Critical Singular Riemannian Metrics

• Critical singular Riemannian metrics are defined by the L^(n/2)-curvature functional and exhibit singularities that form natural critical points in geometric analysis.
• The methodology involves computing the Hessian for conformal deformations, where Laplacian eigenvalues determine stability and diagnose saddle points in symmetric spaces.
• Investigations show that while metrics on round spheres and hyperbolic spaces can be stable, many compact symmetric spaces (e.g., SU(q)) become variationally unstable under conformal perturbations.

Critical singular Riemannian metrics are Riemannian metrics exhibiting singularities (e.g., points or submanifolds where the metric degenerates or fails to be smooth) that arise as critical points for natural geometric functionals, most notably those of the form $\int_M |R(g)|^p\, dv_g$ with $p = n/2$ on an $n$-dimensional closed manifold. These metrics are tightly linked to the structure of the $L^{n/2}$-norm of the curvature tensor, which is both variationally and analytically critical in Riemannian geometry. Their paper lies at the intersection of regularity theory for geometric PDE, conformal invariance, the calculus of variations, and the theory of singularities in geometric analysis.

1. The $L^{n/2}$-Curvature Functional and its Variational Significance

The central object of paper is the functional

$$\mathcal{R}_{\frac{n}{2}}(g) = \int_M |R(g)|^{\frac{n}{2}}\, dv_g,$$

where $R(g)$ is the Riemann curvature tensor of the metric $g$, and integration is over a closed smooth $n$-manifold $M$ of unit volume. The exponent $p=n/2$ is critical for several reasons:

• Conformal invariance: In dimension $n$, $\mathcal{R}_{n/2}(g)$ is invariant under conformal rescaling of the metric, analogous to the critical Sobolev exponent.
• Regularity threshold: $L^{n/2}$-bounds for the curvature are minimal for precompactness and regularity in key compactness and singularity-removal theorems (notably in the Cheeger-Gromov and Anderson-Tian PDE theories).
• Analytical rigidity: Blow-ups and singular structures at this scale often model geometric singularities or degenerations, such as in limit spaces and critical flows.

A metric is called critical for $\mathcal{R}_{n/2}$ (or more generally for $\mathcal{R}_p$ for any $p$) if it satisfies the Euler–Lagrange equation for the functional under the constraint $\operatorname{vol}(g) = 1$.

2. Classification of Critical Metrics: Stability and Instability

Critical metrics for $\mathcal{R}_{n/2}$ are natural generalizations of Einstein metrics, especially in cases where the variational problem is neither linear nor weakly regular. The main results (Bhattacharya et al., 2012) yield strong classification and stability phenomena:

• Locally symmetric spaces: Metrics on locally symmetric spaces (for example, round spheres, real/complex/quaternionic projective spaces, Grassmannians) are always critical for $\mathcal{R}_p$, as $R(g)$ is parallel (covariant derivative vanishes).
• Stability for $p>2$: For $p > 2$, round spheres $(S^n)$ and their quotients, as well as hyperbolic spaces, are stable critical points—the Hessian of the functional is positive-definite in all directions modulo diffeomorphism and scaling.
• Instability in the critical case: For $p = n/2$, there exist important classes of classical, compact, irreducible, locally symmetric spaces that are unstable critical points for $\mathcal{R}_{n/2}$, i.e., saddle points.

The explicit instability proven for certain symmetric spaces—e.g., $SU(q)$ ($q\geq 3$), $Sp(q)$ ($q\geq 2$), $E_6/F_4$, etc.—means that in the space of unit-volume metrics, there exist directions (notably conformal deformations linked to Laplacian eigenfunctions) along which the functional decreases. Thus, these metrics do not correspond to local minima and are variationally unstable.

3. Second Variation: Structure of the Hessian and Conformal Instability

The instability phenomenon is elucidated by computing the explicit Hessian in the direction of conformal deformations. For a locally symmetric metric $g$ and conformal variation $h=fg$, the second variation of $\mathcal{R}_p$ at $g$ is given by

$$H(fg, fg) = p|R|^{p-2}\big(a|\Delta f|^2 - b|df|^2 + c|f|^2\big),$$

where the coefficients $a, b, c$ are explicit and depend on dimension, structure constants, and the curvature of $g$ (see, for example, [(Bhattacharya et al., 2012), Proposition 2.1]).

