Rodrigo Ristow Montes Jose A. Verderesi
We provide a characterization of the Clifford Torus in S3 via moving frames and contact structure equations. More precisely, we prove that minimal surfaces in S3 with constant contact angle must be the Clifford Torus. Some applications of this result are then given, and some examples are discussed.
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We provide a congruence theorem for minimal surfaces in $S^5$ with constant contact angle using Gauss-Codazzi-Ricci equations. More precisely, we prove that Gauss-Codazzi-Ricci equations for minimal surfaces in $S^5$ with constant contact angle satisfy an equation for the Laplacian of the holomorphic angle. Also, we will give a characterization of flat minimal surfaces in $S^5$ with constant contact angle.
We show that an immersed minimal annulus, with two planar boundary curves along which the surface meets these planes with constant contact angle, is part of the catenoid.
In this paper we introduce the notion of contact angle. We deduce formulas for Laplacian and Gaussian curvature of a minimal surface in $S^{2n+1}$ and give a characterization of the generalized Clifford Torus as the only non-legendrian minimal surface in $S^5$ with constant Contact and Kaehler angles.
Parallel mean curvature surfaces with constant contact angle along free boundaries
本文研究了具有常接触角的平行平均曲率曲面，通过分类和构造例子验证了相关定理的准确性。
Assuming that there exists a translating soliton $u_\infty$ with speed $C$ in a domain $Ω$ and with prescribed contact angle on $\partialΩ$, we prove that a graphical solution to the mean curvature flow with the same prescribed contact angle converges to $u_\infty +Ct$ as $t\to\infty$. We also generalize the recent existence result of Gao, Ma, Wang and Weng to non-Euclidean settings under suitable bounds on convexity of $Ω$ and Ricci curvature in $Ω$.
We introduce a notion of the noncommutative integrability within a framework of contact geometry.
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.
Consistency of minimizing movements with smooth mean curvature flow of droplets with prescribed contact-angle in $\mathbb R^3$
该研究证明了三维空间中带预定接触角液滴的最小化运动解与光滑平均曲率流等价，为两类平均曲率运动描述建立了一致性。
In this paper we study nonparametric mean curvature type flows in $M\times\mathbb{R}$ which are represented as graphs $(x,u(x,t))$ over a domain in a Riemannian manifold $M$ with prescribed contact angle. The speed of $u$ is the mean curvature speed minus an admissible function $ψ(x,u,Du)$. Long time existence and uniformly convergence are established if $ψ(x,u, Du)\equiv 0$ with vertical contact angle and $ψ(x,u,Du)=h(x,u)ω$ with $h_u(x,u)\geq h_0>0$ and $ω=\sqrt{1+|Du|^2}$. Their applications include mean curvature type equations with prescribed contact angle boundary condition and the asymp…
A boundary maximum principle for stationary pairs of varifolds with fixed contact angle
本文研究带光滑边界的紧 Riemannian 流形中满足固定接触角条件的平稳 varifold 对，并为其建立边界极大值原理。
The minimal surface equation $Q$ in the second order contact bundle of $R^3$, modulo translations, is provided with a complex structure and a canonical vector-valued holomorphic differential form $Omega$ on $Q\0$. The minimal surfaces $M$ in $R^3$ correspond to the complex analytic curves $C$ in $Q$, where the derivative of the Gauss map sends $M$ to $C$, and $M$ is equal to the real part of the integral of $Ω$ over $C$. The complete minimal surfaces of finite topological type and with flat points at infinity correspond to the algebraic curves in $Q$.
Two infinite sequences of minimal surfaces in space are constructed using symmetry analysis. In particular, explicit formulas are obtained for the self-intersecting minimal surface that fills the trefoil knot.
We consider an evolving plane curve with two endpoints that can move freely on the $x$-axis with generating constant contact angles. We discuss the asymptotic behavior of global-in-time solutions when the evolution of this plane curve is governed by area-preserving curvature flow equation. The main result shows that any moving curve converges to a traveling wave if the moving curve starts from an embedded convex curve and remains bounded in global time.
We prove a blow-up criterion in terms of an $L_2$-bound of the curvature for solutions to the curve diffusion flow if the maximal time of existence is finite. In our setting, we consider an evolving family of curves driven by curve diffusion flow, which has free boundary points supported on a line. The evolving curve has fixed contact angle $α\in (0, π)$ with that line and satisfies a no-flux condition. The proof is led by contradiction: A compactness argument combined with the short time existence result enables us to extend the flow, which contradicts the maximality of the solution.
We consider the motion by mean curvature of an $n$-dimensional graph over a time-dependent domain in $\mathbb{R}^n$, intersecting $\mathbb{R}^n$ at a constant angle. In the general case, we prove local existence for the corresponding quasilinear parabolic equation with a free boundary, and derive a continuation criterion based on the second fundamental form. If the initial graph is concave, we show this is preserved, and that the solution exists only for finite time. This corresponds to a symmetric version of triple junction motion of hypersurfaces by mean curvature, with constant angles at th…
Gradient flow of phase transitions with fixed contact angle
该研究针对能量有界初始数据，分析固定边界接触角下Allen-Cahn方程的相变梯度流，建立解与能量测度的收敛性质，推导Ilmanen型单调性公式，完善了边界接触角相关的梯度流理论。
Complex contact manifolds arise naturally in differential geometry, algebraic geometry and exterior differential systems. Their classification would answer an important question about holonomy groups. The geometry of such manifold $X$ is governed by the contact lines contained in $X$. These are related to the notion of a variety of minimal rational tangents. In this review we discuss the partial classification theorems of projective complex contact manifolds. Among such manifolds one finds contact Fano manifolds (which include adjoint varieties) and projectivised cotangent bundles. In the firs…
Existence of weak mean curvature flow with prescribed contact angle via elliptic regularization
本文通过Ilmanen正则化，将其扩展到毛细现象领域并推导对应一阶变分估计，证明了满足θ∈(0,π)指定接触角的Brakke型弱平均曲率流的存在性。
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。
K-contact manifolds with minimal closed Reeb orbits
该研究利用Boothby-Wang纤维化构造单连通K-切触流形，给出其同胚于奇维球面的充要条件，构造了具最少闭Reeb轨道但不同胚于奇维球面的流形实例，丰富了K-切触流形的分类研究。
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