William H. Meeks, Giuseppe Tinaglia
We derive intrinsic curvature and radius estimates for compact disks embedded in $\mathbb{R}^3$ with nonzero constant mean curvature and apply these estimates to study the global geometry of complete surfaces embedded in $\mathbb{R}^3$ with nonzero constant mean curvature.
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We consider surfaces with parallel mean curvature vector field and finite total curvature in product spaces of type $\mathbb{M}^n(c)\times\mathbb{R}$, where $\mathbb{M}^n(c)$ is a space form, and characterize certain of these surfaces. When $n=2$, our results are similar to those obtained in \cite{bds} for surfaces with constant mean curvature in space forms.
In this article, we study constant mean curvature isometric immersions into $\mathbb{S}^2 \times \mathbb{R}$ and $\mathbb{H}^2 \times \mathbb{R}$ and we classify these isometric immersions when the surface has constant intrinsic curvature. As applications, we use the sister surface correspondence to classify the constant mean curvature surfaces with constant intrinsic curvature in the $3-$dimensional homogenous manifolds $\mathbb{E}(κ, τ)$ and we use the Torralbo-Urbano correspondence to classify the parallel mean curvature surfaces in $\mathbb{S}^2 \times \mathbb{S}^2$ and $\mathbb{H}^2 \time…
The goal of this article is to study minimal surfaces in $\mathbb{M}^2 \times \mathbb{R}$ having finite total curvature, where $\mathbb{M}^2$ is a Hadamard manifold. The main result gives a formula to compute the total curvature in terms of topological, geometrical and conformal data of the minimal surface. In particular, we prove the total curvature is an integral multiple of $2π$.
Minimal hypersurfaces in $\mathbb{S}^5$ with constant scalar curvature and zero Gauss curvature are totally geodesic
本文证明五维球面中常数量曲率且零高斯曲率的闭极小超曲面必为全测地超曲面。
We prove a Simons type equation for non-minimal surfaces with parallel mean curvature vector (pmc surfaces) in $M^3(c)\times\mathbb{R}$, where $M^3(c)$ is a 3-dimensional space form. Then, we use this equation in order to characterize certain complete non-minimal pmc surfaces.
We study a half-space problem related to graphs in $\mathbb{H}^2\times\mathbb{R}$, where $\mathbb{H}^2$ is the hyperbolic plane, having constant mean curvature $H$ defined over unbounded domains in $\mathbb{H}^2$.
We find a Simons type formula for submanifolds with parallel mean curvature vector (pmc submanifolds) in product spaces $M^n(c)\times\mathbb{R}$, where $M^n(c)$ is a space form with constant sectional curvature $c$, and then we use it to prove a gap theorem for the mean curvature of certain complete proper-biharmonic pmc submanifolds, and classify proper-biharmonic pmc surfaces in $\mathbb{S}^n(c)\times\mathbb{R}$.
In this paper we study rotational surfaces in the space $\mathbb{H}^2\times\mathbb{R}$ whose mean curvature is given as a prescribed function of their angle function. These surfaces generalize, among others, the ones of constant mean curvature and the translating solitons of the mean curvature flow. Using a phase plane analysis we construct entire rotational graphs, catenoid-type surfaces, and exhibit a classification result when the prescribed function is linear.
In this paper we prove that a properly embedded constant mean curvature surface in $\mathbb{H}^2\times\mathbb{R}$ which has finite topology and stays at a finite distance from a vertical geodesic line is invariant by rotation around a vertical geodesic line.
In this paper, we classify the hypersurfaces in $\mathbb{S}^{n}\times \mathbb{R}$ and $\mathbb{H}^{n}\times\mathbb{R}$, $n\neq 3$, with $g$ distinct constant principal curvatures, $g\in\{1,2,3\}$, where $\mathbb{S}^{n}$ and $\mathbb{H}^{n}$ denote the sphere and hyperbolic space of dimension $n$, respectively. We prove that such hypersurfaces are isoparametric in those spaces. Furthermore, we find a necessary and sufficient condition for an isoparametric hypersurface in $\mathbb{S}^{n}\times \mathbb{R}\subset \mathbb{R}^{n+2}$ and $\mathbb{H}^{n}\times\mathbb{R}\subset \mathbb{L}^{n+2}$ with f…
There are examples of complete spacelike surfaces in the Lorentzian product $\mathbb{H}^2\times\mathbb{R}_1$ with constant Gaussian curvature $K\leq -1$. In this paper, we show that there exists no complete spacelike surface in $\mathbb{H}^2\times\mathbb{R}_1$ with constant Gaussian curvature $K>-1$.
