Nonlinear stability of continuously self-similar naked singularities for the Einstein-scalar field equations I: main results

Abstract: This is the first part of a series of papers proving the nonlinear stability of a one-parameter family of continuously self-similar $C<sup>{1,α}$ naked singularity solutions, with $0&lt;α\ll1$, to the spherically symmetric Einstein-scalar field equations. The stability holds for initial perturbations lying in a small open neighborhood of the data generating these naked singularity solutions, measured in a localized Hölder topology. These continuously self-similar naked singularity spacetimes were previously constructed by Christodoulou [D. Christodoulou, Examples of naked singularity formation in the gravitational collapse of a scalar field, Ann. of Math. 140 (1994), 607--653], who also proved their instability to black hole formation under sufficiently rough perturbations [D. Christodoulou, The instability of naked singularities in the gravitational collapse of a scalar field, Ann. of Math. 149 (1999), 183--217], thereby verifying weak cosmic censorship within a rough functional framework. In complete contrast, in this paper, we obtain the first nonlinear stability of these naked singularity spacetimes under general perturbations of the same regularity as the background. We rely on the linearized stability result established in the companion paper [J. Singh and W. Zheng, Nonlinear stability of continuously self-similar naked singularities for the Einstein--scalar field equations II: linearized stability]. Our result underscores the decisive role of the functional framework in formulating the Weak Cosmic Censorship conjecture.

• The paper proves the nonlinear stability of a family of self-similar naked singularity solutions within the threshold C^(1,α) framework.
• It employs robust linear decay estimates to control small perturbations, ensuring the solutions remain free of trapped surfaces.
• The findings challenge the Weak Cosmic Censorship by demonstrating that naked singularity formation is generic in a localized Hölder topology.

Nonlinear Stability of Continuously Self-Similar Naked Singularities: Main Results and Implications

Introduction and Background

This paper establishes the nonlinear stability of a one-parameter family of continuously self-similar, $C^{1,\alpha}$ naked singularity solutions to the spherically symmetric Einstein-scalar field equations (SSESFEs) for small $\alpha>0$ (2605.16235). The analysis is conducted in the context of initial data with localized Hölder regularity, specifically in a neighborhood of the data generating the known $k$-self-similar naked singularities first constructed by Christodoulou [chris94].

Historically, Christodoulou demonstrated that these self-similar naked singularities are unstable to black hole formation under generic, sufficiently rough (BV-class) perturbations [chris99]. This was seen as strong support for the Weak Cosmic Censorship (WCC) conjecture, which predicts that singularities visible to infinity (naked singularities) should not form from generic, physically meaningful initial data. However, these earlier instability results relied on considering perturbations of rougher regularity than the background solution itself, raising the question of whether instability persists when both the solution and perturbations are matched in regularity.

Statement of Main Results

This paper proves that the family of $k$-self-similar naked singularity solutions, constructed for $k^2\in (0, 1/3)$, are nonlinearly stable with respect to perturbations in a small open neighborhood of initial data in the localized Hölder class $C^{1,\alpha}$ with $\alpha = k^2/(1-k^2)$. That is, if the initial scalar field data is perturbed within this threshold regularity class, the resultant solution to the Einstein-scalar field system in spherical symmetry:

• Remains free of trapped surfaces,
• Contains a future-incomplete null infinity,
• Asymptotically converges (in the region of interest) to a self-similar naked singularity spacetime.

The theorem contradicts the $C^{1,\alpha}$-topology version of the WCC in spherical symmetry: in this topology, the naked singularity solutions are stable and hence generic within the small neighborhood defined by the topology.

Strong quantitative estimates are given for the decay rates of deviations from exact self-similarity, in terms of the time coordinate approaching the singularity. This is supported by the companion linear analysis [zhenglinear], which establishes robust decay for solutions to the linearized SSESFEs around the naked singularity background in this regularity regime.

Threshold Regularity and the Functional Framework

A central insight is the decisive role of regularity in singularity (in-)stability. The paper introduces the notion of a threshold regularity—the minimal smoothness at which the self-similar naked singularity becomes stable to small perturbations. This threshold is precisely the localized Hölder class intrinsic to the Christodoulou solution: $C^{1,k^2/(1-k^2)}$.

Below this threshold (i.e., rougher perturbations, BV class, or Hölder exponents $\alpha < k^2/(1-k^2)$), the naked singularities are unstable and lead to black hole formation—consistent with prior works [chris99, liuli, an_highcodim, li2025interior]. At or above threshold, the “blue-shift” instability mechanism, which amplifies incoming perturbations near the singularity, is suppressed—a phenomenon confirmed by both previous linear analyses [singh2] and the nonlinear results of this paper.

This dichotomy underscores that properly formulating the WCC requires specifying the underlying functional spaces—the statement “generic data do not form naked singularities” is ill-posed without reference to the function space topology.

Mathematical and Physical Context

The Einstein-scalar field system in spherical symmetry, when restricted by self-similarity ($\alpha>0$0-self-similarity), admits a powerful ODE reduction, allowing precise analysis of both the background solution and the evolution of perturbations. The current nonlinear arguments crucially exploit:

• The companion linear decay estimates [zhenglinear], available for small $\alpha>0$1, to control the propagation and nonlinear coupling of perturbations.
• Intrinsic symmetries, including scaling and translation of the scalar field, to decompose the solution space and isolate potential non-decaying modes.

