Properties of hypersurface singular sets of solutions to the $σ_k$-Yamabe equation in the negative cone

1. Problem Statement and Context

The Equation:
The paper investigates the $\sigma_k$-Yamabe equation in the negative cone on a domain $\Omega \subset \mathbb{R}^n$ ($n \geq 3$). The equation is given by:
$$\sigma_k(\lambda(-A_w)) = 1, \quad \lambda(-A_w) \in \Gamma_k^+ \quad \text{in } \Omega$$
where:

• $w$ is a positive conformal factor defining the metric $g = w^{-2}|dx|^2$.
• $A_w = w\nabla^2 w - \frac{1}{2}|\nabla w|^2 I$ is the Schouten tensor (type $(1,1)$).
• $\sigma_k$ is the $k$-th elementary symmetric polynomial of the eigenvalues of $-A_w$.
• $\Gamma_k^+$ is the Gårding cone defined by $\sigma_j(\lambda) > 0$ for $1 \leq j \leq k$.

The Singularity:
The authors focus on conformally flat Lipschitz viscosity solutions that admit a hypersurface of singular points $\Sigma$. Unlike the boundary of the domain (where solutions to the related Loewner-Nirenberg problem are smooth), $\Sigma$ is an internal hypersurface where the solution $w$ is:

• Lipschitz continuous ($C^{0,1}$) everywhere.
• $C^1$ and $C^2$ on either side of $\Sigma$ ($B_\varepsilon^+$ and $B_\varepsilon^-$).
• Not $C^1$ across $\Sigma$ (the normal derivative $\nabla_\nu w$ has a jump discontinuity).

Motivation:
This study is motivated by recent results on the $\sigma_k$-Loewner-Nirenberg problem (where $w=0$ on $\partial\Omega$). For $k \geq 2$ and domains with multiple boundary components, solutions are known to develop such internal singular hypersurfaces (e.g., in annuli). The paper aims to characterize the geometric and analytic properties of these singular sets.

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2. Methodology

The authors employ a combination of viscosity solution theory, asymptotic analysis, and geometric PDE techniques.

• Viscosity Test Functions: To handle the lack of classical differentiability across $\Sigma$, the authors construct specific $C^2$ test functions ($\xi_{\alpha, \beta, \gamma}$) that touch the solution $w$ from above or below at points on $\Sigma$. By applying the definition of viscosity supersolutions/subsolutions to these test functions, they derive necessary conditions on the eigenvalues of the Schouten tensor.
• Local Coordinate Analysis: The analysis is performed in a local coordinate system where $\Sigma$ is the graph of a function $\phi$. They decompose derivatives into tangential ($\nabla_\Sigma$) and normal ($\nabla_\nu$) components.
• Asymptotic Expansion (Formal Series): For the case $k=2$, the authors perform a detailed asymptotic expansion of the solution near the singularity. They assume an ansatz of the form:
  $$w(x) \approx w_0(\pi(x)) + w_1(\pi(x))d(x) + w_*(\pi(x))d(x)^p$$
  where $d(x)$ is the distance to $\Sigma$, $\pi(x)$ is the projection onto $\Sigma$, and $p$ is an exponent to be determined.
• Eigenvalue Interlacing: They utilize Cauchy's interlacing theorem to relate the eigenvalues of the full Hessian to the eigenvalues of the tangential sub-matrix, allowing them to deduce conditions on the geometry of $\Sigma$.

