Starshaped compact hypersurfaces with prescribed m-th mean curvature in hyperbolic space

一般共形诱导平均曲率流

General Conformally Induced Mean Curvature Flow

本文提出一种新颖的共形诱导平均曲率型流，将等周不等式结果推广到相对于受等距 Killing 场扰动的一般共形 Killing 向量场呈星形的超曲面。

空间形式中带毛细边界超曲面的约束平均曲率型流

A constrained mean curvature type flow for capillary boundary hypersurfaces in space forms

本文为空间形式中的毛细边界超曲面引入一种新的约束平均曲率型流，证明其全局收敛于球冠且保持体积、降低能量，并以此给出星形毛细边界超曲面毛细等周不等式的流证明。

The Dirichlet problem for prescribed curvature equations of $p$-convex hypersurfaces

In this paper, we study the Dirichlet problem for $p$-convex hypersurfaces with prescribed curvature. We prove that there exists a graphic hypersurface satisfying the prescribed curvature equation with homogeneous boundary condition. An interior curvature estimate is also obtained.

Curvature estimates for $p$-convex hypersurfaces of prescribed curvature

In this paper, we establish the curvature estimates for $p$-convex hypersurfaces in $\mathbb{R}^{n+1}$ of prescribed curvature with $p\geq \frac{n}{2}$. The existence of a star-shaped hypersurface of prescribed curvature is obtained. We also prove a type of interior $C^2$ estimates for solutions to the Dirichlet problem of the corresponding equation.

Hypersurfaces of Prescribed Gauss Curvature in Exterior Domains

We prove an existence theorem for convex hypersurfaces of prescribed Gauss curvature in the complement of a compact set in Euclidean space which are close to a cone.

Entire spacelike hypersurfaces of prescribed Gauss curvature in Minkowski space

We are concerned with spacelike convex hypersurfaces of positive constant (K-hypersurfaces) or prescribed Gauss curvature in Minkowski space. Our main purpose is to study entire solutions as well as the Dirichlet problem in bounded domains of the related Monge-Ampere equation.

Entire spacelike hypersurfaces of prescribed scalar curvature in Minkowski space

We prove existence and uniqueness of entire spacelike hypersurfaces in the Minkowski space with prescribed negative scalar curvature, and with given values at infinity which stay at a bounded distance of a lightcone.

On the global existence of generalized rotational hypersurfaces with prescribed mean curvature in the Euclidean spaces, II

In the previous paper, it has been proved that the generalized rotational hypersurfaces of O(n-1)-type and O (l+1) x O(m+1)-type, for which the mean curvature is any prescribed continuous function. This paper is a sequel, and a similar existence result is shown for any type.

具有指定梯度像的常平均曲率超曲面

The constant mean curvature hypersurfaces with prescribed gradient image

该文研究具指定梯度像的常平均曲率超曲面的存在性，在ℝⁿ中一致凸有界光滑域的条件下，证明常平均曲率方程第二边值问题存在唯一凸解。

Strictly Locally Convex Hypersurfaces with Prescribed Curvature and Boundary in Space Forms

This paper is devoted to $C^2$ a priori estimates for strictly locally convex radial graphs with prescribed Weingarten curvature and boundary in space forms. By constructing two-step continuity process and applying degree theory arguments, existence results in space forms are established for prescribed Gauss curvature equation under the assumption of a strictly locally convex subsolution.

规定曲率问题的凸毛细超曲面

Convex capillary hypersurfaces of prescribed curvature problem

本文将闭凸超曲面的规定$k$-阶Weingarten曲率问题扩展到毛细情形，通过将其转化为球冠上带Robin边界条件的Hessian商方程可解性问题，在自然充分条件下证明了严格凸毛细超曲面的存在性，扩展了毛细Minkowski问题的相关研究。

Horo-convex hypersurfaces with prescribed shifted Gauss curvatures in $\mathbb{H}^{n+1}$

In this paper, we consider prescribed shifted Gauss curvature equations for horo-convex hypersurfaces in $\mathbb{H}^{n+1}$. Under some sufficient condition, we obtain an existence result by the standard degree theory based on the a prior estimates for the solutions to the equations. Different from the prescribed Weingarten curvature problem in space forms, we do not impose a sign condition for radial derivative of the functions in the right-hand side of the equations to prove the existence due to the horo-covexity of hypersurfaces in $\mathbb{H}^{n+1}$.

