Bernstein Theorems for Space-like Graphs with Parallel Mean Curvature and Controlled Growth

各向同性三维空间中常平均曲率曲面的伯恩斯坦型定理

Bernstein-type theorem for constant mean curvature surfaces in the isotropic 3-space

针对各向同性三维空间$\boldsymbol{\text{I}}^3$中的常平均曲率曲面，通过证明其高斯曲率值分布定理，得到了对应常平均曲率图的伯恩斯坦型定理，拓展了该领域的理论成果。

关于R^4中具有有限全曲率的二维平移孤立子的伯恩斯坦型定理

A Bernstein type theorem for a two dimensional translating soliton in ${\mathbb R}^4$ with finite total curvature

本文证明R^4中具有有限全曲率的二维完备平移孤立子必为平面，该结论是其主要结果的推论。

A Bernstein theorem for complete spacelike constant mean curvature hypersurfaces in Minkowski space

We obtain a gradient estimate for the Gauss maps from complete spacelike constant mean curvature hypersurfaces in Minkowski space into the hyperbolic space. As applications, we prove a Bernstein theorem which says that if the image of the Gauss map is bounded from one side, then the spacelike constant mean curvature hypersurface must be linear. This result extends the previous theorems obtained by B. Palmer and Y.L. Xin where they assume that the image of the Gauss map is bounded. We also proved a Bernstein theorem for spacelike complete surfaces with parallel mean curvature vector in four-dim…

具有零高斯曲率的超曲面的三种Bernstein型定理

Three Bernstein type theorems for hypersurfaces with zero Gaussian curvature

本文研究了具有零高斯曲率的凸超曲面在欧几里得和闵氏背景下，证明了在特定条件下这些超曲面必须为超平面，并通过例子说明了额外条件的重要性。

Constant mean curvature spacelike hypersurfaces in Lorentzian warped products and Calabi-Bernstein type problems

In this paper we provide several uniqueness and non-existence results for complete parabolic constant mean curvature spacelike hypersurfaces in Lorentzian warped products under appropriate geometric assumptions. As a consequence of this parametric study, we obtain very general uniqueness and non-existence results for a large family of uniformly elliptic EDP's, so solving the Calabi-Bernstein problem in a wide family of spacetimes.

A simple proof for Bernstein type theorems in Gauss space

A weighted area estimate for entire graphs with bounded weighted mean curvature in Gauss space is given by a simple proof. Bernstein type theorems for self shrinkers (\cite {wa}) as well as for graphic $λ$-hypersurfaces (\cite{ chwe2}) follow immediately as consequences.

Complete Constant Mean Curvature surfaces and Bernstein type Theorems in $\mathbb{M}^2\times \mathbb{R}$

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…

Bernstein type theorems for spacelike stationary graphs in Minkowski spaces

For entire spacelike stationary 2-dimensional graphs in Minkowski spaces, we establish Bernstein type theorems under specific boundedness assumptions either on the W-function or on the total (Gaussian) curvature. These conclusions imply the classical Bernstein theorem for minimal surfaces in 3-dimensional Euclidean space and Calabi's theorem for spacelike maximal surfaces in 3-dimensional Minkowski space.

特定正曲率流形中的稳定Bernstein问题

Stable Bernstein Problem in certain positively curved manifolds

该研究针对特定正曲率流形的稳定Bernstein问题，证明了不同曲率条件下完备双侧稳定极小浸入的全测地性与不存在性，推广了已有低维结果至高维情形。

A note on two-dimensional minimal surface graphs in R^n and a theorem of Bernstein-Liouville type

Using Schauder's theory for linear elliptic partial differential equations in two independent variables and fundamental estimates for univalent mappings due to E. Heinz we establish an upper bound of the Gaussian curvature of two-dimensional minimal surface graphs in R^n. This leads us to a theorem of Bernstein-Liouville type.

