论文标题
Minkowski空间中整个Hypersurfaces的规定曲率问题
The prescribed curvature problem for entire hypersurfaces in Minkowski space
论文作者
论文摘要
我们在本文中证明了三个结果。首先，我们证明了C^2中的一类功能$φ\（\ Mathbb {s}^{n-1}）$和$ψ（x，x，v）in c^2（\ MathBB {r}^{r}^{n+1} {n+1} \ times \ times \ times \ times \ times \ times \ mathbb {h}^n） $ m_u $满足$σ_k（κ[m_u]）=ψ（x，x，x，n）$和$ u（x）\ rightarrow | x |+φ\ left（\ frac {x} {x | x | x |} \ right）$ as $ | x | x | x | \ rightArlow \ rightarrow \ rightrow \ infty。 $ k $ -convex，spacelike hypersurface $ m_u $满足$σ_k（κ[m_u]）=ψ（x，x，u（x））$和$ u（x）\ rightArrow | x |+φ+φ\ left（\ frac {\ frac {x}}}全部，严格凸出的独特性，向下翻译solitons $ m_u $，以$σ_K$曲率流动方程为$σ_k$ curvation Flow方程的无限渐近行为。此外，我们证明了向下翻译solitons $ m_u $具有有限的主曲线。
We prove three results in this paper. First, we prove for a wide class of functions $φ\in C^2(\mathbb{S}^{n-1})$ and $ψ(X, ν)\in C^2(\mathbb{R}^{n+1}\times\mathbb{H}^n),$ there exists a unique, entire, strictly convex, spacelike hypersurface $M_u$ satisfying $σ_k(κ[M_u])=ψ(X, ν)$ and $u(x)\rightarrow |x|+φ\left(\frac{x}{|x|}\right)$ as $|x|\rightarrow\infty.$ Second, when $k=n-1, n-2,$ we show the existence and uniqueness of entire, $k$-convex, spacelike hypersurface $M_u$ satisfying $σ_k(κ[M_u])=ψ(x, u(x))$ and $u(x)\rightarrow |x|+φ\left(\frac{x}{|x|}\right)$ as $|x|\rightarrow\infty.$ Last, we obtain the existence and uniqueness of entire, strictly convex, downward translating solitons $M_u$ with prescribed asymptotic behavior at infinity for $σ_k$ curvature flow equations. Moreover, we prove that the downward translating solitons $M_u$ have bounded principal curvatures.