Hypersurfaces of Prescribed Curvature Measure

We consider the corresponding Christoffel-Minkowski problem for curvature measures. The existence of star-shaped $(n-k)$-convex bodies with prescribed $k$-th curvature measures ($k>0$) has been a longstanding problem. This is settled in this paper through the establishment of a crucial $C^2$ a priori estimate for the corresponding curvature equation on $\mathbb S^n$.

Prescribed mean curvature flow of non-compact space-like Cauchy hypersurfaces

In this paper we consider the prescribed mean curvature flow of a non-compact space-like Cauchy hypersurface of bounded geometry in a generalized Robertson-Walker space-time. We prove that the flow preserves the space-likeness condition and exists for infinite time. We also prove convergence in the setting of manifolds with boundary. Our discussion generalizes previous work by Ecker, Huisken, Gerhardt and others with respect to a crucial aspects: we consider any non-compact Cauchy hypersurface under the assumption of bounded geometry. Moreover, we specialize the aforementioned works by conside…

Non-scale-invariant inverse curvature flows in hyperbolic space

We consider inverse curvature flows in hyperbolic space with starshaped initial hypersurface, driven by positive powers of a homogeneous curvature function. The solutions exist for all time and, after rescaling, converge to a sphere.

Entire spacelike constant $σ_k$ curvature hypersurfaces with prescribed boundary data at infinity

In this paper, we investigate the existence and uniqueness of convex, entire, spacelike hypersurfaces of constant $σ_k$ curvature with prescribed set of lightlike directions $\mathcal{F}\subset\mathbb{S}^{n-1}$ and perturbation $q$ on $\mathcal{F}$. We prove that given a closed set $\mathcal{F}$ in the ideal boundary at infinity of hyperbolic space and a perturbation $q$ that satisfies some mild conditions, there exists a complete entire spacelike constant $σ_k$ curvature hypersurface $\mathcal{M}_u$ with prescribed set of lightlike directions $\mathcal{F}$ satisfying when $\frac{x}{|x|}\in\ma…

双曲空间中具有指定无穷远渐近边界的常高阶平均曲率超曲面

Hypersurfaces of constant higher order mean curvature in hyperbolic space with prescribed asymptotic boundary at infinity

本文针对双曲空间中具有指定无穷远渐近边界的常高阶平均曲率超曲面，通过新论证将证明简化至半凸情形，证明了其存在性，结果适用于次临界范围内大部分指标。

conformal product manifolds 中的 prescribed mean curvature 超曲面

Prescribed mean curvature hypersurfaces in conformal product manifolds

本文在 conformal product manifolds 中研究 prescribed mean curvature 超曲面的存在性，扩展了 Gerhardt 的工作，并获得高维 PMC Plateau 问题的新解。

预定曲率方程的特征值问题

An eigenvalue problem for prescribed curvature equations

本文研究星形k-凸闭超曲面的预定σ_k-曲率方程特征值问题，在偶数据条件下证明了唯一特征值及唯一相关超曲面的存在性，并证明该超曲面严格凸，关键方法是对L_p型方程建立关于p的一致估计。

Deforming convex hypersurfaces to a hypersurface with prescribed harmonic mean curvature

A heat flow method is used to deform convex hypersurfaces in a ring domain to a hypersurface whose harmonic mean curvature is a prescribed function.

