Stable free boundary minimal hypersurfaces in locally wedge-shaped manifolds
Zetian Yan
本文证明了在4维楔形区域中，稳定的C^{1,1}-到边正常嵌入自由边界极小超曲面必然是平坦的。
我们证明了角度θ∈(0,π]的4维楔形区域Ω^4_θ的稳定C^{1,1}-到边正常嵌入自由边界极小超曲面Σ^3是平坦的。
We prove that a stable $C^{1,1}$-to-edge properly embedded free boundary minimal hypersurface $Σ^3$ of a $4$-dimensional wedge domain $Ω^4_θ$ with angle $θ\in (0,π]$ is flat.
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Curvature estimates for stable free boundary minimal hypersurfaces in locally wedge-shaped manifolds
本文研究局部楔形流形中的自由边界极小超曲面，证明其紧性定理与Bernstein型定理，建立满足楔角假设的正则自由边界极小超曲面的曲率估计，可用于楔形空间面积泛函的极小极大理论构建。
Heintze-Karcher inequality and capillary hypersurfaces in a wedge
本文通过改进的Heintze-Karcher方法，证明了楔形中毛细超曲面的最佳不等式，并分类了与边缘相交的常曲率超曲面。
Let $Σ$ be a compact immersed stable capillary hypersurface in a wedge bounded by two hyperplanes in $\mathbb R^{n+1}$. Suppose that $Σ$ meets those two hyperplanes in constant contact angles and is disjoint from the edge of the wedge. It is proved that if $\partial Σ$ is embedded for $n=2$, or if $\partialΣ$ is convex for $n\geq3$, then $Σ$ is part of the sphere. And the same is true for $Σ$ in the half-space of $\mathbb R^{n+1}$ with connected boundary $\partialΣ$.
Min-max theory for free boundary minimal hypersurfaces in locally wedge-shaped manifolds
本文为局部楔形流形的面积泛函建立极小极大理论，证明锐角情形存在内部与各面光滑、棱边处 $C^{2,\alpha}$ 的自由边界极小超曲面。
Anisotropic capillary hypersurfaces in a wedge
本文将 Minkowski 范数推广为单位球面上的规范，研究楔形区域中的各向异性毛细超曲面，建立新的 Minkowski 公式与 Heintze-Karcher 型不等式，并证明 Alexandrov 型定理。
Anisotropic weighted isoperimetric inequalities for star-shaped and $F$-mean convex hypersurface
本文证明了适用于星形和$F$-平均凸超曲面的各向异性加权等周不等式，并指出Wulff形状是唯一极小化子，同时给出了基于Hausdorff距离的定量版本及Weinstock不等式的稳定性结果。
Given a smooth convex cone in the Euclidean $(n+1)$-space ($n\geq2$), we consider strictly mean convex hypersurfaces with boundary which are star-shaped with respect to the center of the cone and which meet the cone perpendicularly. If those hypersurfaces inside the cone evolve by a class of inverse curvature flows, then, by using the convexity of the cone in the derivation of the gradient and Hölder estimates, we can prove that this evolution exists for all the time and the evolving hypersurfaces converge smoothly to a piece of a round sphere as time tends to infinity.
Embedded cylindrical and doughnut-shaped $λ$-hypersurfaces
本文构造了圆柱状非凸完备嵌入 $\lambda$-超曲面以及两个不等距的紧致甜甜圈形嵌入 $\lambda$-超曲面，说明 $\lambda$-超曲面不满足自收缩子的平面区域猜想相关结论。
In this paper we introduce a Guan-Li type volume preserving mean curvature flow for free boundary hypersurfaces in a ball. We give a concept of star-shaped free boundary hypersurfaces in a ball and show that the Guan-Li type mean curvature flow has long time existence and converges to a free boundary spherical cap, provided the initial data is star-shaped.
In 1998 Smoczyk [Smo98] showed that, among others, the blowup limits at singularities are convex for the mean curvature flow starting from a closed star-shaped surface in $\mathbf{R}^3$. We prove in this paper that this is true for the mean curvature flow of star-shaped hypersurfaces in $\mathbf{R}^{n+1}$ in arbitrary dimension $n\geq 2$. In fact, this holds for a much more general class of initial hypersurfaces. In particular, this implies that the mean curvature flow of star-shaped hypersurfaces is generic in the sense of Colding-Minicozzi [CM12].