We obtain compact orientable embedded surfaces with constant mean curvature $0<H<\frac{1}{2}$ and arbitrary genus in $\mathbb{S}^2\times\mathbb{R}$. These surfaces have dihedral symmetry and desingularize a pair of spheres with mean curvature $\frac{1}{2}$ tangent along an equator. This is a particular case of a conjugate Plateau construction of doubly periodic surfaces with constant mean curvature in $\mathbb{S}^2\times\mathbb{R}$, $\mathbb{H}^2\times\mathbb{R}$, and $\mathbb{R}^3$ with bounded height and enjoying the symmetries of certain tessellations of $\mathbb{S}^2$, $\mathbb{H}^2$, and …
Einstein submanifolds with parallel mean curvature into $\mathbb{S}^n\times\mathbb{R}$
本文证明 $\mathbb{S}^n\times\mathbb{R}$ 中满足平坦法丛与平行平均曲率条件的 Einstein 子流形可表示为等距浸入的扭曲积。
Spheres with parallel mean curvature in $\mathbb{S}^2 \times \mathbb{H}^2$
本文研究球面与双曲平面乘积空间中平行平均曲率曲面，分类了因拓扑原因使全纯二次微分消失的球面。
In this paper, we obtain a complete list of all self-similar solutions of inverse mean curvature flow in $\mathbb{R}^2$.
We prove that the normal curvatures of hyperspheres, the Rund curvature, and the Finsler curvature of circles in Hilbert geometry tend to 1 as the radii tend to infinity
We give an estimate of the Gauss curvature for minimal surfaces in ${\mathbb R}^m$ whose Gauss map omits more than $m(m+1)/2$ hyperplanes in ${\mathbb P}^{m-1}({\mathbb C})$.
We show that constant mean curvature hypersurfaces in $\mathbb H^n\times\mathbb R$, with small and pinched boundary contained in a horizontal slice $P$ are topological disks, provided they are contained in one of the two halfspaces determined by $P$. This is the analogous in $\mathbb H^n\times\mathbb R$ of a result in $\mathbb R^3$ by A. Ros and H. Rosenberg.
In this paper we present a local description for complete minimal hypersurfaces in $\mathbb{S}^5$ with zero Gauss-Kronecker curvature, zero $3$-mean curvature and nowhere zero second fundamental form.
The purpose of this paper is to study immersed surfaces in the product spaces $\mathbb{M}^2(κ)\times\mathbb{R}$, whose mean curvature is given as a $C^1$ function depending on their angle function. This class of surfaces extends widely, among others, the well-known theory of surfaces with constant mean curvature. In this paper we give necessary and sufficient conditions for the existence of prescribed mean curvature spheres, and we describe complete surfaces of revolution proving that they behave as the Delaunay surfaces of CMC type.
We prove that any complete immersed two-sided mean convex translating soliton $Σ\subset \mathbb{R}^3$ for the mean curvature flow is convex. As a corollary it follows that an entire mean convex graphical translating soliton in $\mathbb{R}^3$ is the axisymmetric "bowl soliton". We also show that if the mean curvature of $Σ$ tends to zero at infinity, then $Σ$ can be represented as an entire graph and so is the "bowl soliton". Finally we classify all locally strictly convex graphical translating solitons defined over strip regions.
We prove that proper biharmonic hypersurfaces with constant scalar curvature in Euclidean sphere $\mathbb S^5$ must have constant mean curvature. Moreover, we also show that there exist no proper biharmonic hypersurfaces with constant scalar curvature in Euclidean space $\mathbb E^5$ or hyperbolic space $\mathbb H^5$, which give affirmative partial answers to Chen's conjecture and Generalized Chen's conjecture.
In this paper, we prove that the translating solitons of the mean curvature flow in $\mathbb{R}^4$ which arise as blow up limit of embedded, mean convex mean curvature flow must have $SO(2)$ symmetry.
Given $H\in [0,\infty),$ some sufficient conditions for existence of CMC $H$ graphs with boundary in two parallel planes of $\mathbb{H}^2\times\mathbb{R}$ are presented. Height estimates for outwards-oriented CMC surfaces (horo)cyllindrically bounded are also exhibited.