A Gluing Construction for Prescribed Mean Curvature

The gluing technique is used to construct hypersurfaces in Euclidean space having approximately constant prescribed mean curvature. These surfaces are perturbations of unions of finitely many spheres of the same radius assembled end-to-end along a line segment. The condition on the existence of these hypersurfaces is the vanishing of the sum of certain integral moments of the spheres with respect the prescribed mean curvature function.

Entire spacelike hypersurfaces with constant $σ_k$ curvature in Minkowski space

In this paper, we prove the existence of smooth, entire, strictly convex, spacelike, constant $σ_k$ curvature hypersurfaces with prescribed lightlike directions in Minkowski space. This is equivalent to prove the existence of smooth, entire, strictly convex, spacelike, constant $σ_k$ curvature hypersurfaces with prescribed Gauss map image. We also show that there doesn't exist any entire, convex, strictly spacelike, constant $σ_k$ curvature hypersurfaces. Moreover, we generalize the result in \cite{RWX} and construct strictly convex, spacelike, constant $σ_k$ curvature hypersurface with bounde…

Entire spacelike hypersurfaces of constant Gauss curvature in Minkowski space

We prove existence and stability of smooth entire strictly convex spacelike hypersurfaces of prescribed Gauss curvature in Minkowski space. The proof is based on barrier constructions and local a priori estimates.

对给定平均曲率超曲面的拓扑和奇点集的定量估计

Quantitative Estimates on the Topology and Singular Set of Prescribed Mean Curvature Hypersurfaces

本文研究了给定平均曲率超曲面的拓扑和奇点性质，建立了面积和曲率约束下的定量估计，扩展了极小超曲面的相关研究。

Modified mean curvature flow of star-shaped hypersurfaces in hyperbolic space

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…

具有给定梯度像的均曲率类型超曲面

The mean curvature type hypersurfaces with prescribed gradient image

研究均曲率类型超曲面的存在性，证明在给定梯度像条件下，第二边界值问题有唯一凸解。

具有孤立奇点的常平均曲率超曲面的光滑逼近

Smooth approximations for constant-mean-curvature hypersurfaces with isolated singularities

本文针对具有孤立奇点的常平均曲率超曲面，证明了在切锥正则且超曲面为最小化Caccioppoli集边界时，存在一列同平均曲率的光滑超曲面收敛到它，并指出维数8时锥条件可省略。

Minkowski空间中完备超曲面的预定曲率问题

The prescribed curvature problem for entire hypersurfaces in Minkowski space

本文证明了Minkowski空间中完备类空超曲面在预定曲率方程及向下平移解下的存在性、唯一性与有界性，涵盖严格凸与$k$-凸情形。

The boundary value Minkowski problem for Weingarten curvatures

In this paper, we prove the existence of hypersurfaces in the Euclidean space with prescribed boundary and whose k-th Weingarten curvature equals a given function that depends on the normal of the hypersurface. The proof is based on the solvability of a fully nonlinear elliptic PDE. The required a priori estimates are established under the natural assumptions that the prescribed boundary is strictly convex and the prescribed function satisfies a Serrin type condition.

具有预设光锥的极大超曲面在洛伦兹-闵氏空间中的研究

Maximal hypersurfaces with prescribed light-like cones in Lorentz-Minkowski space

本文研究洛伦兹-闵氏空间中具有多重光锥的极大超曲面，通过考虑带有多重狄拉克质量的均值曲率方程弱解来构建解。

Convex hypersurfaces of prescribed curvatures in hyperbolic space

For a smooth, closed and uniformly $h$-convex hypersurface $M$ in $\mathbb{H}^{n+1}$, the horospherical Gauss map $G: M \rightarrow \mathbb{S}^n$ is a diffeomorphism. We consider the problem of finding a smooth, closed and uniformly $h$-convex hypersurface $M\subset \mathbb{H}^{n+1}$ whose $k$-th shifted mean curvature $\widetilde{H}_{k}$ ($1\leq k\leq n$) is prescribed as a positive function $\tilde{f}(x)$ defined on $\mathbb{S}^n$, i.e. \begin{eqnarray*} \widetilde{H}_{k}(G^{-1}(x))=\tilde{f}(x). \end{eqnarray*} We can prove the existence of solution to this problem if the given function $\t…