双曲空间中常高阶平均曲率超曲面的唯一性

Uniqueness of Hypersurfaces of Constant Higher Order Mean Curvature in Hyperbolic Space

本文研究了双曲空间中极球面与等距球面的唯一性，将 Bernstein 定理推广至常高阶平均曲率的嵌入超曲面，证明了浸入超曲面的 Bernstein 型结果，并确立了极球面与等距球面的刚性。

欧几里得空间中子流形的谱伯恩斯坦定理

Spectral Bernstein theorems for submanifolds in Euclidean spaces

本文研究欧几里得空间中子流形的本质谱，探讨了有限总均曲率、外向距离梯度收敛性、外向体积增长或曲率pinching等几何条件，并证明了当子流形的二阶基本形式满足L^p有界时，其本质谱为[0, +∞)。

The Gauss image of entire graphs of higher codimension and Bernstein type theorems

Under suitable conditions on the range of the Gauss map of a complete submanifold of Euclidean space with parallel mean curvature, we construct a strongly subharmonic function and derive a-priori estimates for the harmonic Gauss map. The required conditions here are more general than in previous work and they therefore enable us to improve substantially previous results for the Lawson-Osseman problem concerning the regularity of minimal submanifolds in higher codimension and to derive Bernstein type results.

Complete Spacelike Hypersurfaces in Generalized Robertson-Walker and the Null Convergence Condition. Calabi-Bernstein problems

We study constant mean curvature spacelike hypersurfaces in generalized Robertson-Walker spacetimes which are spatially parabolic covered (i.e. its fiber F is a (non- compact) complete Riemannian manifold whose universal covering is parabolic) and satisfy the null convergence condition. In particular, we provide several rigidity results under appro- priate mathematical and physical assumptions. We pay special attention to the case where the GRW spacetime is Einstein. As an application, some Calabi-Bernstein type results are given.

A Bernstein Theorem for Minimal Maps with Small Second Fundamental Form

We consider minimal maps $f:M\to N$ between Riemannian manifolds $(M,\mathrm{g}_M)$ and $(N,\mathrm{g}_N)$, where $M$ is compact and where the sectional curvatures satisfy $\sec_N\le σ\le \sec_M$ for some $σ>0$. Under certain assumptions on the differential of the map and the second fundamental form of the graph $Γ(f)$ of $f$, we show that $f$ is either the constant map or a totally geodesic isometric immersion.

关于整体图曲面的Bernstein型定理综述

A survey on Bernstein-type theorems for entire graphical surfaces

本文综述欧几里得与Lorentz-Minkowski空间中图曲面的Bernstein型定理，涵盖极小图、Heinz型平均曲率估计、常平均曲率图及Calabi-Bernstein定理。

Ricci 下界条件下极小图的 Bernstein 性质与半空间性质

Bernstein and half-space properties for minimal graphs under Ricci lower bounds

本文在 Ricci 曲率下界条件下建立极小图的新梯度估计，证明非负 Ricci 曲率流形上的正整体极小图必为常数，并证明完备抛物流形具有半空间性质。

Complete Vertical Graphs with Constant Mean Curvature in Semi-Riemannian Warped Products

We obtain necessary conditions for the existence of complete vertical graphs of constant mean curvature in the Hyperbolic and Steady State spaces. In the two-dimensional case we prove Bernstein-type results in each of these ambient spaces.

A Bernstein type result for graphical self-shrinkers in $\mathbb{R}^4$

Self-shrinkers are important geometric objects in the study of mean curvature flows, while the Bernstein Theorem is one of the most profound results in minimal surface theory. We prove a Bernstein type result for graphical self-shrinker surfaces with codimension two in $\mathbb{R}^4$. Namely, under certain natural conditions on the Jacobian of any smooth map from $\mathbb{R}^2$ to $\mathbb{R}^2$, we show that the self-shrinker which is the graph of this map must be affine linear. The proof relies on the derivation of structure equations of graphical self-shrinkers in terms of the parallel form…

Gauss map of translating solitons of mean curvature flow

In this short note we study Bernstein's type theorem of translating solitons whose images of their Gauss maps are contained in compact subsets in an open hemisphere of the standard $\mathbf{S}^n$ (see Theorem 1.1). As a special case we get a classical Bernstein's type theorem in minimal submanifolds in $\mathbf{R}^{n+1}$ (see Corollary 1.2).

Curvature estimates for minimal submanifolds of higher codimension

We derive curvature estimates for minimal submanifolds in Euclidean space for arbitrary dimension and codimension via Gauss map. Thus, Schoen-Simon-Yau's results and Ecker-Huisken's results are generalized to higher codimension. In this way we improve Hildebrandt-Jost-Widman's result for the Bernstein type theorem.

