Viscosity solution to complex Hessian quotient equations
Jingrui Cheng, Yulun Xu
TL;DR
This work develops a viscosity-theoretic framework for complex Hessian quotient equations on compact Hermitian manifolds. By introducing and exploiting a viscosity strict subsolution, the authors prove the existence of viscosity solutions to $f(\lambda[\chi+dd^c\varphi])=e^{G(x)+c}$ under a precise constant $c$ defined by $e^{c}=\inf_{u\in\mathcal{E}_{\Gamma,\chi}}\max_M e^{-G}f(\lambda[\chi+dd^cu])$, generalizing beyond determinant-domination assumptions. The core technique combines regularization of the asymptotic operator $f_{\infty}$ with Richberg gluing to obtain smooth subsolutions, plus a stability estimate that upgrades $L^1$ convergence to uniform convergence to yield a viscosity solution. The paper also establishes uniqueness results under strict monotonicity in the right-hand side and discusses conditional uniqueness when the RHS is $\varphi$-independent, outlining a conjecture on the monotonicity of the canonical constant $c(G)$. Overall, the work extends the viscosity approach to Hessian quotient equations and provides a robust pathway from subsolutions to full viscosity solvability on Hermitian manifolds, with potential implications for complex geometric PDEs and related convexity structures.
Abstract
In this paper, we prove the existence of viscosity solutions to complex Hessian equations on compact Hermitian manifolds, assuming the existence of a strict subsolution in the viscosity sense. The results cover the complex Hessian quotient equations. This generalized our previous results where the equation needs to satisfy a determinant domination condition.