Entire scalar curvature flow and hypersurfaces of constant scalar curvature in Minkowski space

We prove existence in the Minkowski space of entire spacelike hypersurfaces with constant negative scalar curvature and given set of lightlike directions at infinity; we also construct the entire scalar curvature flow with prescribed set of lightlike directions at infinity, and prove that the flow converges to a spacelike hypersurface with constant scalar curvature. The proofs rely on barriers construction and a priori estimates.

The inhomogeneous Allen--Cahn equation and the existence of prescribed-mean-curvature hypersurfaces

We prove that for any given compact Riemannian manifold $N$ of dimension $n+1 \geq 3$ and any non-negative Lipschitz function $g$ on $N$, there exists a quasi-embedded, boundaryless hypersurface $M \subset N,$ of class $C^{2, α}$ for any $α\in (0,1),$ such that $M$ is the image of a two-sided immersion whose mean curvature is given by $gν$ for an appropriate choice of continuous unit normal $ν$ to the immersion; and moreover, the singular set $Σ= \overline{M} \setminus M$ is empty if $2 \leq n \leq 6,$ finite if $n=7$ and satisfies ${\mathcal H}^{n-7 + γ}(Σ) = 0$ for every $γ>0$ if $n \geq 8$.…

Hypersurfaces with mean curvature prescribed by an ambient function: compactness results

We consider, in a first instance, the class of boundaries of sets with locally finite perimeter whose (weakly defined) mean curvature is $g ν$, for a given continuous positive ambient function $g$, and where $ν$ denotes the inner normal. It is well-known that taking limits in the sense of varifolds within this class is not possible in general, due to the appearence of "hidden boundaries", that is, portions (of positive measure with even multiplicity) on which the (weakly defined) mean curvature vanishes, so that $g$ does not prescribe the mean curvature in the limit. As a special instance of a…

Strictly Locally Convex Radial Graphs of Prescribed Curvature and Boundary in Space Forms

We obtain $C^2$ a priori estimates for solutions of the nonlinear second-order elliptic equation related to the geometric problem of finding a strictly locally convex hypersurface with prescribed curvature and boundary in a space form. Under the assumption of a strictly locally convex subsolution, we establish existence results by using degree theory arguments.

曲率估计对于具有常曲率的超曲面在双曲空间中的存在性 II

Curvature estimates for hypersurfaces of constant curvature in hyperbolic space II

本文研究双曲空间中具有常(n-2)曲率和指定无穷远处渐近边界的光滑完备超曲面的存在性，通过推导曲率估计，扩展了曲率值的范围。

从Clifford超曲面到赤道的Allen--Cahn与平均曲率流

Allen--Cahn and mean curvature flows from Clifford hypersurfaces to equators

该研究借助OpenAI对Solomon--Yau猜想的解决，将与Pedro Gaspar的合作成果推广至n≥2的所有维度，在S^{n+1}上构造了连接Clifford超曲面近似值与赤道球面的永恒Allen--Cahn流及无穷多永恒积分Brakke流。

具有给定奇点集的均曲率流

Mean curvature flows with prescribed singular sets

研究构造了在R^{m+n}中相对于近欧几里得度量的光滑度量的均凸古老解，其首次奇点集恰好为K×{0}。

warped product 流形中具有给定 Weingarten 曲率的 $k$-凸超曲面

$k$-convex hypersurfaces with prescribed Weingarten curvature in warped product manifolds

针对 warped product 流形中 $n<2k$ 的 $k$-凸超曲面，本文基于 Ren-Wang 猜想直接证明 Weingarten 曲率方程的曲率估计，并利用度理论建立星形紧致解的存在性。

Phase transitions and minimal hypersurfaces in hyperbolic space]

The purpose of this paper is to investigate the Cahn-Hillard approximation for entire minimal hypersurfaces in the hyperbolic space. Combining comparison principles with minimization and blow-up arguments, we prove existence results for entire local minimizers with prescribed behaviour at infinity. Then, we study the limit as the length scale tends to zero through a $Γ$-convergence analysis, obtaining existence of entire minimal hypersurfaces with prescribed boundary at infinity. In particular, we recover some existence results proved in M. Anderson and U. Lang using geometric measure theory.