完全中心仿射极值超曲面与中心仿射Bernstein猜想

Complete centroaffine extremal hypersurfaces and centroaffine Bernstein conjecture

本文通过构造两类完全中心仿射极值超曲面，反驳了中心仿射Bernstein猜想，表明在欧氏完备性（$n\ge2$）和中心仿射完备性（$n\ge3$）下该猜想均不成立。

A remark on a Bernstein type theorem for entire Willmore graphs in R^3

In this note we prove that every two-dimensional entire Willmore graph in $R^3$ with square integrable mean curvature is a plane.

On stable hypersurfaces with constant mean curvature in Euclidean spaces

In this paper, we derive curvature estimates for strongly stable hypersurfaces with constant mean curvature immersed in $\mathbb{R}^{n+1}$, which show that the locally controlled volume growth yields a globally controlled volume growth if $\partial M=\emptyset$. Moreover, we deduce a Bernstein-type theorem for complete stable hypersurfaces with constant mean curvature of arbitrary dimension, given a finite $L^p$-norm curvature condition.

Compact spacelike surfaces whose mean curvature function satisfies a nonlinear inequality in a 3-dimensional Generalized Robertson-Walker spacetime

Spacelike surfaces in Generalized Robertson-Walker spacetimes whose mean curvature function satisfies a natural nonlinear inequality are analyzed. Several uniqueness and nonexistence results for such compact spacelike surfaces are proved. In the nonparametric case, new Calabi-Bernstein type problems are solved as a consequence.

Curvature estimates for minimal submanifolds of higher codimension and small G-rank

We obtain new curvature estimates and Bernstein type results for minimal $n-$submanifolds in $\ir{n+m},\, m\ge 2$ under the condition that the rank of its Gauss map is at most 2. In particular, this applies to minimal surfaces in Euclidean spaces of arbitrary codimension.

Graphic Bernstein Results in Curved Pseudo-Riemannian Manifolds

We generalize a Bernstein-type result due to Albujer and Alías, for maximal surfaces in a curved Lorentzian product 3-manifold of the form $Σ_1\times \mathbb{R}$, to higher dimension and codimension. We consider $M$ a complete spacelike graphic submanifold with parallel mean curvature, defined by a map $f: Σ_1\to Σ_2$ between two Riemannian manifolds $(Σ_1^m, g_1)$ and $(Σ^n_2, g_2)$ of sectional curvatures $K_1$ and $K_2$, respectively. We take on $Σ_1\times Σ_2$ the pseudo-Riemannian product metric $g_1-g_2$. Under the curvature conditions, $\mathrm{Ricci}_1 \geq 0$ and $K_1\geq K_2$, we pro…

The geometry of Grassmannian manifolds and Bernstein type theorems for higher codimension

We identify a region $\Bbb{W}_{\f{1}{3}}$ in a Grassmann manifold $\grs{n}{m}$, not covered by a usual matrix coordinate chart, with the following important property. For a complete $n-$submanifold in $\ir{n+m} \, (n\ge 3, m\ge2)$ with parallel mean curvature whose image under the Gauss map is contained in a compact subset $K\subset\Bbb{W}_{\f{1}{3}}\subset\grs{n}{m}$, we can construct strongly subharmonic functions and derive a priori estimates for the harmonic Gauss map. While we do not know yet how close our region is to being optimal in this respect, it is substantially larger than what co…

平均曲率流的平移解的伯恩斯坦问题

A Bernstein problem for translating solutions to the mean curvature flow

该研究针对平均曲率流的平移解的伯恩斯坦问题，在N≥8时构造出非凸的平均凸整体图平移子，完善了相关刚性理论。

The Bernstein Problem in the Heisenberg Group

We establish the following theorem of Bernstein type for the first Heisenberg group: Let S be a C^2 connected H-minimal surface which is a graph over some plane P, then S is either a non-characteristic vertical plane, or its generalized seed curve satisfies a type of constant curvature condition.

Spacelike hypersurfaces in twisted product spacetimes with complete fiber and Calabi-Bernstein-type problems

In this article spacelike hypersurfaces immersed in twisted product spacetimes $I\times_f F$ with complete fiber are studied. Several conditions ensuring global hyperbolicity are presented, as well as a relation that needs to hold on each spacelike hypersurface in $I\times_f F$ for it to be a simple warped product. When the fiber is assumed to be closed (compact and without boundary) and the ambient spacetime has a suitable expanding behaviour, non-existence results for constant mean curvature hypersurfaces are obtained. Under the same hypothesis, a characterization of compact maximal hypersur…

On graphic Bernstein type results in higher codimension

Let Σbe a minimal submanifold of \R^{n+m} that can be represented as the graph of a smooth map f:\R^n-->\R^m. We apply a formula we derived in the study of mean curvature flow to obtain conditions under which Σmust be an affine subspace. Our result covers all known ones in the general case. The conditions are stated in terms of the singular values of $df$.

