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arXiv · 2408.14850

An Integral Approach to Prescribing Scalar Curvature Equations

Ruosi Chen·Huaiyu Jian·Xingchen Zhou
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Abstract

We develop an integral approach to obtain interior a priori $C^{1,1}$ estimates for convex solutions of prescribing scalar curvature equations $σ_2(κ) = f(x)$ as well as the Hessian equations $σ_2(D^2u) = f(x)$. This new approach can deal with the case when $f$ is of weaker regularity. As a result, we prove that the $C^{1,1}$ modules of the solutions depend only on the Lipschitz modules of $f(x)$, instead of the $\|f\|_{C^k}$ for some $k\geq 2$ in all the papers we have known up to now.

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Ruosi Chen, Huaiyu Jian, Xingchen Zhou. 2024-08-29. An Integral Approach to Prescribing Scalar Curvature Equations. https://arxiv.org/abs/2408.14850

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