The sign of the Hessian is governed by the spectrum of the (negative) Laplacian acting on functions. Instability arises precisely when the lowest nonzero Laplacian eigenvalue is small enough to violate positivity, combined with large enough coefficients $b, c$ (i.e., when conformal perturbations along Laplacian eigenfunctions reduce the value of the functional).

This computation is made precise using concrete data on Laplacian spectra and curvature tensors from sources such as Besse's monograph and Urakawa’s comparison results.

4. Examples and Rigidity Table

A summary of the stability analysis is encapsulated in the following table:

Space Type	Stability for $\mathcal{R}_p$	Remarks
Sphere ($S^n$), hyperbolic space	Stable ($p>2$)	Local minimizer
Certain noncompact locally symmetric spaces	Stable (conformal directions)	Guarantee holds for $p>2$
Compact symmetric spaces in Theorem 1 (e.g., $SU(q)$, ...)	Unstable ($p=n/2$)	Saddle point under conformal deformations
Other compact locally symmetric spaces	Stable (conformal)	Not in the unstable class

Hence, the set of unstable critical metrics for the $L^{n/2}$-norm encompasses a wide array of symmetric spaces, which are otherwise canonical in many geometric variational problems.

5. Context: Critical Singular Metrics and Relation to Curvature Blowing-Up

The $L^{n/2}$-criticality is also the threshold for the analytic formation of singularities in geometric flows and in the paper of metric degenerations:

• Singularity formation: Metrics with controlled $L^{n/2}$-norm of curvature serve as models for singular Einstein and Yamabe metrics and for the blow-up analysis in geometric PDE.
• Concentration-compactness: Essential results in Cheeger-Gromov convergence, regularity for Einstein metrics, and the paper of bubbling phenomena depend critically on $L^{n/2}$-curvature bounds.
• Notion of critical singular metric: A metric that minimizes or is critical for $L^{n/2}$-curvature, possibly allowed to have controlled singularities (e.g., isolated points or regions where the metric degenerates or is only in $W^{1,p}$), is often termed a critical singular metric.

This broader context links the paper of variational criticality to the analysis of singular spaces, regularity thresholds, and the behavior of solutions under degeneration.

6. Broader Implications and Open Problems

• Sharpness of instability: The existence of unstable critical metrics for $\mathcal{R}_{n/2}$ on classic symmetric spaces places important constraints on geometric flows, such as $L^p$-gradient flows or conformally invariant flow evolutions, suggesting bifurcation and nonuniqueness phenomena.
• Interplay with conformal geometry: The conformal invariance of the functional at $p=n/2$ makes the structure of critical metrics especially rich, with intricate moduli and possible singularity formation.
• Extension to singular and critical metrics: The paper extends the understanding of critical points for curvature functionals to settings that naturally include singularities and lay the groundwork for further paper of "critical singular" metrics, their formation, and their role in geometric analysis.

7. Key Formulas

The essential analytic formulas are:

• Curvature functional:

$\mathcal{R}_p(g) = \int_M |R(g)|^p\, dv_g$

• Second variation (conformal):

$$H(fg, fg) = p|R|^{p-2}[a|\Delta f|^2 - b|df|^2 + c|f|^2]$$

where $a, b, c$ are explicit in terms of geometric data of $g$.

• Instability and explicit classification (Bhattacharya et al., 2012)
• Laplacian eigenvalue and curvature data [Besse, "Einstein Manifolds"; Urakawa, Compositio Math 1986]
• Prior stability results for $L^p$-norms [Maity, (Maity, 2012)]

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In summary, critical singular Riemannian metrics with respect to the $L^{n/2}$-norm of the curvature exhibit a rich variational landscape in which certain classical locally symmetric spaces act as unstable saddle points, while others are variationally rigid. The structural analysis of the Hessian and the interplay with the Laplacian spectrum pinpoints the precise sources of instability, with profound implications for the formation and structure of singularities and for the calculus of variations in Riemannian geometry.