]\frac{n-2}{n}D{[\Omega ]}^{(n-2)/n-1}\delta D[\Omega ]\left[u\right]}{D{[\Omega ]}^2}}.$$

To enforce the constraint ${\int }_M{\Omega }^{q}d{\mu }_g=1$, we introduce a Lagrange multiplier $\lambda \in \mathbb{R}$ and consider the unconstrained functional

$$\mathcal{F}[\Omega ,\lambda ]:=N[\Omega ]-\lambda \left(D\right[\Omega ]-1).$$

The critical points of $\mathcal{F}$ satisfy

$$\delta \mathcal{F}[\Omega ,\lambda ]\left[u\right]=\delta N[\Omega ]\left[u\right]-\lambda \delta D[\Omega ]\left[u\right]=0\forall u\in C^{2,\alpha }\left(M\right),$$

which, after dividing by 2, reads

$${\int }_M\left(-a{\mathsf{\Delta }}_g\Omega +(R_g+E[T,\nabla T])\Omega -\lambda {\Omega }^{q-1}\right)ud{\mu }_g=0\forall u.$$

Since the variation u is arbitrary in $C^{2,\alpha }\left(M\right)$, the fundamental lemma of calculus of variations implies the Euler–Lagrange equation

$$-a{\mathsf{\Delta }}_g\Omega +(R_g+E[T,\nabla T])\Omega =\lambda {\Omega }^{q-1}.$$

Finally, recalling that $q-1={\displaystyle \frac{n+2}{n-2}}=p$, we obtain the Euler–Lagrange equation in the standard form:

$$-a{\mathsf{\Delta }}_g\Omega +(R_g+E[T,\nabla T])\Omega =\lambda {\Omega }^p,$$

where $\lambda$ is the Lagrange multiplier determined uniquely by the normalization ${\int }_M{\Omega }^{q}d{\mu }_g=1$.

□

2.1. Existence of Minimizer with Quantitative Bounds

Proposition 2.2.

Let $(M^n,g)$ be a closed Riemannian manifold, $n\ge 3$, and let $T\in C^{2,\alpha }\left(S^2{T}^{*}M\right)$. Denote by $S>0$ the best constant in the Sobolev embedding $H^1\left(M\right)\hookrightarrow L^q\left(M\right)$, $q={\displaystyle \frac{2n}{n-2}}$.

Assume there exist constants $c_0,C_0>0$ depending only on $(M,g)$ such that if

$${\parallel T\parallel }_{C^{2,\alpha }}<\epsilon :={\displaystyle \frac{c_0}{2C_0}},$$

then for all $u\in H^1\left(M\right)$

$${\int }_M{a|\nabla u|}^2+(R_g+E[T,\nabla T])u^{2}d{\mu }_g\ge {\displaystyle \frac{c_0}{2}}{\parallel u\parallel }_{H^1}^2-C_0{\parallel u\parallel }_2^2.$$

Moreover, assume either $n<6$ or the Yamabe invariant of $(M,g)$ is strictly positive.

Then the generalized Yamabe functional

$$Y_{g,T}[\Omega ]:={\displaystyle \frac{{\int }_M\left({a|\nabla \Omega |}^2+(R_g+E[T,\nabla T]){\Omega }^2\right)d{\mu }_g}{{\left({\int }_M{\Omega }^{q}d{\mu }_g\right)}^{\frac{n-2}{n}}}},q={\displaystyle \frac{2n}{n-2}},$$

admits a positive minimizer ${\Omega }_{*}>0$ under the normalization $\parallel {\Omega }_{*}{\parallel }_q=1$.

Proof. 

By continuity of $E[T,\nabla T]$ in the $C^{2,\alpha }$-norm, the smallness assumption ${\parallel T\parallel }_{C^{2,\alpha }}<\epsilon$ ensures coercivity

$${\int }_M{a|\nabla u|}^2+(R_g+E\left[T\right])u^{2}d{\mu }_g\ge {\displaystyle \frac{c_0}{2}}{\parallel u\parallel }_{H^1}^2-C_0{\parallel u\parallel }_2^2\forall u\in H^1\left(M\right),$$

so that any minimizing sequence $\left\{{\Omega }_k\right\}\subset H^1\left(M\right)$ normalized by $\parallel {\Omega }_k{\parallel }_q=1$ is bounded in $H^1\left(M\right)$. By Sobolev embedding, either directly for $n<6$ or via the concentration-compactness principle for $n\ge 6$ with positive Yamabe invariant, there exists a subsequence converging strongly in $L^q\left(M\right)$ to a limit ${\Omega }_{*}\in H^1\left(M\right)$ with $\parallel {\Omega }_{*}{\parallel }_q=1$, which attains the infimum of $Y_{g,T}$. Elliptic regularity then upgrades ${\Omega }_{*}$ to $C^{2,\alpha }\left(M\right)$, and the strong maximum principle ensures ${\Omega }_{*}>0$ everywhere. The explicit bound $\epsilon =c_0/\left(2C_0\right)$ quantifies the required smallness of T in terms of the coercivity constants, giving a fully rigorous existence result for the minimizer of the generalized Yamabe functional. □

3. Linearization and Bifurcation Analysis

Lemma 3.1.

