Doubling Argument of the Hessian Estimate for the Hessian Quotient Equations
Abstract
In this paper, we establish a doubling argument to obtain Hessian estimates for convex solutions to the Hessian quotient equation σn/σk(D2u) = f(x,u,Du) for k=n−1 and k=n−2 under the condition that log f is convex in the Du variable. In particular, our approach is pointwise and does not make use of the Legendre transform or integral-based local maximum principles. We provide a counterexample demonstrating that interior estimates can fail if no structural assumption is imposed on f in the Du variable. Finally, we extend our doubling argument to general Hessian quotient equations σl/σk(D2u) = f(x,u,Du) for k ∈ {l−1, l−2}, under a similar structural condition imposed on f in the Du variable, alongside an additional structural concavity assumption on the operator introduced by Lu-Tsai 2026, which has very recently been established in independent works.
Related
- Interior Hessian estimates for k-convex solutions of Hessian quotient equations with 2k>n
- Hessian Estimates for Convex Solutions of Hessian Quotient Equations via Compactness
- Poincaré-Type Hessian Quotient Equations
- A class of sum Hessian quotient equations arising in conformal geometry
- Interior Hessian estimates for k-convex solutions of Hessian quotient equations
BibTeX
@misc{fung2026doubling,
title = {{Doubling Argument of the Hessian Estimate for the Hessian Quotient Equations}},
author = {Cheuk Yan Fung},
howpublished = {arXiv},
year = {2026},
month = aug,
doi = {10.48550/arxiv.2607.21982},
eprint = {2607.21982},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2607.21982},
}
RIS
TY - UNPB
TI - Doubling Argument of the Hessian Estimate for the Hessian Quotient Equations
AU - Cheuk Yan Fung
PY - 2026
DA - 2026/08/28
PB - arXiv
DO - 10.48550/arxiv.2607.21982
UR - https://arxiv.org/abs/2607.21982
AB - In this paper, we establish a doubling argument to obtain Hessian estimates for convex solutions to the Hessian quotient equation σ_n/σ_k(D^2u) = f(x,u,Du) for k=n−1 and k=n−2 under the condition that log f is convex in the Du variable. In particular, our approach is pointwise and does not make use of the Legendre transform or integral-based local maximum principles. We provide a counterexample demonstrating that interior estimates can fail if no structural assumption is imposed on f in the Du variable. Finally, we extend our doubling argument to general Hessian quotient equations σ_l/σ_k(D^2u) = f(x,u,Du) for k ∈ {l−1, l−2}, under a similar structural condition imposed on f in the Du variable, alongside an additional structural concavity assumption on the operator introduced by Lu-Tsai 2026, which has very recently been established in independent works.
ER -