Rotators-Translators to Mean Curvature Flow in $\mathbb H^2\times\mathbb R$
本文证明在双曲空间与实直线的乘积空间中，存在单参数螺旋面族，其在平均曲率流下同时绕垂直轴旋转并垂直平移。
We prove that a properly embedded annular end of a surface in $\mathbb H^2\times\mathbb R$ with constant mean curvature $0<H\leq \frac{1}{2}$ can not be contained in any horizontal slab. Moreover, we show that a properly embedded surface with constant mean curvature $0<H\leq \frac{1}{2}$ contained in $\mathbb H^2\times[0,+\infty)$ and with finite topology is necessarily a graph over a simply connected domain of $\mathbb H^2$. For the case $H=\frac{1}{2}$, the graph is entire.
Complete minimal hypersurfaces in $\mathbb H^5$ with constant scalar curvature and zero Gauss-Kronecker curvature
本文证明了五维双曲空间 $\mathbb H^5$ 中具有常数量曲率和零 Gauss-Kronecker 曲率的完备极小超曲面必定全测地，从而在五维情形下部分证实了 Cheng-Peng 关于 $\mathbb H^4$ 的相关猜想。
Geodesic boundary of constant mean curvature surfaces in $\mathbb{H}^2\times \mathbb{R}$
将极小曲面测地边界结果推广至常平均曲率 $0\le H\le 1/2$ 的曲面，研究其在 $\mathbb{H}^2\times \mathbb{R}$ 中的边界性质。
In this paper we consider Lorentzian surfaces in the 4-dimensional pseudo-Riemannian sphere $\mathbb S^4_2(1)$ with index 2 of curvature one. We obtain the complete classification of minimal Lorentzian surfaces $\mathbb S^4_2(1)$ whose Gaussian and normal curvatures are constants. We conclude that such surfaces have the Gaussian curvature $1/3$ and the absolute value of normal curvature $2/3$. We also give some explicit examples.
As is well known, self-similar solutions to the mean curvature flow, including self-shrinkers, translating solitons and self-expanders, arise naturally in the singularity analysis of the mean curvature flow. Recently, Guo \cite{Guo} proved that $n$-dimensional compact self-shrinkers in $\mathbb{R}^{n+1}$ with scalar curvature bounded from above or below by some constant are isometric to the round sphere $\mathbb{S}^n(\sqrt{n})$, which implies that $n$-dimensional compact self-shrinkers in $\mathbb{R}^{n+1}$ with constant scalar curvature are isometric to the round sphere $\mathbb{S}^n(\sqrt{n}…
It is known that a complete immersed minimal surface with finite total curvature in $\mathbb H^2\times\mathbb R$ is proper, has finite topology and each one of its ends is asymptotic to a geodesic polygon at infinity (Hauswirth and Rosenberg, 2006; Hauswirth, Nelli, Sa Earp and Toubiana, 2015). In this paper we prove that these three properties characterize complete immersed minimal surfaces with finite total curvature in $\mathbb H^2\times\mathbb R$. As corollaries of this theorem we obtain characterizations for minimal Scherk-type graphs and horizontal catenoids in $\mathbb H^2\times\mathbb …
The aim of this paper is to extend classic results of the theory of CMC surfaces in the product spaces to the class of immersed surfaces in $\mathbb{M}^2(κ)\times\mathbb{R}$ whose mean curvature is given as a $C^1$ function depending on their angle function. We cover topics such as the existence of a priori curvature estimates for graphs, height estimates for horizontal and vertical compact graphs and a structure-type result, which classifies all the simply connected, properly embedded surfaces with finite topology.
Height estimates for surfaces with some constant curvature in $\mathbb{r} \times_{f} \mathbb{r}^{2}$
本文利用共形参数推导翘积空间中正外在或平均曲率曲面的必要方程，给出高度估计并分类紧致极小图，扩展了先前研究。
Given $a,b\in\mathbb{R}$ and $Φ\in C^1(\mathbb{S}^2)$, we study immersed oriented surfaces $Σ$ in the Euclidean 3-space $\mathbb{R}^3$ whose mean curvature $H$ and Gauss curvature $K$ satisfy $2aH+bK=Φ(N)$, where $N:Σ\rightarrow\mathbb{S}^2$ is the Gauss map. This theory widely generalize some of paramount importance such as the ones constant mean and Gauss curvature surfaces, linear Weingarten surfaces and self-translating solitons of the mean curvature flow. Under mild assumptions on the prescribed function $Φ$, we exhibit a classification result for rotational surfaces in the case that the …
Let $M$ be a compact constant mean curvature surface either in $\mathbb{S}^3$ or $\mathbb{R}^3$. In this paper we prove that the stability index of $M$ is bounded below by a linear function of the genus. As a by product we obtain a comparison theorem between the spectrum of the Jacobi operator of $M$ and those of Hodge Laplacian of $1$-forms on $M$.
