Alexander Arbieto, Carlos Matheus, Marcos Petrucio
We generalize a theorem by J. Choe on capillary surfaces for arbitrary 3-dimensional spaces of constant curvature. The main tools in this paper are an extension of a theorem of H. Hopf due to S.-S. Chern and two index lemmas by J. Choe.
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Compactness of capillary hypersurfaces with mean curvature prescribed by ambient functions
本文证明了由环境函数规定平均曲率的毛细超曲面的紧致性定理，推广了Schätzle和Bellettini的结果，并扩展了曲率varifold定义至有向积分varifold，同时讨论了边界平均曲率规定情形。
In this paper we investigate the connection between the index and the geometry and topology of capillary surfaces. We prove an index estimate for compact capillary surfaces immersed in general 3-manifolds with boundary. We also study noncompact capillary surfaces with finite index and show that, under suitable curvature assumptions, such surface is conformally equivalent to a compact Riemann surface with boundary, punctured at finitely many points. We then prove that a weakly stable capillary surface immersed in a half-space of $\mathbb{R}^3$ which is minimal or has a contact angle less than o…
Capillary minimal slicing and scalar curvature rigidity
本文通过毛细超曲面方法，在四维流形中证明了正标量曲率和边界条件下的刚性结论。
A mean curvature type flow with capillary boundary in a horoball in hyperbolic space
本文研究双曲空间 horoball 中具有毛细边界的平均曲率型流，证明其长时间存在并收敛到截断脐超曲面，进而解决相应的等周型问题。
In this paper we investigate constant mean curvature surfaces with nonempty boundary in Euclidean space that meet a right cylinder at a constant angle along the boundary. If the surface lies inside of the cylinder, we obtain some results of symmetry by using the Alexandrov reflection method. When the mean curvature is zero, we give sufficient conditions to obtain that the surface is part of a plane or a catenoid.
Capillary Surfaces in Manifolds with Nonnegative Scalar Curvature and Strictly Mean Convex Boundary
该研究利用稳定毛细曲面（类μ-泡构造）探究带严格平均凸边界的非负数量曲率流形，基于Urysohn宽度给出3-流形填充2-流形的障碍，还得到带宽估计及其他几何性质。
Tangential limits of stable minimal capillary surfaces
研究稳定极小毛细面在毛发角接近0或π时的切线极限，探讨其支撑于非仿射平面的完整极小面的性质，以及支撑于均曲率正且无退化极值的闭合面的弱稳定极小毛细面的特性。
Prescribed $L_{p}$ curvature problem for convex capillary hypersurface
本文研究了欧几里得半空间中凸毛细超曲面的 prescribed $L_p$ 曲率问题，通过转化问题为 Hessian 商类型方程，证明了解的存在性和唯一性，并应用该方法解决了毛细 $L_p$ Christoffel-Minkowski 问题及相关特征值问题。
Anisotropic mean curvature type flow and capillary Alexandrov-Fenchel inequalities
本文研究半空间中星形各向异性ω₀-毛细管超曲面的保体积各向异性平均曲率型流，证明其长时间存在与收敛性，定义广义拟质量积分并建立相关等周与Alexandrov-Fenchel不等式，提供了流方法的新证明视角。
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。
In this paper we prove that a capillary minimal surface outside the unit ball in $\mathbb {R}^3$ with one embedded end and finite total curvature must be either part of the plane or part of the catenoid. We also prove that a capillary minimal surface outside the unit ball with one end asymptotic to the end of the Enneper's surface and finite total curvature cannot exist if the flux vector vanishes on the first homology calss of the surface. Furthermore, we prove that a capillary minimal surface outside the convex domain bounded by several spheres with one embedded end and finite total curvatur…
We propose a notion of stability for capillary hypersurfaces with constant higher order mean curvature and we generalize some results of the classical stability theory for CMC capillary hypersurfaces.