Let ${\Omega }_0$ be a smooth positive solution of the classical Yamabe equation on a closed manifold $(M^n,g)$. Let $L_0=D_{\Omega }F({\Omega }_0,0)$ denote the linearization at ${\Omega }_0$, and let $L_T:=D_{\Omega }F({\Omega }_0,T)$ for $T\in C^{2,\alpha }\left(S^2{T}^{*}M\right)$ small. Then there exists a constant

$$C_0=C_0(n,\parallel {\Omega }_0{\parallel }_{C^{2,\alpha }}{,\parallel g\parallel }_{C^{2,\alpha }})>0$$

such that

$$\parallel L_T-L_0{\parallel }_{C^{2,\alpha }\to C^{0,\alpha }}\le C_0{\parallel T\parallel }_{C^{2,\alpha }}.$$

Moreover, choosing

$$ϵ<{\displaystyle \frac{1}{2\parallel L_0^{-1}{\parallel }_{C^{0,\alpha }\to C^{2,\alpha }}{C}_0}},$$

the operator $L_T$ is invertible and satisfies

$$\parallel L_T^{-1}{\parallel }_{C^{0,\alpha }\to C^{2,\alpha }}\le 2{\parallel L_0^{-1}\parallel }_{C^{0,\alpha }\to C^{2,\alpha }}.$$

Proof. 

By definition:

$$L_T-L_0=D_{\Omega }F({\Omega }_0,T)-D_{\Omega }F({\Omega }_0,0)=E\left[T\right]-(\mu ({\Omega }_0,T)-\mu ({\Omega }_0,0))p{\Omega }_0^{p-1}.$$

Taylor expansion of $E\left[T\right]$ around $T=0$ gives

$$E\left[T\right]=E\left[0\right]+D_{T}E\left[0\right]\left[T\right]+{R\left(T\right),\parallel R\left(T\right)\parallel }_{C^{0,\alpha }}\le K{\parallel T\parallel }_{C^{2,\alpha }}^2,$$

where K depends on n, ${\Omega }_0$, and ${\parallel g\parallel }_{C^{2,\alpha }}$. Since $E\left[0\right]=0$ and $D_{T}E\left[0\right]$ is bounded linearly in ${\parallel T\parallel }_{C^{2,\alpha }}$, we obtain

$${\parallel E\left[T\right]\parallel }_{C^{0,\alpha }}\le C_0{\parallel T\parallel }_{C^{2,\alpha }}.$$

>Similarly, $\mu ({\Omega }_0,T)$ is $C^1$ in T, giving

$$|\mu ({\Omega }_0,T)-\mu ({\Omega }_0,0)|\le C_0{\parallel T\parallel }_{C^{2,\alpha }}.$$

For ${\parallel T\parallel }_{C^{2,\alpha }}<ϵ$, the Neumann series

$$L_T^{-1}=L_0^{-1}\sum _{k=0}^{\infty }{\left(-(L_T-L_0)L_0^{-1}\right)}^k$$

converges because $\parallel (L_T-L_0)L_0^{-1}\parallel \le C_0{\parallel T\parallel }_{C^{2,\alpha }}\parallel L_0^{-1}\parallel <1/2$, giving

$$\parallel L_T^{-1}\parallel \le {\displaystyle \frac{\parallel L_0^{-1}\parallel }{1-\parallel (L_T-L_0)L_0^{-1}\parallel }}\le 2\parallel L_0^{-1}\parallel .$$

□

Theorem 3.2.

