The existence of $(\mathbf{p}, k)$-convex hypersurfaces for a class of Hessian quotient type curvature equations
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866913033635233792 |
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| author | Gong, Jiabao Tu, Qiang |
| author_facet | Gong, Jiabao Tu, Qiang |
| contents | This article investigates the existence of closed, star-shaped hypersurfaces for a class of Hessian quotient type curvature equations, in which the operator $\frac{σ_k}{σ_l}(Λ)$ arising in these equations can be viewed as a generalization of the classical Hessian quotient operator. By combining a priori estimates with the continuity method, we establish the existence and uniqueness of $(\mathbf{p}, k)$-convex hypersurfaces for both nonhomogeneous and homogeneous equations of this type. Furthermore, by exploiting the recently discovered ``inverse convexity'' property of the operator $\frac{σ_k}{σ_l}(Λ)$, we prove a constant rank theorem and thereby obtain the existence and uniqueness of strictly convex solutions to these curvature equations. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_13578 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | The existence of $(\mathbf{p}, k)$-convex hypersurfaces for a class of Hessian quotient type curvature equations Gong, Jiabao Tu, Qiang Analysis of PDEs This article investigates the existence of closed, star-shaped hypersurfaces for a class of Hessian quotient type curvature equations, in which the operator $\frac{σ_k}{σ_l}(Λ)$ arising in these equations can be viewed as a generalization of the classical Hessian quotient operator. By combining a priori estimates with the continuity method, we establish the existence and uniqueness of $(\mathbf{p}, k)$-convex hypersurfaces for both nonhomogeneous and homogeneous equations of this type. Furthermore, by exploiting the recently discovered ``inverse convexity'' property of the operator $\frac{σ_k}{σ_l}(Λ)$, we prove a constant rank theorem and thereby obtain the existence and uniqueness of strictly convex solutions to these curvature equations. |
| title | The existence of $(\mathbf{p}, k)$-convex hypersurfaces for a class of Hessian quotient type curvature equations |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2604.13578 |