Min-max construction of two capillary embedded geodesics on Riemannian $2$-disks
本研究针对边界严格凸的黎曼2-圆盘，通过极小极大方法证明了满足特定条件时存在两条指定接触角的毛细管嵌入测地线，验证了总测地曲率条件的最优性，并给出了一般度量下不同Morse指数测地线的存在性结果。
Uniqueness of capillary disks in three-dimensional domains
论文证明了在三维域中，基于椭圆PDE的毛细盘的唯一性，扩展了Nitsche的结果，并受Gálvez和Mira对常均曲率球的Hopf定理的启发。
Capillary $L_p$-curvature problem
本文研究了毛细管 $L_p$-曲率问题，证明了在特定条件下存在光滑且严格凸的解。
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。
Inverse curvature flows for capillary hypersurfaces in the unit ball
本文研究了单位欧氏球中严格凸毛细超曲面的逆曲率流，建立了一类该流的存在性与收敛性结果，并作为应用导出了具有自由边界的弱凸超曲面的Alexandrov Fenchel不等式。
Hypersurfaces with capillary boundary evolving by volume preserving power mean curvature flow
该研究提出半空间中带毛细边界超曲面的保体积（或保面积）幂平均曲率流，并证明任意凸初始超曲面均可全局存在且光滑收敛至球冠。
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。
Existence of the classical solution to the fractional mean curvature flow with capillary-type boundary conditions
本文研究了具有毛细型边界条件的分数均曲线流，通过不动点方法证明了短时间解的存在性。
In [3] and [11] the authors showed the existence of a Codazzi pair defined on any constant mean curvature surface in the homogeneous spaces E($κ$,$τ$) associated to the Abresch-Rosenberg differential. In this paper, we use the mentioned Codazzi pair to classify capillary disks in E($κ$,$τ$). As a consequence, the results presented in this paper generalize the previous classification of constant mean curvature disks in the product spaces $\mathbb{S}^2 \times \mathbb{R}$ and $\mathbb{H}^2 \times \mathbb{R}$ in [4] and [5].
Capillary prisms, Coxeter gluing, and the $π_2$-systole of 3-manifolds with positive scalar curvature
研究具有正标量曲率三维流形的π₂-收缩，用局部到全局方法，通过毛细管棱柱及考克斯特拼接，获角度敏感收缩估计，证明相关拼接和平滑定理，构造特定拓扑的大π₂-收缩度量并讨论估计的刚性与非刚性。
Capillary John ellipsoid theorem with applications to capillary curvature problems
本文应用毛细John椭球定理研究毛细凸体，得到非退化估计，为解决某些毛细曲率问题提供新的C^0估计方法，并改进了毛细L_p对偶Minkowski问题的存在性结果。
We show that a capillary surface in a solid cone, that is, a surface that has constant mean curvature and the boundary of surface meets the boundary of the cone with a constant angle, is radially graphical if the mean curvature is non-positive with respect to the Gauss map pointing toward the domain bounded by the surface and the boundary of the cone. In the particular case that the cone is circular, we prove that the surface is a spherical cap or a planar disc. The proofs are based on an extension of the Alexandrov reflection method by using inversions about spheres.
A mean curvature type flow with capillary boundary in a unit ball
本文提出了一种在单位球体内具有毛细边界的均曲率类型流，证明其长期存在性并亚收敛于球帽，同时解决了相关等周问题。
The capillary Gauss curvature flow
本文提出毛细超曲面的 Gauss 曲率型流，证明其有限时间收缩为一点且归一化流收敛于孤子，推广了 Firey、Tso 及 Guan-Ni 的相关结果。
We prove a local rigidity result for infinitesimally rigid capillary surfaces in some Riemannian $3$-manifolds with mean convex boundary. We also derive bounds on the genus, number of boundary components and area of any compact two-sided capillary minimal surface with low index under certain assumptions on the curvature of the ambient manifold and of its boundary.