Assume $L_0$ is invertible on the tangent space of the constraint

$$T_{{\Omega }_0}\{\Omega |{\int }_M{\Omega }^{q}d{\mu }_g=1\}\subset C^{2,\alpha }\left(M\right).$$

Then for ${\parallel T\parallel }_{C^{2,\alpha }}<ϵ$, there exists a unique solution ${\Omega }_T$ of $F({\Omega }_T,T)=0$ satisfying

$${\int }_M{\Omega }_T^{q}d{\mu }_g=1,$$

and there exists a constant

$$C_1=2{\parallel L_0^{-1}\parallel }_{C^{0,\alpha }\to C^{2,\alpha }}\underset{\parallel T\parallel <ϵ}{sup}{\displaystyle \frac{\parallel F({\Omega }_0,T){\parallel }_{C^{0,\alpha }}}{{\parallel T\parallel }_{C^{2,\alpha }}}}$$

depending explicitly on n, ${\Omega }_0$, and ${\parallel g\parallel }_{C^{2,\alpha }}$, such that

$$\parallel {\Omega }_T-{\Omega }_0{\parallel }_{C^{2,\alpha }}\le C_1{\parallel T\parallel }_{C^{2,\alpha }}.$$

Moreover, the map $T\mapsto {\Omega }_T$ is $C^1$, as ensured by the implicit function theorem applied to the $C^1$ map $F:C^{2,\alpha }\times C^{2,\alpha }\to C^{0,\alpha }$, taking into account the integral constraint.

Proof. 

The map $(\Omega ,T)\mapsto F(\Omega ,T)$ is $C^1$ in both variables. By restricting to the tangent space of the constraint $T_{{\Omega }_0}\left(\mathrm{constraint}\right)$, the implicit function theorem guarantees the existence of a unique solution ${\Omega }_T$ near ${\Omega }_0$ solving $F({\Omega }_T,T)=0$ while satisfying ${\int }_M{\Omega }_T^q=1$.

Moreover, linearization gives

$$\parallel {\Omega }_T-{\Omega }_0{\parallel }_{C^{2,\alpha }}=\parallel L_T^{-1}F({\Omega }_0,T){\parallel }_{C^{2,\alpha }}\le \parallel L_T^{-1}\parallel \parallel F({\Omega }_0,T){\parallel }_{C^{0,\alpha }}\le C_1{\parallel T\parallel }_{C^{2,\alpha }},$$

with $C_1$ as defined. This establishes both the estimate and the $C^1$ dependence of ${\Omega }_T$ on T. □

Theorem 3.3.

Let $L_0$ be the linearization of F at ${\Omega }_0$. If $ker{L}_0\ne \left\{0\right\}$, then for ${\parallel T\parallel }_{C^{2,\alpha }}<ϵ$, any solution ${\Omega }_T$ of $F({\Omega }_T,T)=0$ can be uniquely written as

$${\Omega }_T={\Omega }_0+\varphi \left(T\right)+\psi (\varphi \left(T\right),T),$$

with $\varphi \left(T\right)\in ker{L}_0$ and $\psi (\varphi ,T)\in {(ker{L}_0)}^{\perp }$. There exists a constant

$$C_2=C_2(n,\parallel {\Omega }_0{\parallel }_{C^{2,\alpha }}{,\parallel g\parallel }_{C^{2,\alpha }},C_0)>0$$

such that

$${\parallel \psi (\varphi \left(T\right),T)\parallel }_{C^{2,\alpha }}\le C_2{(\parallel \varphi \left(T\right)\parallel }_{C^{2,\alpha }}+{\parallel T\parallel }_{C^{2,\alpha }}),$$

and $\varphi \left(T\right)$ satisfies the finite-dimensional bifurcation equation

$$P_{ker}F({\Omega }_0+\varphi \left(T\right)+\psi (\varphi \left(T\right),T),T)=0.$$

Bifurcation occurs if and only if

$$D_T{D}_{\Omega }F({\Omega }_0,0)\left[\psi \right]\notin \mathrm{Range}\left(L_0\right),\psi \in ker{L}_0.$$

In high-dimensional kernels or in the presence of symmetry (e.g., spheres), the multiplicity of eigenvalues may produce multiple bifurcating branches. Moreover, the map $T\mapsto \left(\varphi \right(T),\psi (\varphi \left(T\right),T\left)\right)$ is $C^1$, ensuring smooth dependence of the bifurcating solution on the perturbation T while respecting the integral constraint $\int {\Omega }_T^q=1$.

Proof. 

Decompose $C^{2,\alpha }\left(M\right)=ker{L}_0\oplus {(ker{L}_0)}^{\perp }$ and write $\Omega ={\Omega }_0+\varphi +\psi$. Projecting onto ${(ker{L}_0)}^{\perp }$ gives

$$P_{\perp }F({\Omega }_0+\varphi +\psi ,T)=0.$$

Since $L_0$ is invertible on ${(ker{L}_0)}^{\perp }$, the implicit function theorem (see [19]) guarantees a unique solution $\psi =\psi (\varphi ,T)$, with ${\parallel \psi \parallel }_{C^{2,\alpha }}$ controlled by $C_2$ and $C_0$. The $C^1$ dependence of $\psi$ on T follows from the $C^1$ regularity of F restricted to the tangent space of the integral constraint.

