Existence of solution to modified Gursky-Streets equation

TL;DR

This paper proves the existence of solutions, including Lipschitz viscosity solutions, for the modified Gursky-Streets equation in conformal geometry under specific conditions, advancing understanding of fully nonlinear geometric PDEs.

Contribution

It establishes existence results and uniform estimates for solutions to the modified Gursky-Streets equation in certain parameter regimes.

Findings

Existence of solutions with uniform $C^{1,1}$ estimates for $eta > 0$ or $r > 0$ under specified conditions.

Existence of Lipschitz continuous viscosity solutions when $r eq 0$.

Advances in solving fully nonlinear conformal geometric equations.

Abstract

We solve the modified Gursky-Streets equation, which is a fully nonlinear equation arising in conformal geometry with uniform $C^{1, 1}$ estimates when (i) $\gamma > 0$ and $1 \leq k \leq n$ or (ii) $r > 0$ and $2 s k \leq r n$. We also prove the existence of a Lipschitz continuous viscosity solution when $r \neq 0$.