Four closed characteristics on compact star-shaped hypersurfaces in $\mathbb{R}^{8}$
研究 $\mathbb{R}^8$ 中紧星形超曲面的闭特征多重性问题，证明在无非退化 $C^3$ 且无 Maslov 型指标 $-1$ 本原闭特征的条件下，至少存在四条本原闭特征。
We define a new version of modified mean curvature flow (MMCF) in hyperbolic space $\mathbb{H}^{n+1}$, which interestingly turns out to be the natural negative $L^2$-gradient flow of the energy functional defined by De Silva and Spruck in \cite{DS09}. We show the existence, uniqueness and convergence of the MMCF of complete embedded star-shaped hypersurfaces with fixed prescribed asymptotic boundary at infinity. As an application, we recover the existence and uniqueness of smooth complete hypersurfaces of constant mean curvature in hyperbolic space with prescribed asymptotic boundary at infini…
In this paper, we use the inverse curvature flow to prove a sharp geometric inequality on star-shaped and two-convex hypersurface in hyperbolic space.
In this article, we prove a geometric inequality for star-shaped and mean-convex hypersurfaces in hyperbolic space by inverse mean curvature flow. This inequality can be considered as a generalization of Willmore inequality for closed surface in hyperbolic $3$-space.
In this paper we complete the study started in [Pi2] of evolution by inverse mean curvature flow of star-shaped hypersurface in non-compact rank one symmetric spaces. We consider the evolution by inverse mean curvature flow of a closed, mean convex and star-shaped hypersurface in the quaternionic hyperbolic space. We prove that the flow is defined for any positive time, the evolving hypersurface stays star-shaped and mean convex. Moreover the induced metric converges, after rescaling, to a conformal multiple of the standard sub-Riemannian metric on the sphere defined on a codimension 3 distrib…
In this paper we consider a star-shaped hypersurface flow by mean curvature. Without any assumption on the convexity, we give a new proof of gradient estimate for a short time. As an application, we also give a lower bound for the blowing up time.
In this paper, we show that the inverse anisotropic mean curvature flow in $\mathbb{R}^{n+1}$, initiating from a star-shaped, strictly $F$-mean convex hypersurface, exists for all time and after rescaling the flow converges exponentially fast to a rescaled Wulff shape in the $C^\infty$ topology. As an application, we prove a Minkowski type inequality for star-shaped, $F$-mean convex hypersurfaces.
In this paper, we first study the behavior of inverse mean curvature flow in Schwarzschild manifold. We show that if the initial hypersurface $Σ$ is strictly mean convex and star-shaped, then the flow hypersurface $Σ_t$ converges to a large coordinate sphere as $t\rightarrow \infty$ exponentially. We also describe an application of this convergence result. In the second part of this paper, we will analyse the inverse mean curvature flow in Kottler-Schwarzchild manifold. By deriving a lower bound for the mean curvature on the flow hypersurface independently of the initial mean curvature, we can…
We prove two weighted geometric inequalities that hold for strictly mean convex and star-shaped hypersurfaces in Euclidean space. The first one involves the weighted area and the area of the hypersurface and also the volume of the region enclosed by the hypersurface. The second one involves the total weighted mean curvature and the area of the hypersurface. Versions of the first inequality for the sphere and for the adS-Reissner-Nordström manifold are proven. We end with an example of a convex surface for which the ratio between the polar moment of inertia and the square of the area is less th…
A Constrained Mean Curvature Flow On Capillary Hypersurface Supported On Totally Geodesic Plane
针对双曲空间中支撑于全测地超平面的毛细超曲面，本文提出新的 Minkowski 型公式并构造保体积约束平均曲率流，证明其长时间存在、一致收敛到 θ-全脐帽，并验证该帽在固定体积下极小化能量。
We consider the evolution by inverse mean curvature flow of a closed, mean convex and star-shaped hypersurface in the complex hyperbolic space. We prove that the flow is defined for any positive time, the evolving hypersurface stays star-shaped and mean convex. Moreover the induced metric converges, after rescaling, to a conformal multiple of the standard sub- Riemannian metric on the sphere. Finally we show that there exists a family of examples such that the Webster curvature of this sub-Riemannian limit is not constant.