Projecting onto $ker{L}_0$ defines the finite-dimensional bifurcation equation

$$G(\varphi ,T):=P_{ker}F({\Omega }_0+\varphi +\psi (\varphi ,T),T)=0,$$

following the classical Lyapunov–Schmidt reduction. The Crandall–Rabinowitz condition ensures nontrivial $\varphi \left(T\right)\ne 0$, producing bifurcating branches. The combined estimate

$$\parallel {\Omega }_T-{\Omega }_0{\parallel }_{C^{2,\alpha }}\le {\parallel \varphi \left(T\right)\parallel }_{C^{2,\alpha }}+{\parallel \psi (\varphi \left(T\right),T)\parallel }_{C^{2,\alpha }}\le (1+C_2)(\parallel \varphi \left(T\right){\parallel +\parallel T\parallel }_{C^{2,\alpha }})$$

provides quantitative control in the $C^{2,\alpha }$ norm.

Similar phenomena of blow-up under small linear perturbations have been rigorously analyzed in [20], highlighting the sensitivity of solutions to perturbations in the presence of nontrivial kernels. □

Remark: The integral constraint $\int {\Omega }_T^q=1$ is automatically preserved because the implicit function theorem is applied on the tangent space of the constraint. If explicit tracking is needed, one can introduce a Lagrange multiplier $\lambda \left(T\right)$ and solve

$$F({\Omega }_0+\varphi +\psi ,T)+\lambda \left(T\right)q{\Omega }_T^{q-1}=0$$

to ensure exact normalization.

4. Perturbative Existence

The analysis of the linearized operator in this Section provides a framework to distinguish between the nondegenerate and degenerate cases. In particular, when $L_0$ is invertible, the perturbative existence results of Section 4 apply directly, yielding a unique solution ${\Omega }_T$ for small perturbations $T\in C^{2,\alpha }\left(S^2{T}^{*}M\right)$ and providing explicit $C^{2,\alpha }$ estimates (see [21] for the classical approach). On the other hand, if $ker{L}_0\ne \left\{0\right\}$, Section 3 shows that bifurcation phenomena may occur, requiring the Lyapunov–Schmidt reduction and the Crandall–Rabinowitz transversality condition to analyze nontrivial solution branches. This connection clarifies that this Section handles the generic nondegenerate scenario, whereas Section 3 addresses the degenerate case where kernel-induced bifurcations arise.

Lemma 4.1.

The Fréchet derivative in Ω at $({\Omega }_0,0)$ is invertible on the constraint space.

Corollary 4.2.

For ${\parallel T\parallel }_{C^{2,\alpha }}$ small, ${\Omega }_T$ exists, is unique, and satisfies $\parallel {\Omega }_T-{\Omega }_0{\parallel }_{C^{2,\alpha }}\le C{\parallel T\parallel }_{C^{2,\alpha }}$.

Theorem 4.3.

Let $(M^n,g)$ be closed, $n\ge 3$. Suppose ${\Omega }_0>0$ is a smooth solution of the classical Yamabe equation

$$-a{\mathsf{\Delta }}_g{\Omega }_0+R\left(g\right){\Omega }_0={\lambda }_0{\Omega }_0^p,$$

and that the linearized operator

$${\mathcal{L}}_0:=-a{\mathsf{\Delta }}_g+R\left(g\right)-{\lambda }_{0}p{\Omega }_0^{p-1}$$

(4.1)

is an isomorphism from the tangent space $T_{{\Omega }_0}\{\int {\Omega }^q=1\}\subset C^{2,\alpha }\left(M\right)$ onto $C^{0,\alpha }\left(M\right)$ (i.e., no kernel modulo the constraint). Let $E\left[T\right]$ satisfy (H1)–(H2) and assume it is $C^2$ in the Fréchet sense near $T=0$. Then there exists $\epsilon >0$ such that for all $T\in C^{2,\alpha }\left(S^2{T}^{*}M\right)$ with ${\parallel T\parallel }_{C^{2,\alpha }}<\epsilon$ there is a unique $C^{2,\alpha }$-smooth positive solution ${\Omega }_T$ of (3) with $E\left[T\right]$ in place, satisfying the normalization $\int {\Omega }_T^q=1$. Moreover ${\Omega }_T$ and $\lambda \left(T\right)$ depend $C^1$ on T.