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3. Key Contributions and Results

A. General Case ($k \geq 2$): The PDE on the Singular Set

Theorem 1.2: If $w$ is a viscosity solution with a singular hypersurface $\Sigma$, then the trace $w_0 = w|_\Sigma$ and the normal derivatives $\nabla_\nu w^\pm$ satisfy a specific polynomial equation on $\Sigma$.
Specifically, for $\alpha \in \{\nabla_\nu w^+, \nabla_\nu w^-\}$, the tensor $T_\alpha$ defined by:
$$T_\alpha = -w_0(\nabla_\Sigma^2 w_0 - \alpha \text{II}_\Sigma) + \frac{1}{2}(|\nabla_\Sigma w_0|^2 + \alpha^2)g_\Sigma$$
satisfies the condition:
$$\lambda(g_\Sigma^{-1} T_\alpha) \in \partial \Gamma_{k-1}^+ \quad \text{on } \Sigma.$$
Significance: This result proves that the failure of $C^1$ regularity is not arbitrary. The jump in the normal derivative is constrained by a precise nonlinear PDE relating the geometry of $\Sigma$ (via the second fundamental form $\text{II}_\Sigma$) and the solution's values.

B. The Case $k=2$: Minimality of the Singular Set

Theorem 1.3: For $k=2$, the singular hypersurface $\Sigma$ is minimal with respect to the metric $g = w^{-2}|dx|^2$ restricted to either side of the singularity.

• The mean curvature of $\Sigma$ with respect to the metric $g|_{B_\varepsilon^\pm}$ and the normal $\pm \nu$ is negative.
• In the variational sense, $\Sigma$ is stationary for the area functional associated with the Lipschitz metric.
  Significance: This connects the analytic singularity of the PDE solution directly to a geometric property (minimality). It echoes previous non-existence results for $C^1$ solutions in multi-component domains, where the existence of a minimal hypersurface was a key obstruction.

C. Asymptotic Regularity and Expansion

Proposition 1.6: The authors investigate the Hölder regularity of the gradient near the singularity. They search for an expansion where the next term scales as $d(x)^p$.

• They prove that such an expansion satisfying the equation $\sigma_2(\lambda(-A_{\bar{w}})) = 1$ exists if and only if the exponent is $p = 3/2$.
• The coefficient $w_*$ of this term is uniquely determined and negative.
  Significance: This confirms the observed regularity $C^{1, 1/2}$ (since $p=3/2$ implies the second derivative behaves like $d^{-1/2}$) for the $\sigma_2$-Loewner-Nirenberg problem on annuli, suggesting this regularity holds in more general settings where ODE methods fail.

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4. Technical Highlights

1. Ordering of Normal Derivatives (Lemma 2.1): The authors first establish that $\nabla_\nu w^+ < \nabla_\nu w^-$ on $\Sigma$. This ordering is crucial for the subsequent analysis of the viscosity inequalities.
2. Degenerate Ellipticity: The paper notes that near the singular hypersurface, the equation becomes degenerate elliptic, unlike near the boundary where it remains uniformly elliptic. This complicates standard regularity theory but is handled via the specific ansatz in Proposition 1.6.
3. Connection to Minimal Hypersurfaces: The proof of Theorem 1.3 relies on the specific structure of the equation for $k=2$, showing that the differential inequality derived from the PDE forces the mean curvature to be negative, implying minimality in the variational sense.

5. Significance and Impact

• Regularity Theory: The paper provides a rigorous framework for understanding the structure of singularities in fully nonlinear elliptic equations of Monge-Ampère type (specifically $\sigma_k$-Yamabe). It moves beyond existence to describe the nature of the singularity.
• Geometric Analysis: By linking the singular set to minimal hypersurfaces (for $k=2$), the work bridges the gap between nonlinear PDEs and differential geometry.
• Generalization: While previous results on singularities were largely restricted to annuli (where symmetry allows ODE analysis), this paper extends these insights to general domains using viscosity solution techniques, suggesting that the $C^{1, 1/2}$ regularity and the geometric constraints on $\Sigma$ are universal features of these problems.
• Future Directions: The results open questions about the geometric nature of singular sets for $k \geq 3$ and the precise Hölder regularity of gradients in higher dimensions.

In summary, Duncan and Nguyen demonstrate that the singularities in the negative $\sigma_k$-Yamabe equation are highly structured, governed by a specific boundary PDE on the singular set, and geometrically characterized as minimal surfaces (for $k=2$) with a specific $C^{1, 1/2}$ regularity profile.