The isoperimetric problem in the Riemannian manifold admitting a non-trivial conformal vector field
本文在具有非平凡共形向量场的黎曼流形中引入保体积、降面积的平均曲率型流，证明其长时间存在与收敛，并建立星形区域等周不等式。
Let $M$ be a Hadamard manifold with curvature bounded above by a negative constant $-α$, satisfying the "strict convexity condition", and assume that $M$ admits a "helicoidal" one-parameter subgroup $G$ of isometries of $M$. Then, given a compact topological $G-$shaped hypersurface $Γ$ in the asymptotic boundary of $M,$ and $|H|<\sqrtα$, we prove the existence of a complete properly embedded hypersurface whose mean curvature is equal to $H$ and whose asymptotic boundary is $Γ$. We are able, this way, to extend a previous theorem of B.Guan and J.Spruck on the hyperbolic space to any rank 1 symm…
We consider the inverse curvature flows in the anti-de Sitter-Schwarzschild manifold with star-shaped initial hypersurface, driven by the 1-homogeneous curvature function. We show that the solutions exist for all time and the principle curvatures of the hypersurface converges to 1 exponentially fast.
We show that the constant mean curvature hypersurfaces in the hyperbolic n-space spanning the boundary of a star shaped C^{1,1} domain in the asymptotic sphere give a foliation of the hyperbolic n-space. We also show that if C is a closed codimension-1 C^{2,a} submanifold in the asymptotic sphere bounding a unique constant mean curvature hypersurface S_H in the hyperbolic n-space with asymptotic boundary C for any -1<H<1, then the constant mean curvature hypersurfaces {S_H} foliates the hyperbolic n-space.
An isoperimetric type inequality in De Sitter space
本文证明了de Sitter空间中类空、紧致、星形且$2$-凸超曲面的最优等周不等式。
Inverse Mean Curvature Flow of Rotationally Symmetric Hypersurfaces
研究了旋转对称超曲面的逆均值流，证明其收敛性并扩展了Minkowski不等式，同时建立了与极小曲面理论的联系。
A Minkowski type inequality in warped cylinders
该研究证明了渐近平坦或双曲弯曲柱体中弱平均凸星形超曲面的闵可夫斯基型不等式，并借助逆平均曲率流弱解验证其对Schwarzschild流形、双曲空间的向外极小超曲面成立。
Some properties of hyperbolic Yamabe solitons
该研究定义双曲Yamabe流，探究其定常解双曲Yamabe孤立子的性质，分析浸入子流形情形下的超曲面类型，刻画各类积流形的因子流形特征并给出分类，还确定了对应条件并以Robertson–Walker时空验证。
We consider a compact, star-shaped, mean convex hypersurface $Σ^2\subset \mathbb{R}^3$. We prove that in some cases the flow exists until it shrinks to a point in a spherical manner, which is very typical for convex surfaces as well (see \cite{An1}). We also prove that in the case we have a surface of revolution which is star-shaped and mean convex, a smooth solution always exists up to some finite time $T < \infty$ at which the flow shrinks to a point asymptotically spherically.
In this paper, we consider the inverse hessian quotient curvature flow with star-shaped initial hypersurface in anti-de Sitter-Schwarzschild manifold. We prove that the solution exists for all time, and the second fundamental form converges to identity exponentially fast.
A constrained mean curvature flow and Alexandrov-Fenchel inequalities
本文研究了具有毛细边界的星形超曲面的均曲率流，证明了其长期存在性和收敛性，并建立了新的Alexandrov-Fenchel不等式。
We prove the existence and local uniqueness of asymptotically flat, static vacuum metrics with arbitrarily prescribed Bartnik boundary data that are close to the induced boundary data on any star-shaped hypersurface or a general family of perturbed hypersurfaces in the Euclidean space. It confirms the existence part of the Bartnik static extension conjecture for large classes of boundary data, and the static vacuum metric obtained is geometrically unique in a neighborhood of the Euclidean metric.
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