k-Hessian curvature type equations in space forms

Zhou, Jundong

doi:10.58997/ejde.2022.18

Cited by 3 publications

(2 citation statements)

References 20 publications

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“…Inspired by the proof of Theorem 1.3, we can use the flow method to prove the existence of solution to the equation (1.2). To obtain the existence of (η, k)-convex hypersurface satisfying the prescribed curvature equation (1.2), we need two additional conditions on Ψ as in [4,5,33]. The first condition is that there exist two positive constants r 1 < 1 < r 2 such that…”

Section: If We Setmentioning

confidence: 99%

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A Class of inverse curvature type flows in Rn+1

Sheng,

Xue

2024

Preprint

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In this paper, we consider an expanding flow of smooth, closed, $(\eta,k)$-convex hypersurfaces in Euclidean $\mathbb{R}^{n+1}$ with speed $u^{\alpha}\rho^{\delta}\sigma_k^{-\frac{\beta}{k}}(\lambda(\eta))$, where $u, \rho$ are the support function and radical function of the hypersurface, respectively, $\alpha,\delta\in\mathbb{R}^1$, $\beta>0$, $k$ is an integer and $1 \leq k \leq n$, $\eta=Hg-h$, the first Newton transformation of the second fundamental form $h$, $\lambda(\eta)$ denote the eigenvalues of $g^{-1}\eta$. For $\alpha+\delta+\beta\leq 1$, we prove that the flow has a unique smooth and $(\eta,k)$-convex solution for all time, and converges smoothly after normalisation, to a sphere centered at the origin. Moreover, for $\alpha+\delta+\beta< 1$, we prove that the flow with the speed $fu^{\alpha}\rho^{\delta}\sigma_k^{-\frac{\beta}{k}}(\lambda(\eta))$ exists for all time, and converges smoothly after normalisation to a soliton, which is a solution of $fu^{\alpha-1}\rho^{\delta}\sigma_k^{-\frac{\beta}{k}}(\lambda(\eta))=\gamma$, provided that $f$ is a smooth positive function on $\mathbb{S}^n$ and $\gamma$ is a positive constant. What's more, we can use a more general flow to prove the existence of solution to a class of Hessian quotient equations again.

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Section: If We Setmentioning

confidence: 99%

“…For more related literatures, see [8,9,30,32] and reference therein. In [4,5,33], the authors studied the problem of prescribed Weingarten curvature with (k, l)-Hessian quotient equation of λ(η),…”

Section: Introductionmentioning

confidence: 99%

A Class of inverse curvature type flows in Rn+1

Sheng,

Xue

2024

Preprint

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A class of Hessian quotient equations in warped product manifolds

Sheng,

Xue

2024

Sci. China Math.

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Starshaped Compact Hypersurfaces in Warped Product Manifolds II: A Class of Hessian Type Equations

Wang

2024

J Geom Anal

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In this note, we prove the existence of one particular class of starshaped compact hypersurfaces, by deriving global curvature estimates for such hypersurfaces; this generalizes a recent result on this problem from the Euclidean space to Riemannian warped products. Moreover, we show that interior second order a priori estimates for admissible solutions to the associated fully nonlinear elliptic partial differential equations can be readily established by similar arguments.

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k-Hessian curvature type equations in space forms

Cited by 3 publications

References 20 publications

A Class of inverse curvature type flows in Rn+1

A Class of inverse curvature type flows in Rn+1

A class of Hessian quotient equations in warped product manifolds

Starshaped Compact Hypersurfaces in Warped Product Manifolds II: A Class of Hessian Type Equations

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