It is also critical that the threshold regularity incorporates the matching of the analytic class of the solution and its perturbations; earlier instability analyses considered lower-regularity data, thus did not strictly demonstrate non-genericity in the sense illuminated here.

Contradictory and Strong Claims

Contradictory to earlier expectations and the typical interpretation of the WCC—which asserts non-genericity of naked singularities—the main theorem demonstrates genericity (in an open set) of the self-similar naked singularity in the $\alpha>0$2 topology. This runs in direct opposition to the predicted scenario in the BV setting and earlier works focused on black hole formation from generic rough data [chris99, liuli].

Numerically, the existence of a “critical value” for one-parameter families of initial data, as in the critical collapse literature [choptuik1, gund_understandingcritcollapse], is also addressed. The Christodoulou solution does not correspond to a critical value dividing dispersion and black hole formation when data is scaled within the $\alpha>0$3 neighborhood. Rather, there exists an open interval of parameter values all producing naked singularities, further revealing the nuanced structure of the solution space.

However, the result is not universal: for large or rough perturbations (those with sufficiently large norm in threshold spaces, or lower regularity), black hole formation resumes—see the appendix for a rigorous black hole formation theorem under large smooth perturbations, emphasizing the non-criticality of the naked singularity as an attractor.

Broader Implications and Future Directions

Theoretical Implications

• Weak Cosmic Censorship is not universally valid without precise functional specifications. This work sharpens the focus: in $\alpha>0$4, $\alpha>0$5 is stable and generic; in $\alpha>0$6 or rougher, it is unstable. The question of what constitutes a “physically meaningful” space for initial data is central.
• The existence of a threshold regularity matches the rigorous behavior of self-similar and other nonlinear wave equations near blowup, where high regularity or codimension-one phenomena (critical attractors) play a crucial role [donninger2012stable, costin2016stability].

Practical and Mathematical Consequences

• The methods and functional framework invite transfer to other PDEs exhibiting self-similar singularity formation, including wave maps, Yang--Mills, fluid, and Schroedinger equations [donninger2024spectral, costin2016stability, glogic2025globally].
• The approach influences future studies of naked singularity formation in less symmetric settings (e.g., vacuum Einstein, Euler, Einstein-Maxwell-scalar), where the function space topology may play an analogous role [yakov_rod23, guo_hadzic_jang21].

Open Problems

Several substantial open problems are identified:

• Characterization of transition regimes: For initial data one-parameter families near the threshold regularity, identifying the sharp transition from dispersive to singular or black hole solutions.
• BV vs. Hölder frameworks: Determining whether dispersion is possible within the $\alpha>0$7 class but not in $\alpha>0$8.
• Nonlinear stability for the full parameter range: Extending this analysis beyond the small $\alpha>0$9 regime, or to other matter models, is an open technical challenge.
• Strong cosmic censorship and genericity in less symmetric (axisymmetric, vacuum) cases: These remain largely unresolved and require advances in both forming and analyzing singularities in low-regularity, non-symmetric settings.

Conclusion

This paper rigorously establishes that the continuously self-similar naked singularity solutions of the SSESFEs are nonlinearly stable at threshold regularity, revealing a sharp dependence of the Weak Cosmic Censorship conjecture on the topology of initial data. The generic formation of naked singularities within an open set of $k$0 data is both a technical and conceptual advance, highlighting the need for precise functional formulations in mathematical relativity. This development connects with a wide range of nonlinear PDE phenomena—especially those characterized by self-similar singularity formation and critical thresholds—and sets the stage for further analytical and conceptual progress in the mathematical foundations of General Relativity and related nonlinear wave equations.

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References

• "Nonlinear stability of continuously self-similar naked singularities for the Einstein-scalar field equations I: main results" (2605.16235)
• D. Christodoulou, "Examples of naked singularity formation in the gravitational collapse of a scalar field," Ann. Math. 140 (1994), 607–653 [chris94]
• D. Christodoulou, "The instability of naked singularities in the gravitational collapse of a scalar field," Ann. Math. 149 (1999), 183–217 [chris99]
• J. Singh and W. Zheng, "Nonlinear stability of continuously self-similar naked singularities for the Einstein-scalar field equations {2}: linearized stability" [zhenglinear]
• J. Liu and J. Li, “A Robust Proof of the Instability of Naked Singularities of a Scalar Field in Spherical Symmetry,” Commun. Math. Phys. 363 (2018), 561–578 [liuli]
• X. An, "Naked singularity censoring with anisotropic apparent horizon," Ann. Math., to appear [an_highcodim]
• J. Li, "Interior instability of naked singularities of a scalar field," (Li, 11 Aug 2025) [li2025interior]
• J. Singh, "A Construction of Approximately Self-Similar Naked Singularities for the Spherically Symmetric Einstein-Scalar Field System," Ann. Henri Poincaré, 2024 [singh1]
• J. Singh, "Regimes of (in-)stability for self-similar naked singularities," PhD thesis, 2025 [singhphdtheis]
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