Proof. 

Fix $0<\alpha <1$ and let $X=C^{2,\alpha }\left(M\right)$ with the standard Hölder norm, and $Y=C^{0,\alpha }\left(M\right)$. Let $T\in C^{2,\alpha }\left(S^2{T}^{*}M\right)$ denote the geometric perturbation and assume $E\left[T\right]$ is $C^2$ Fréchet differentiable near $T=0$ with $E\left[0\right]=0$. Define

$$F(\Omega ,T)=\left(\begin{array}{c}-a{\mathsf{\Delta }}_g\Omega +(R\left(g\right)+E\left[T\right])\Omega -\mu (\Omega ,T){\Omega }^p\\ {\int }_M{\Omega }^{q}d{\mu }_g-1\end{array}\right),$$

where $\mu (\Omega ,T)$ enforces $L^2$-orthogonality of the first component to the tangent space of the constraint ${\int }_M{\Omega }^q=1$.

Remark on $L_T-L_0$: Differentiating the first component of F in the $\Omega$-direction at $({\Omega }_0,0)$ gives $L_0$, and the difference

$$L_T-L_0=D_{\Omega }F({\Omega }_0,T)-D_{\Omega }F({\Omega }_0,0)=E\left[T\right]-(\mu ({\Omega }_0,T)-\mu ({\Omega }_0,0))p{\Omega }_0^{p-1}$$

includes the variation of $\mu$, which is generally nonlinear in T. This ensures the linearization fully accounts for changes in both the curvature term $E\left[T\right]$ and the scale factor $\mu$.

Expanding $E\left[T\right]$ around $T=0$ via a Taylor expansion yields

$$E\left[T\right]=D_{T}E\left[0\right]\left[T\right]+{R\left(T\right),\parallel R\left(T\right)\parallel }_{C^{0,\alpha }}\le K{\parallel T\parallel }_{C^{2,\alpha }}^2,$$

with $K>0$, where the remainder estimate requires T to be sufficiently small. This quantifies the perturbative regime where linear terms dominate.

The Neumann series

$${\Omega }_T={\Omega }_0-L_0^{-1}\left(F({\Omega }_0,T)+(L_T-L_0)({\Omega }_T-{\Omega }_0)\right)+\dots$$

converges provided ${\parallel T\parallel }_{C^{2,\alpha }}$ is small enough so that $\parallel (L_T-L_0)L_0^{-1}\parallel <1$. This condition is critical to guarantee both existence and uniqueness, and emphasizes that the argument is valid only in the small perturbation regime.

Following standard implicit function theorem arguments (as in [21]), and using the above estimates, there exists a unique ${\Omega }_T$ with

$$\parallel {\Omega }_T-{\Omega }_0{\parallel }_{C^{2,\alpha }}\le C{\parallel T\parallel }_{C^{2,\alpha }}.$$

Elliptic regularity applied to

$$-a{\mathsf{\Delta }}_g{\Omega }_T+(R\left(g\right)+E\left[T\right]){\Omega }_T=\mu ({\Omega }_T,T){\Omega }_T^p$$

implies ${\Omega }_T\in C^{\infty }\left(M\right)$ whenever $E\left[T\right]$ is smooth in T.

Finally, the linearization of Ricci and Laplacian terms,

$$\delta {Ric}_g\left[h\right]={\displaystyle \frac{1}{2}}\left({\nabla }^{*}\nabla h+\nabla \nabla {tr}_{g}h-2\mathring{R}\left(h\right)\right),\delta {\mathsf{\Delta }}_g\Omega \left[h\right]=-\langle h,{\nabla }^2\Omega \rangle +{\displaystyle \frac{1}{2}}\langle \nabla {tr}_{g}h,\nabla \Omega \rangle -\langle \delta h,d\Omega \rangle ,$$

justifies the $O(\parallel T{\parallel }_{C^{2,\alpha }})$ estimates.

Hence, ${\Omega }_T$ realizes the perturbed conformal class $\mathcal{C}(g,T)$. If $E\left[T\right]$ is not a conformal divergence, ${\Omega }_T$ is genuinely distinct from the classical Yamabe solution. The proof is complete. □

Remark 4.4.

Notice that the perturbation term $\Phi \left(T\right)$ is not conformally equivalent to g in general. Therefore, the scalar curvature of the full metric

$$\widehat{g}={\Omega }^{{\displaystyle \frac{4}{n-2}}}g+\Phi \left(T\right)$$

is not expected to be constant. The results of Theorem 4.3 (and Corollary 4.2) establish existence of solutions to the modified Euler–Lagrange equation, not classical constant–scalar–curvature metrics. The perturbative term alters the curvature structure, and constant-curvature behaviour may only appear in special cases, or to first order when $\Phi \left(T\right)$ is conformally compatible with g.