Classification of hypersurfaces with constant principal curvatures in $\mathbb{H}^m\times \mathbb{H}^n$
本文对双曲空间乘积中具有常主曲率的超曲面进行分类，证明了常乘积角函数情形下主曲率个数至多为3，并给出完整分类及等参超曲面的推论。
In this paper, we systematically investigate the geometry and topology of manifolds with integral radial curvature bounds, and obtain many interesting and important conclusions.
We study the fundamental properties of curvature in groupoids within the framework of synthetic differential geometry. As is usual in synthetic differential geometry, its combinatorial nature is emphasized. In particular, the classical Bianchi identity is deduced from its combinatorial one.
In this short paper, we study a symmetric covariant tensor in Finsler geometry, which is called the mean Berwald curvature. We first investigate the geometry of the fibres as the submanifolds of the tangent sphere bundle on a Finsler manifold. Then we prove that if the mean Berwald curvature is isotropic along fibres, then the Berwald scalar curvature is constant along fibres.
In this work we give a new lower bound on the Morse index for constant mean curvature tori of revolution immersed in the three-sphere $\mathbb{S}^3$, by computing some explicit negative eigenvalues for the corresponding Jacobi operator.
In this paper we study constant mean curvature surfaces $Σ$ in a product space, $\mathbb{M}^2\times \mathbb{R}$, where $\mathbb{M}^2$ is a complete Riemannian manifold. We assume the angle function $ν= \meta{N}{\partial_t}$ does not change sign on $Σ$. We classify these surfaces according to the infimum $c(Σ)$ of the Gaussian curvature of the projection of $Σ$. When $H \neq 0$ and $c(Σ)\geq 0$, then $Σ$ is a cylinder over a complete curve with curvature 2H. If H=0 and $c(Σ) \geq 0$, then $Σ$ must be a vertical plane or $Σ$ is a slice $\mathbb{M}^2 \times {t}$, or $\mathbb{M}^2 \equiv \mathbb{R…
We introduce fourth fundamental form $IV,$ and $i$-th curvature formulas of hypersurfaces in the four dimensional Euclidean geometry ${\mathbb{E}}^{4}$. Defining fourth fundamental form and $i$-th curvatures for hypersurfaces, we calculate them on rotational hypersurface. In addition we study rotational hypersurface satisfying $Δ^{IV}\mathbf{x=Ax}$ for some $4\times 4$ matrix $\mathbf{A.}$
We construct the first examples of complete, properly embedded minimal surfaces in $\mathbb{H}^2 \times \mathbb{R}$ with finite total curvature and positive genus. These are constructed by gluing copies of horizontal catenoids or other nondegenerate summands. We also establish that every horizontal catenoid is nondegenerate. Finally, using the same techniques, we are able to produce properly embedded minimal surfaces with infinitely many ends. Each annular end has finite total curvature and is asymptotic to a vertical totally geodesic plane.
Douglas curvature in finsler geometry: A review of the notion and its applications in geometry and physics
本文综述了芬斯勒几何中Douglas曲率的概念、性质、历史发展、特征化、推广及其在几何与物理中的应用，并展望了未来研究方向。
In this paper, we study the interplay of geometry and positive scalar curvature on a complete, non-compact manifold with non-negative Ricci curvature. In three-dimensional manifold, we prove a minimal volume growth, an estimate of integral of scalar curvature and width. In higher dimensional manifold, we obtain a volume growth with a stronger condition.
We classify the homogeneous and isoparametric hypersurfaces of $\mathbb{S}^2\times\mathbb{S}^2$. In the classification, besides the hypersurfaces $\mathbb{S}^1(r)\times\mathbb{S}^2,\,r\in (0,1]$, it appears a family of hypersurfaces with three different constant principal curvatures and zero Gauss-Kronecker curvature. Also we classify the hypersurfaces of $\mathbb{S}^2\times\mathbb{S}^2$ with at most two constant principal curvatures and, under certain conditions, with three constant principal curvatures.