Closed Weingarten hypersurfaces in warped product manifolds

Given a compact Riemannian manifold $M$, we consider a warped product $\bar M = I \times_h M$ where $I$ is an open interval in $\Rr$. We suppose that the mean curvature of the fibers do not change sign. Given a positive differentiable function $ψ$ in $\bar M$, we find a closed hypersurface $Σ$ which is solution of an equation of the form $F(B)=ψ$, where $B$ is the second fundamental form of $Σ$ and $F$ is a function satisfying certain structural properties. As examples, we may exhibit examples of hypersurfaces with prescribed higher order mean curvature.

$3$-凸超曲面在$\mathbb{H}^5$中的渐近平台问题

Asymptotic Plateau problem for $3$-convex hypersurface in $\mathbb{H}^5$

本文研究了5维双曲空间中$3$-凸超曲面的渐近平台问题，通过拉格朗日乘数法证明了满足特定曲率方程和渐近边界的超曲面存在性。

Kummer configurations and $S_m-$reflector problems: Hypersurfaces in $\Rn$ with given mean intensity

For a congruence of straight lines defined by a hypersurface in $R^{n+1}, n \geq 1,$ and a field of reflected directions created by a point source we define the notion of intensity in a tangent direction and introduce elementary symmetric functions $S_m, m=1, 2,...,n,$ of {\it principal intensities}. The problem of existence and uniqueness of a closed hypersurface with prescribed $S_n$ is the "reflector problem" extensively studied in recent years. In this paper we formulate and give sufficient conditions for solvability of an analogous problem in which the mean intensity $S_1$ is a given func…

Radial graphs of constant curvature and prescribed boundary

In this paper we are concerned with the problem of finding hypersurfaces of constant curvature and prescribed boundary in the Euclidean space, using the theory of fully nonlinear elliptic equations. We prove that if the given data admits a suitable radial graph as a subsolution, then we can find a radial graph with constant curvature and that realizes the prescribed boundary. As an application we prove that if $Ω\subset\mathbb{S}^n$ is a mean convex domain whose closure is contained in an open hemisphere of $\mathbb{S}^n$ then, for $0<R<n(n-1),$ there exists a radial graph of constant scalar c…

Entire spacelike radial graphs in the Minkowski space, asymptotic to the light-cone, with prescribed scalar curvature

Existence and uniqueness in ${\Bbb R}^{n,1}$ of entire spacelike hypersurfaces contained in the future of the origin $O$ and asymptotic to the light-cone, with scalar curvature prescribed at their generic point $M$ as a negative function of the unit vector $\overrightarrow{Om}$ pointing in the direction of $\overrightarrow{OM}$, divided by the square of the norm of $\overrightarrow{OM}$ (a dilation invariant problem). The solutions are seeked as graphs over the future unit-hyperboloid emanating from $O$ (the hyperbolic space); radial upper and lower solutions are constructed which, relying on …

双曲空间中的极限球面p-Christoffel-Minkowski问题与给定p-偏移Weingarten曲率问题

The horospherical $p$-Christoffel-Minkowski and prescribed $p$-shifted Weingarten curvature problems in hyperbolic space

本文研究双曲空间中凸超曲面的对应几何问题，通过建立新满秩定理证明极限球面p-Christoffel-Minkowski问题存在光滑原点对称严格凸解，并提出给定p-偏移Weingarten曲率问题且证明其存在性结果。