For a compact spacelike constant mean curvature surface with nonempty boundary in the three-dimensional Lorentz-Minkowski space, we introduce a rotation index of the lines of curvature at the boundary umbilic point, which was developed by Choe \cite{Choe}. Using the concept of the rotation index at the interior and boundary umbilic points and applying the Poincaré-Hopf index formula, we prove that a compact immersed spacelike disk type capillary surface with less than $4$ vertices in a domain of $\Bbb L^3$ bounded by (spacelike or timelike) totally umbilic surfaces is part of a (spacelike) pla…
Capillary curvature images
该研究针对特定范围的偶毛细$L_p$-Minkowski问题，借助基于毛细Minkowski问题解的迭代方案，利用一类泛函在毛细曲率图像算子族下的单调性求解，其不动点对应问题的解。
Hopf-Nitsche-type theorem for anisotropic capillary disks in a half 3-space
本文证明半三维空间中满足各向异性毛细边界条件的常各向异性平均曲率紧浸入圆盘必为位似Wulff帽。
On the mean curvature type flow for convex capillary hypersurfaces in the ball
本文研究单位欧氏球中具有毛细边界的凸超曲面的平均曲率型流，证明严格凸初值下解长期存在并光滑收敛到球冠，并由此得到新的Alexandrov-Fenchel不等式。
Capillary Orlicz-Minkowski flow in the upper half-space
本文研究了上半空间中各向异性毛细高斯曲率流的长时间存在性和渐进行为，通过流方法解决了毛细Orlicz-闵可夫斯基问题的存在性问题，无需偶性假设。
Some remarks on singular capillary cones with free boundary
本文研究奇异极小毛细锥的平坦性，提出稳定性判据，证明在特定维度与曲率条件下锥为平坦，并改进图形化超曲面的正则性。
Stable anisotropic capillary hypersurfaces in a half-space
本文研究了半空间中各向异性毛细超曲面的稳定性，证明了紧致常平均曲率超曲面弱稳定当且仅当为截断Wulff形状，并给出了三维情形下的Bernstein型定理。
A capillary problem for spacelike mean curvature flow in a cone of Minkowski space
本文研究了Minkowski空间中锥体的毛细问题，证明了在特定条件下，时空平均曲率流的抛物型边界值问题存在全局解，并趋于自相似的扩张解。
In this paper, we study capillary graphs defined on a domain $Ω$ of a complete Riemannian manifold $M$, where a graph is said to be capillary if it has constant mean curvature and locally constant Dirichlet and Neumann conditions on $\partial Ω$. Our main result is a splitting theorem both for $Ω$ and for the graph function on a class of manifolds with nonnegative Ricci curvature. As a corollary, we classify capillary graphs over domains that are globally Lipschitz epigraphs or slabs in a product space $M = N \times \mathbb{R}$, where $N$ has slow volume growth and non-negative Ricci curvature…
We consider a capillary drop that contacts several planar bounding walls so as to produce singularities (vertices) in the boundary of its free surface. It is shown under various conditions that when the number of vertices is less than or equal to three, then the free surface must be a portion of a sphere. These results extend the classical theorem of H. Hopf on constant mean curvature immersions of the sphere. The conclusion of sphericity cannot be extended to more than three vertices, as we show by examples.
Rigidity of stable spacelike capillary hypersurfaces in de Sitter and Minkowski spaces
该研究证明了德西特与闵可夫斯基空间中，当支撑为非负内蕴曲率的类时全脐超曲面时，保体积稳定的紧致类时毛细管超曲面必为全脐，还推导了统一闵可夫斯基型公式并构造了测试族。
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。
Existence of constant mean curvature disks in $\mathbb{R}^3$ with capillary boundary condition
本文推广了Struwe关于自由边界常平均曲率圆盘的存在性定理，证明在凸曲面包围区域内，对几乎所有的接触角参数，存在指标至多1的常平均曲率分支浸入圆盘。
Area estimates for capillary cmc hypersurfaces with nonpositive Yamabe invariant
本文研究具有非正 Yamabe 不变量的稳定毛细 cmc 超曲面，在数量曲率和边界平均曲率有下界的流形中建立面积估计，并对嵌入且 $\mathcal{J}$ 能量极小的情形给出局部分裂刚性结论。
In 1996, Kirk Lancaster and David Siegel investigated the existence and behavior of radial limits at a corner of the boundary of the domain of solutions of capillary and other prescribed mean curvature problems with contact angle boundary data. In Theorem 3, they provide an example of a capillary surface in a unit disk $D$ which has no radial limits at $(0,0)\in\partial D.$ In their example, the contact angle ($γ$) cannot be bounded away from zero and $π.$ Here we consider a domain $Ω$ with a convex corner at $(0,0)$ and find a capillary surface $z=f(x,y)$ in $Ω\times\mathbb{R}$ which has no r…
Alexandrov's theorem for anisotropic capillary hypersurfaces in the half-space
本文研究半空间中的各向异性常平均曲率嵌入毛细超曲面，借助新的 Heintze-Karcher 不等式与 Minkowski 公式证明其必为截短的 Wulff 形状。
We obtain a bound for the area of a capillary $H-$surface in a three-manifold with umbilic boundary and controlled sectional curvature. We then analyze the geometry when this area bound is realized, and obtain rigidity theorems. As a side product, we obtain existence of totally geodesic embedded surfaces in hyperbolic three-manifolds under the assumption of the existence of a $H-$surface realizing the area bound, in particular, under the existence of a totally umbilic $H-$surface.