Hypersurfaces of $\mathbb{H}^2\times\mathbb{H}^2$ with constant sectional curvature
本文分类了 $\mathbb{H}^2\times\mathbb{H}^2$ 中具有常截面曲率的超曲面，与 $\mathbb{S}^2\times\mathbb{S}^2$ 相比，例子更多样，并构造了一个具有非常值乘积角函数的特例。
The purpose of this paper is, first, to give an algorithm that enables to obtain the lines of curvature on parametric hypersurfaces in Euclidean 4-space, and then, to obtain the curvatures of such lines by using the extended Darboux frame along the curve.
For $0\leq H< 1/2$, we construct entire $H$-graphs in $\mathbb{H}^2\times\mathbb{R}$ that are parabolic and not invariant by one parameter groups of isometries of $\mathbb{H}^2\times\mathbb{R}$. Their asymptotic boundaries are $(\partial_\infty\mathbb{H}^2)\times\mathbb{R}$; they are dense at infinity. When $H=0$ the examples are minimal graphs constructed by P. Collin and the second author [2].
In this paper we obtain height estimates for compact, constant mean curvature vertical graphs in the homogeneous spaces $\mathrm{Nil}_3$ and $\widetilde{PSL}_2(\mathbb{R})$. As a straightforward consequence, we announce a structure-type result for proper graphs defined on relatively compact domains.
Let $X_{\mathbb{C}}$ be a smooth real affine variety with compact real points $X_{\mathbb{R}}$. We show that $X_{\mathbb{C}}$ is diffeomorphic to the normal bundle of $X_{\mathbb{R}}$ provided that $X_{\mathbb{C}}$ admits a complete Riemannian metric of nonnegative sectional curvature which is also invariant under the conjugation.
In this paper we aim at identifying the level sets of the gauge norm in the Heisenberg group $\mathbb{H}^n$ via the prescription of their (non-constant) horizontal mean curvature. We establish a uniqueness result in $\mathbb{H}^1$ under an assumption on the location of the singular set, and in $\mathbb{H}^n$ for $n\geq 2$ in the proper class of horizontally umbilical hypersurfaces
In this paper we study the theory of self translating solitons of the mean curvature flow of immersed surfaces in the product space $\mathbb{H}^2\times\mathbb{R}$. We relate this theory to the one of manifolds with density, and exploit this relation by regarding these translating solitons as minimal surfaces in a conformal metric space. Explicit examples of these surfaces are constructed, and we study the asymptotic behaviour of the existing rotationally symmetric examples. Finally, we prove some uniqueness and non-existence theorems.
We construct almost complex algebraic curvature tensors for pseudo Hermitian inner products whose skew-symmetric curvature operator has constant Jordan normal form on the set of non-degenerate complex lines.
We find a new monotone increasing quantity along smooth solutions to the inverse mean curvature flow in $\mathbb R^n$. As an application, we derive a sharp geometric inequality for mean convex, star-shaped hypersurfaces which relates the volume enclosed by a hypersurface to a weighted total mean curvature of the hypersurface.
We define a notion of mean curvature flow with surgery for two-dimensional surfaces in $\mathbb{R}^3$ with positive mean curvature. Our construction relies on the earlier work of Huisken and Sinestrari in the higher dimensional case. One of the main ingredients in the proof is a new estimate for the inscribed radius established by the first author.
We classify translation surfaces in isotropic geometry with arbitrary constant isotropic Gaussian and mean curvature under the condition that at least one of translating curves lies in a plane.
In this note, we study the prescribed mean curvature equation with Neumann boundary conditions on Riemannian product manifold $ M^n\times \mathbb{R}$. The main goal is to establish the boundary gradient estimates for solutions by the maximum principle. As a consequence, we obtain an existence result.
In this paper, we classify the hypersurfaces of $\mathbb{S}^2\times\mathbb{S}^2$ with constant sectional curvature. By applying the so-called Tsinghua principle, which was first discovered by the first three authors in 2013 at Tsinghua University, we prove that the constant sectional curvature can only be $\frac{1}{2}$ and the product angle function $C$ defined by Urbano is identically zero. We show that any such hypersurface is a parallel hypersurface of a minimal hypersurface in $\mathbb{S}^2\times\mathbb{S}^2$ with $C=0$, and we establish a one-to-one correspondence between the involving mi…
In this work, we study a class of rotational surfaces in the pseudo-Euclidean space $\mathbb{E}_2^4$ whose profile curves lie in two-dimensional planes. We solve the differential equation that characterizes the rotational surfaces with zero mean curvature to determine the profile curves of such rotational surfaces. Then, we give some explicit parametrization of maximal rotational surfaces and the timelike surfaces with zero mean curvature in $\mathbb{E}_2^4$.
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