Curvature Estimates and Prescribed Curvature Problems for Sum Hessian Equations on Star-Shaped Hypersurfaces
Abstract
Let M be a closed star-shaped hypersurface in ℝn+1 and let η=Hg−h be the first Newton transformation of its second fundamental form. For a fixed constant α>0 and 2≤ k≤ n−1, we study the prescribed sum Hessian curvature equation σk(λ(η))+ασk−1(λ(η))=f(X,ν). Under the natural admissibility condition λ(η)∈Γk−1∩ {λ:σk(λ)+ασk−1(λ)>0}, we establish a global curvature estimate for closed star-shaped solutions with a general positive right-hand side depending on the position vector and the unit normal. A key point is that the sum Hessian operator is nonhomogeneous; the resulting lower-order term in the maximum-principle calculation is controlled by a partial curvature estimate. We also use the augmented-variable identity σk(λ)+ασk−1(λ) =σk(λ1,…,λn,α) to recover the relevant ellipticity and concavity properties. As an application of the a priori estimates, we prove existence and uniqueness for the corresponding prescribed curvature problem under suitable spherical barrier and radial monotonicity assumptions.
Related
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BibTeX
@misc{ma2026curvature,
title = {{Curvature Estimates and Prescribed Curvature Problems for Sum Hessian Equations on Star-Shaped Hypersurfaces}},
author = {Dongjing Ma and Peihe Wang and Hua Zhu},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.05032},
eprint = {2611.05032},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.05032},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Curvature Estimates and Prescribed Curvature Problems for Sum Hessian Equations on Star-Shaped Hypersurfaces
AU - Dongjing Ma
AU - Peihe Wang
AU - Hua Zhu
PY - 2026
DA - 2026/11/03
PB - arXiv
DO - 10.48550/arxiv.2611.05032
UR - https://arxiv.org/abs/2611.05032
AB - Let M be a closed star-shaped hypersurface in ℝ^n+1 and let η=Hg−h be the first Newton transformation of its second fundamental form. For a fixed constant α>0 and 2≤ k≤ n−1, we study the prescribed sum Hessian curvature equation σ_k(λ(η))+ασ_k−1(λ(η))=f(X,ν). Under the natural admissibility condition λ(η)∈Γ_k−1∩ {λ:σ_k(λ)+ασ_k−1(λ)>0}, we establish a global curvature estimate for closed star-shaped solutions with a general positive right-hand side depending on the position vector and the unit normal. A key point is that the sum Hessian operator is nonhomogeneous; the resulting lower-order term in the maximum-principle calculation is controlled by a partial curvature estimate. We also use the augmented-variable identity σ_k(λ)+ασ_k−1(λ) =σ_k(λ_1,…,λ_n,α) to recover the relevant ellipticity and concavity properties. As an application of the a priori estimates, we prove existence and uniqueness for the corresponding prescribed curvature problem under suitable spherical barrier and radial monotonicity assumptions.
ER -