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

TL;DR

This paper proves the first nonlinear stability of a family of self-similar naked singularity solutions in Einstein-scalar field equations under small, localized perturbations, highlighting the importance of the functional framework in cosmic censorship.

Contribution

It establishes the nonlinear stability of these naked singularities for the first time under general perturbations with the same regularity as the background.

Findings

Proves nonlinear stability of naked singularities in Einstein-scalar field equations.

Shows stability holds for perturbations in a localized Hölder topology.

Highlights the role of the functional framework in weak cosmic censorship.

Abstract

This is the first part of a series of papers proving the nonlinear stability of a one-parameter family of continuously self-similar $C^{1,\alpha}$ naked singularity solutions, with $0<\alpha\ll1$, 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 H\"older 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.