Variational aspects of phase transitions with prescribed mean curvature

We study the spectrum of phase transitions with prescribed mean curvature in Riemannian manifolds. These phase transitions are solutions to an inhomogeneous semilinear elliptic PDE that give rise to diffuse objects (varifolds) that limit to hypersurfaces, possibly with singularities, whose mean curvature is determined by the "prescribed mean curvature" function and the limiting multiplicity. We establish upper bounds for the eigenvalues of the diffuse problem, as well as the more subtle lower bounds when the diffuse problem converges with multiplicity one. For the latter, we also establish asy…

Embeddedness of Min-Max CMC Hypersurfaces in Manifolds with Positive Ricci Curvature

We prove that on a compact Riemannian manifold of dimension $3$ or higher, with positive Ricci curvature, the Allen--Cahn min-max scheme (implemented by the first author and N. Wickramasekera in 2020), with prescribing function taken to be a non-zero constant $λ$, produces an embedded hypersurface of constant mean curvature $λ$ ($λ$-CMC). More precisely, we prove that the interface arising from said min-max contains no even-multiplicity minimal hypersurface and no quasi-embedded points (both of these occurrences are in principle possible in the conclusions of the aforementioned work by the fir…

Translating solutions for Gauss curvature flows with Neumann boundary conditions

We consider strictly convex hypersurfaces which are evolving by the non-parametric logarithmic Gauss curvature flow subject to a Neumann boundary condition. Solutions are shown to converge smoothly to hypersurfaces moving by translation. In particular, for bounded domains we prove that convex functions with prescribed normal derivative satisfy a uniform oscillation estimate.

扭曲乘积空间上Guan--Li平均曲率流的收敛性结果与爆破示例

Convergence result and blow-up examples for the Guan--Li mean curvature flow on warped product spaces

本文研究扭曲乘积空间上Guan--Li平均曲率流的收敛性，在初始数据最优连续模条件下证明了收敛结果，并通过一维示例说明该条件不可改进。

Minkowski 空间中一类曲率方程的 Dirichlet 问题

The Dirichlet problem for a class of curvature equations in Minkowski space

针对 Minkowski 空间中的一类给定曲率方程，本文通过建立先验 C² 估计，证明了一般边界数据下光滑类空超曲面解的存在性。

Multiplicity one for min-max theory in compact manifolds with boundary and its applications

We prove the multiplicity one theorem for min-max free boundary minimal hypersurfaces in compact manifolds with boundary of dimension between 3 and 7 for generic metrics. To approach this, we develop existence and regularity theory for free boundary hypersurface with prescribed mean curvature, which includes the regularity theory for minimizers, compactness theory, and a generic min-max theory with Morse index bounds. As applications, we construct new free boundary minimal hypersurfaces in the unit balls in Euclidean spaces and self-shrinkers of the mean curvature flows with arbitrarily large …

任意余维数中具有给定平均曲率和接触角的圆盘的存在性

Existence of disks with prescribed mean curvature and contact angle in arbitrary codimension

本文在任意余维数下建立极小极大理论，证明在适当假设下存在具有给定平均曲率和接触角的非恒定分支浸入圆盘或球面，并在欧氏空间中给出自由边界圆盘及维数相关的莫尔斯指标界。

Mean curvature flow of noncompact hypersurfaces with Type-II curvature blow-up

We study the phenomenon of Type-II curvature blow-up in mean curvature flows of rotationally symmetric noncompact embedded hypersurfaces. Using analytic techniques based on formal matched asymptotics and the construction of upper and lower barrier solutions enveloping formal solutions with prescribed behavior, we show that for each initial hypersurface considered, a mean curvature flow solution exhibits the following behavior near the "vanishing" time $T$: (1) The highest curvature concentrates at the tip of the hypersurface (an umbilic point), and for each choice of the parameter $γ>1/2$, the…