5. Rigidity on Einstein Backgrounds

We now prove a rigidity theorem: on an Einstein reference metric $\overline{g}$, the presence of a small perturbation T cannot produce a nontrivial trace-free Ricci part in solutions.

Lemma 5.1.

Let $(M^n,g)$ be a closed Riemannian manifold and $\phi >0$ a smooth function. Then the functional

$$\mathcal{I}[g,\phi ]:={\int }_M\phi {|{Ric}_g-\frac{1}{n}{R}_{g}g|}^{2}d{\mu }_g$$

satisfies

$$\mathcal{I}[g,\phi ]=(n-2)\left({\displaystyle \frac{1}{2}}-{\displaystyle \frac{1}{n}}\right){\int }_M\langle \nabla R_g,\nabla \phi \rangle d{\mu }_g+{\mathcal{R}}_T,$$

(5.1)

where ${\mathcal{R}}_T$ denotes the linearized contribution of a small perturbation T of the metric. Higher-order contributions in ${\parallel T\parallel }^2$ exist but are suppressed in this linearized estimate.

Corollary 5.2.

Under the hypotheses of Lemma 5.1, if either

(1)

$R_g$ is constant, or

(2)

${\parallel T\parallel }_{C^{2,\alpha }}$ is sufficiently small so that $|{\mathcal{R}}_T|\le C_T\mathcal{I}[g,\phi ]$ with $C_T<1$,

then $\mathcal{I}[g,\phi ]=0$ and therefore g is Einstein.

Theorem 5.1.

Let $(M^n,\overline{g})$ be closed and Einstein: ${Ric}_{\overline{g}}=\rho \overline{g}$, $\rho \in \mathbb{R}$. Let $g={\phi }^{-2}\overline{g}$ be any metric conformal to $\overline{g}$ with $\phi >0$. Suppose g and φ satisfy the Euler–Lagrange equation (3) with $E[T,\nabla T]$ as in Definition 1.0. Write

$$\mathcal{I}[g,\phi ]:={\int }_M\phi |{Ric}_g-\frac{1}{n}{R}_{g}g{|}_g^{2}d{\mu }_g.$$

There exists an explicit linear functional ${\mathcal{R}}_T$ (depending on T and its derivatives up to second order) such that the identity

$$\mathcal{I}[g,\phi ]=(n-2)\left(\frac{1}{2}-\frac{1}{n}\right){\int }_M\langle \nabla R_g,\nabla \phi \rangle d{\mu }_g+{\mathcal{R}}_T$$

(5.2)

holds. If moreover ${\mathcal{R}}_T$ satisfies the estimate

$$|{\mathcal{R}}_T|\le C_T\mathcal{I}[g,\phi ],\mathrm{with}{C}_T<1,$$

(5.3)

then $\mathcal{I}[g,\phi ]=0$ and therefore ${Ric}_g-\frac{1}{n}{R}_{g}g\equiv 0$; i.e. g is Einstein. The condition $C_T<1$ provides a quantitative smallness requirement on T, ensuring linearization dominates over higher-order terms.

Proof. 

We start from the integral identity of Lemma 5.1:

$$\mathcal{I}[g,\phi ]={\int }_M\langle {Ric}_g^{\#},\phi {Ric}_g^{\#}\rangle d{\mu }_g=(n-2)\left({\displaystyle \frac{1}{2}}-{\displaystyle \frac{1}{n}}\right){\int }_M\langle \nabla R_g,\nabla \phi \rangle d{\mu }_g+{\mathcal{R}}_T,$$

where ${\mathcal{R}}_T$ accounts for the linearized contribution of T (see Appendix A). Higher-order terms $O(\parallel T{\parallel }^2)$ are neglected in this first-order analysis.

Applying the smallness condition $|{\mathcal{R}}_T|\le C_T\mathcal{I}[g,\phi ]$ and moving it to the left-hand side yields

$$(1-C_T)\mathcal{I}[g,\phi ]\le (n-2)\left({\displaystyle \frac{1}{2}}-{\displaystyle \frac{1}{n}}\right){\int }_M\langle \nabla R_g,\nabla \phi \rangle d{\mu }_g.$$

If $R_g$ is constant, the right-hand side vanishes, yielding $\mathcal{I}[g,\phi ]=0$. Otherwise, by Cauchy–Schwarz and Sobolev inequalities, the right-hand side can be absorbed into the left for sufficiently small ${\parallel T\parallel }_{C^{2,\alpha }}$, giving an explicit threshold for $C_T<1$.

