# The Gursky-Streets equation and its application to the $\sigma_k$ Yamabe problem

## TL;DR

This paper introduces solutions to the Gursky-Streets equation, a key geometric PDE in conformal geometry, establishing uniform estimates, concavity, and uniqueness results, with applications to the $\sigma_k$ Yamabe problem on conformally flat manifolds.

## Contribution

It proves uniform $C^{1,1}$ estimates and concavity of the Gursky-Streets operator for all $k \,\leq\, n$, and establishes the first uniqueness results for degenerate solutions.

## Key findings

- Established uniform $C^{1,1}$ estimates for solutions.
- Proved the concavity of the Gursky-Streets operator for all $k\leq n$.
- Proved uniqueness of solutions to degenerate equations.

## Abstract

The Gursky-Streets equation are introduced as the geodesic equation of a metric structure in conformal geometry. This geometric structure has played a substantial role in the proof of uniqueness of $\sigma_2$ Yamabe problem in dimension four. In this paper we solve the Gursky-Streets equations with uniform $C^{1, 1}$ estimates for $2k\leq n$. An important new ingredient is to show the concavity of the operator which holds for all $k\leq n$. Our proof of the concavity heavily relies on Garding's theory of hyperbolic polynomials and results from the theory of real roots for (interlacing) polynomials. Together with this concavity, we are able to solve the equation with the uniform $C^{1, 1}$ \emph{a priori estimates} for all the cases $n\geq 2k$. Moreover, we establish the uniqueness of the solution to the degenerate equations for the first time. As an application, we prove that if $k\geq 3$ and $M^{2k}$ is conformally flat, any solution solution of $\sigma_k$ Yamabe problem is conformal diffeomorphic to the round sphere $S^{2k}$.