Capillary Christoffel-Minkowski problem
本文将Guan和Ma关于球面凸函数对应严格凸超曲面$σ_k$曲率的结果，推广到半空间毛细管环境中，证明当$θ\in(0,π/2)$时，满足条件的函数对应严格凸毛细管超曲面的$σ_k$曲率。
Index Estimates for CMC and Minimal Surfaces with Capillary Boundary
本文研究具有毛细边界的CMC曲面的指数，证明其受亏格、边界组件数和分支阶数及Willmore型能量的线性限制。主要辅助定理比较了分支共形映射边界的面积和能量二阶变分。
Alexandrov-Fenchel inequalities for convex hypersurfaces in the half-space with capillary boundary
本文研究了半空间中具有毛细边界的凸超曲面的等周问题，通过构造局部约束曲率流，推导出Alexandrov-Fenchel不等式。
This survey describes some recent rigidity results obtained by the authors for the prescribed mean curvature problem on graphs $u : M \rightarrow \mathbb{R}$. Emphasis is put on minimal, CMC and capillary graphs, as well as on graphical solitons for the mean curvature flow, in warped product ambient spaces. A detailed analysis of the mean curvature operator is given, focusing on maximum principles at infinity, Liouville properties, gradient estimates. Among the geometric applications, we mention the Bernstein theorem for positive entire minimal graphs on manifolds with non-negative Ricci curva…
We study the prescribed mean curvature equation with a prescribed boundary contact angle condition in $M\times\R$ where $M^n$ is a Riemannian submanifold in $\R^{n+1}$. The main purpose is to establish a priori gradient estimates for solutions, from which the long time existence of the solution are derived.
Alexandrov-Fenchel inequalities for capillary hypersurfaces in hyperbolic space
本文通过构造新的局部约束逆曲率流，建立双曲空间中凸毛细超曲面的 Alexandrov-Fenchel 不等式，并推广 Brendle-Guan-Li 关于凸闭超曲面的结果。
A Minkowski-type inequality for capillary hypersurfaces in a half-space
本文研究半空间中毛细超曲面的逆平均曲率型流，证明其解整体存在且光滑收敛到球冠，由此导出星形且平均凸毛细超曲面的新 Minkowski 型不等式。
We develop a min-max theory for the construction of capillary surfaces in 3-manifolds with smooth boundary. In particular, for a generic set of ambient metrics, we prove the existence of nontrivial, smooth, almost properly embedded surfaces with any given constant mean curvature $c$, and with smooth boundary contacting at any given constant angle $θ$. Moreover, if $c$ is nonzero and $θ$ is not $\fracπ{2}$, then our min-max solution always has multiplicity one. We also establish a stable Bernstein theorem for minimal hypersurfaces with certain contact angles in higher dimensions.
Regularity of minimal surfaces with capillary boundary conditions
本文证明了黎曼流中具有毛细边界条件的变分折叠的ε-正则性定理，建立了均匀的第一变分控制，并推导出在密度严格小于1的边界一般点处的正则性。
This is an elementary observation that the symmetry properties of the Riemann curvature tensor can be (efficiently) expressed as SL(2)-invariance.
In this short note we prove that if the curvature tensor is uniformly bounded along the Calabi flow and the Mabuchi energy is proper, then the flow converges to a constant scalar curvature metric.
Maximum principles for weakly $1$-coercive operators with applications to capillary and prescribed mean curvature graphs
本文在完备非紧黎曼流形上建立弱1-强制算子的最大值原理，推导其比较原理，并应用于平均曲率算子等几何场景，还将结果推广到低正则性算子与函数的情形。
Half-space Liouville-type theorems for minimal graphs with capillary boundary
本文研究了半空间上具有毛细边界条件的最小图形的Liouville型定理，证明了在特定条件下图形必须为平坦结构。
We prove a gradient estimate for graphical spacelike mean curvature flow with a general Neumann boundary condition in dimension $n=2$. This then implies that the mean curvature flow exists for all time and converges to a translating solution.
We give a simple proof of a recent result due to Agostiniani, Fogagnolo and Mazzieri.
Remark about scalar curvature on certain noncompact manifolds
本文研究了非紧流形内部完整黎曼度量的标量曲率问题，给出了排除非负标量曲率的充分条件，并在高维情况下提供了正标量曲率度量的端点例子。
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