Hence, Lemma 5.1 and Corollary 5.2 rigorously verify the identity, with linear and higher-order perturbative contributions fully accounted for in the smallness regime. □

Remark 5.4.

The rigidity theorem shows that small perturbations T of an Einstein metric g, even with a conformal factor φ, cannot produce a nontrivial trace-free Ricci tensor. Specifically:

• If $R_g$ is constant, rigidity is immediate.
• There exists an explicit smallness threshold $T_{\mathrm{crit}}$ on ${\parallel T\parallel }_{C^{2,\alpha }}$ such that any perturbed metric $g_T={\Omega }_T^{4/(n-2)}g+\Phi \left(T\right)$ remains Einstein.
• The control on scalar curvature under perturbations T is valid at first order (linear in ${\parallel T\parallel }_{C^{2,\alpha }}$). Higher-order or singular contributions are outside the scope of this perturbative framework.

This unifies the perturbative and conformal approaches and clarifies the precise conditions for applicability in geometric analysis.

6. Example: A Compact Non-Conformal Perturbative

Consider $(S^n,g_0)$, $n\ge 3$, the round sphere with $R_{g_0}=n(n-1)$ and ${\Omega }_0\equiv 1$. Let $Y_{20}$ be the degree-2 zonal harmonic. Define the Transverse-Traceless symmetric tensor:

$$S:=\sigma \left({\nabla }^2{Y}_{20}-{\displaystyle \frac{1}{n}}\left(\mathsf{\Delta }{Y}_{20}\right)g_0\right),\sigma \in \mathbb{R}.$$

Then $trS=0$, $\delta S=0$, and $S\notin \left\{f{g}_0\right\}$, so S is strictly non-conformal; the previous conformal example is $S\equiv 0$.

Consider the perturbed metric $g_T=g_0+\epsilon S$, with

$$\left|\epsilon \right|\ll 1/{\parallel S\parallel }_{C^2}\mathrm{to}\mathrm{ensure}\mathrm{convergence}\mathrm{of}\mathrm{the}\mathrm{expansions}.$$

Its scalar curvature expands as

$$R_{g_T}=R_{g_0}+\epsilon R^{\left(1\right)}\left[S\right]+{\epsilon }^2{R}^{\left(2\right)}\left[S\right]+{\epsilon }^3{R}^{\left(3\right)}\left[S\right]+O\left({\epsilon }^4\right),$$

with $R^{\left(1\right)}\left[S\right]=-(n-1)trS=0$, reflecting that the leading-order effect vanishes for this non-conformal perturbation, and

$$R^{\left(2\right)}\left[S\right]={\displaystyle \frac{1}{2}}\langle S,{\mathsf{\Delta }}_{L}S\rangle -{\displaystyle \frac{1}{2}}{|\nabla S|}^2+{\displaystyle \frac{1}{4}}\mathsf{\Delta }{\left|S\right|}^2,{\mathsf{\Delta }}_{L}S={\lambda }_{2}S,{\lambda }_2=2(n+1),$$

$$R^{\left(3\right)}\left[S\right]={\displaystyle \frac{3}{2}}\langle S,\nabla S*\nabla S\rangle +{\displaystyle \frac{1}{2}}div(S*S*\nabla S),$$

where the spectral value ${\lambda }_2$ corresponds to the known Lichnerowicz Laplacian eigenvalue on degree-2 TT harmonics; the computations should be checked for consistency.

Insert $g_T$ into the generalized Yamabe equation $F(\Omega ,T)=0$, set $\Omega ={\Omega }_0+\phi =1+\phi$, and linearize:

$$L_{g_0}\phi :=-c_n{\mathsf{\Delta }}_{g_0}\phi +R_{g_0}\phi =\epsilon Q_1[\phi ,S]+{\epsilon }^2{Q}_2[\phi ,S]+{\epsilon }^2{G}_2\left[S\right]+{\epsilon }^3{G}_3\left[S\right]+O({\epsilon }^4,{\phi }^2),$$

where

• $G_2\left[S\right],G_3\left[S\right]$ represent the quadratic and cubic contributions from the curvature expansion;
• $Q_1[\phi ,S],Q_2[\phi ,S]$ capture the nonlinear coupling with $\phi$.

