[Paper Review] On the $σ_{k}$-Nirenberg problem

Yanyan Li, Luc Nguyen+1 more|arXiv (Cornell University)|Aug 19, 2020

Geometric Analysis and Curvature Flows57 references4 citations

TL;DR

This paper establishes existence and compactness results for the $σ_k$-Nirenberg problem on the $n$-sphere ($n \geq 3$) when $k \geq n/2$, proving that under a non-degeneracy condition on the prescribed curvature function $K$, solutions to the fully nonlinear curvature equation exist and remain uniformly bounded in $C^2$ norm. The key contribution is a sharp $C^2$ estimate for the logarithm of the conformal factor, derived via blow-up analysis and refined decay estimates.

ABSTRACT

We consider the problem of prescribing the $σ_k$-curvature on the standard sphere $\mathbb{S}^n$ with $n \geq 3$. We prove existence and compactness theorems when $k \geq n/2$. This extends an earlier result of Chang, Han and Yang for $n = 4$ and $k = 2$.

Motivation & Objective

• To establish existence and compactness of solutions to the $\sigma_k$-curvature prescription problem on the standard $n$-sphere for $k \geq n/2$.
• To extend earlier results by Chang, Han, and Yang on the $\sigma_2$-Nirenberg problem in dimension 4 to higher dimensions and general $k \geq n/2$.
• To analyze the blow-up behavior of approximate solutions and prove uniform $C^2$ bounds on the logarithm of the conformal factor.
• To develop sharp decay estimates for solutions to the $\sigma_k$-Yamabe equation in the critical and subcritical regimes.
• To characterize the role of critical points of the prescribed curvature $K$ via degree theory, particularly focusing on negative-definite critical points.

Proposed method

• Uses blow-up analysis to study sequences of approximate solutions and derive optimal decay estimates for the conformal factor $v$ in the equation $\sigma_k(\lambda(A_{g_v})) = K$.
• Applies divergence identities and Cacciopoli-type inequalities tailored to the $\sigma_k$-curvature equation to control $L^p$ norms of derivatives.
• Employs Evans-Krylov regularity theory to upgrade pointwise $C^2$ bounds to global $C^{2,\alpha}$ estimates on $\ln v$.
• Leverages the convexity of the conformal Hessian operator (Lemma A.1) to preserve $\Gamma_k$-ellipticity under averaging, aiding in compactness arguments.
• Uses radial symmetry and monotonicity estimates (Lemma A.2) for radially symmetric solutions to derive uniform decay rates in the $k > n/2$ case.
• Relies on degree theory to define and compute $\deg(\nabla K, \mathrm{Crit}_-(K))$, which serves as a topological obstruction to existence.

Experimental results

Research questions

• RQ1 Under what conditions on $K$ does the $\sigma_k$-curvature prescription problem on $\mathbb{S}^n$ admit a solution for $k \geq n/2$?
• RQ2 Can uniform $C^2$ bounds be established for solutions to the $\sigma_k$-Yamabe equation under non-degeneracy assumptions on $K$?
• RQ3 How does the behavior of solutions change when $k > n/2$ versus $k = n/2$, particularly in terms of decay and compactness?
• RQ4 What is the role of the degree $\deg(\nabla K, \mathrm{Crit}_-(K))$ in determining the existence of solutions?
• RQ5 Can sharp decay estimates for the conformal factor $v$ be derived in the blow-up regime using geometric and analytic tools?

Key findings

• For $k \geq n/2$, the $\sigma_k$-Nirenberg problem on $\mathbb{S}^n$ admits a solution if $K \in C^2(\mathbb{S}^n)$ is positive and satisfies the non-degeneracy condition $|\nabla_{g_0}K|_{g_0} + |\Delta_{g_0}K| > 0$.
• The solution set is compact in the $C^2$ topology: any sequence of solutions is uniformly bounded in $C^2(\mathbb{S}^n)$, with $\|\ln v\|_{C^2(\mathbb{S}^n)} \leq C_*$ for a constant $C_*$ depending only on $n$ and $K$.
• When $k > n/2$, the decay of the conformal factor $v$ is controlled via a monotonicity estimate (Lemma A.2), yielding $r^{n-2}u(r) \leq C R_i^{n-2} \lambda_i^{-(n-2)} u_i(0)^{-1}$ for $r \in [R_i/\lambda_i, \rho]$.
• The $C^2$ bound on $\ln v$ is derived from first and second derivative estimates and the Evans-Krylov theorem, ensuring $C^{2,\alpha}$ regularity uniformly in $i$.
• The degree $\deg(\nabla K, \mathrm{Crit}_-(K))$ is a topological invariant under $C^2$ convergence and equals the alternating sum of indices at non-degenerate negative-definite critical points.
• In the case $k = n/2$, the optimal decay estimate is proven via a refined analysis of the $\sigma_k$-equation, extending the method to the critical case.