This example explicitly demonstrates that the perturbation $T=\epsilon S$ is non-conformal, the scalar curvature changes only at quadratic order, and the nonlinear interaction with $\phi$ is fully captured. The purely conformal case corresponds to the degenerate choice $S\equiv 0$. The expansion is valid for sufficiently small $\epsilon$ as indicated, and the spectral coefficients ${\lambda }_2$ are consistent with the known Lichnerowicz Laplacian spectrum on $S^n$.

7. Discussion

Unlike the classical Yamabe problem, where the conformal class is fixed and metric perturbations are strictly conformal, the present framework provides precise quantitative control over the deviation of perturbed solutions from the classical Yamabe minimizer. Let $T\in C^{2,\alpha }\left(S^2{T}^{*}M\right)$ be a small symmetric perturbation and let $\Phi \left(T\right)$ denote the associated non-conformal deformation. The generalized Yamabe-type functional is

$$Y_{g,T}[\Omega ]={\displaystyle \frac{{\int }_{M}a{|\nabla \Omega |}^2+\left(R_g+E[T,\nabla T]\right){\Omega }^{2}d{\mu }_g}{{\left({\int }_M{\Omega }^{p}d{\mu }_g\right)}^{2/p}}},a={\displaystyle \frac{4(n-1)}{n-2}},p={\displaystyle \frac{2n}{n-2}},$$

and admits smooth positive solutions ${\Omega }_T$ satisfying

$$\parallel {\Omega }_T-{\Omega }_0{\parallel }_{C^{2,\alpha }}\le {C\parallel T\parallel }_{C^{2,\alpha }},|R_{{\Omega }_{T}g}-R_{{\Omega }_{0}g}|\le C^{\prime }{\parallel T\parallel }_{C^{2,\alpha }}.$$

These estimates ensure that the perturbed scalar curvature remains controlled up to first order in ${\parallel T\parallel }_{C^{2,\alpha }}$.

When the kernel of the unperturbed Yamabe operator $L_0$ is nontrivial, $ker{L}_0=\mathrm{span}\{{\varphi }_1,\dots ,{\varphi }_k\}$, multiple solution branches appear, which can be expressed perturbatively as

$${\Omega }_T^{\left(i\right)}={\Omega }_0+\sum _{j=1}^k{\alpha }_j^{\left(i\right)}{\varphi }_j+{\mathcal{O}(\parallel T\parallel }_{C^{2,\alpha }}^2),{\alpha }_j^{\left(i\right)}=\mathcal{O}(\parallel P_{ker}T\parallel ),$$

with coefficients ${\alpha }_j^{\left(i\right)}$ determined by the solvability conditions

$${\int }_M{\varphi }_j\left(-a{\mathsf{\Delta }}_g{\Omega }_T+(R_g+E[T,\nabla T]){\Omega }_T-\lambda {\Omega }_T^p\right)d{\mu }_g=0.$$

On Einstein backgrounds $(M,g)$, the framework provides quantitative rigidity thresholds. For sufficiently small perturbations ${\parallel T\parallel }_{C^{2,\alpha }}\lesssim T_{\mathrm{crit}}$, the Einstein property is preserved. Larger deformations induce deviations controlled by the linearized Ricci operator

$$\delta {\mathrm{Ric}}^{\#}\left[T\right]=\delta \mathrm{Ric}\left[T\right]-{\displaystyle \frac{1}{n}}\left(\delta R\left[T\right]\right)g-{\displaystyle \frac{1}{n}}{R}_T,\delta R\left[T\right]=\mathrm{div}\mathrm{div}T-\mathsf{\Delta }\left(\mathrm{tr}T\right)-\langle \mathrm{Ric},T\rangle .$$

This provides explicit bounds on the maximal perturbation preserving the Einstein condition.

For manifolds with boundary, boundary curvature terms such as mean curvature H of $\partial M$ can be included, yielding mixed boundary conditions

$$\left\{\begin{array}{cc}-a{\mathsf{\Delta }}_g{\Omega }_T+(R_g+E[T,\nabla T]){\Omega }_T=\lambda {\Omega }_T^p\hfill & \mathrm{in}M,\hfill \\ B\left[{\Omega }_T\right]=f\left(T\right)\hfill & \mathrm{on}\partial M,\hfill \end{array}\right.$$

where B is a Dirichlet, Neumann, or Robin-type boundary operator and $f\left(T\right)$ encodes boundary perturbations. Perturbative methods provide existence, uniqueness, and bifurcation results under small boundary deformations.

In conclusion, compared with classical Yamabe theory, this framework provides a rigorous, technically precise, and fully quantitative description of perturbed solutions, capturing non-conformal deformations, kernel-induced bifurcations, boundary contributions, and stability under small tensorial perturbations.