Bernstein-Heinz-Chern results in calibrated manifolds

Given $(\bar{M},Ω)$ a calibrated Riemannian manifold with a parallel calibration of rank $m$, and $M^m$ an immersed orientable submanifold with parallel mean curvature $H$ we prove that if $\cos θ$ is bounded away from zero, where $θ$ is the $Ω$-angle of $M$, and if $M$ has zero Cheeger constant, then $M$ is minimal. In the particular case $M$ is complete with $Ricc^M\geq 0$ we may replace the boundedness condition on $\cos θ$ by $\cos θ\geq Cr^{-β}$, when $r\to +\infty$, where $ 0\leqβ<1 $ and $C > 0$ are constants and $r$ is the distance function to a point in $M$. Our proof is surprisingly …

Bernstein定理的短证明：基于Omori-Yau最大值原理

A short proof of Bernstein's theorem via the Omori--Yau maximum principle

本文利用Omori-Yau最大值原理，通过构造显式函数并应用逐点微分不等式，简洁地证明了三维空间中完整极小图必为平面的Bernstein定理。

On the entire self-shrinking solutions to Lagrangian mean curvature flow II

We show Bernstein type results for the entire self-shrinking solutions to Lagrangian mean curvature flow in $(\mathbb{R}^n\times\mathbb{R}^n, g_τ)$. The proofs rely on a priori estimates and barriers construction.

受控曲率浸入讲座

Lectures on Immersions with Controlled Curvatures

探讨黎曼流形间具“小”曲率浸入的构造方法，并指出此类浸入存在的障碍。

Isometric Immersions with Controlled Curvatures

We $δ$-approximate strictly short (e.g. constant) maps between Riemannin manifolds $f_0:X^m\to Y^N$ for $N>>m^2/2$ by $C^\infty$-smooth isometric immersions $f_δ:X^m\to Y^N$ with curvatures $curv(f_δ) < \frac {\sqrt 3}δ$, for $δ\to 0$

RCD空间上的极小曲面方程与Bernstein性质

Minimal Surface Equation and Bernstein Property on RCD spaces

本文在RCD(K,N)空间中研究极小曲面方程，证明其解在图上调和，并在K=0时利用Harnack不等式推出Bernstein性质，进而给出乘积空间中极小图的振荡估计与Bernstein定理。

Zero mean curvature entire graphs of mixed type in Lorentz-Minkowski 3-space

It is classically known that the only zero mean curvature entire graphs in the Euclidean 3-space are planes, by Bernstein's theorem. A surface in Lorentz-Minkowski 3-space $\boldsymbol{R}^3_1$ is called of mixed type if it changes causal type from space-like to time-like. In $\boldsymbol{R}^3_1$, Osamu Kobayashi found two zero mean curvature entire graphs of mixed type that are not planes. As far as the authors know, these two examples were the only known examples of entire zero mean curvature graphs of mixed type without singularities. In this paper, we construct several families of real anal…

自收缩子的高斯加权曲率间隙

Gaussian-Weighted Curvature Gaps for Self-Shrinkers

本文利用法坐标函数和加权Poincaré不等式，证明嵌入自收缩子高斯加权L²曲率积分的下界，并结合熵间隙定理得到非球面闭自收缩子的严格曲率间隙，在二维情形推广到完整嵌入自收缩子。

$\mathbb{H}^n$ 中稳定常平均曲率超曲面

Stable Constant Mean Curvature Hypersurfaces in $\mathbb{H}^n$

本文证明在双曲空间 $\mathbb{H}^n$（$n\geq 4$）中，对平均曲率 $H$ 接近 $n-1$ 时，存在具有无穷多个端、在 Schottky 群作用下不变的强稳定 CMC 超曲面，从而表明该范围内 Bernstein 型刚性不成立。

Stability and curvature estimates for minimal graphs with flat normal bundles

It is well-known that a minimal graph of codimension one is stable, i.e. the second variation of the area functional is non-negative. This is no longer true for higher codimensional minimal graphs. In this note, we prove that a minimal graph of any codimension is stable if its normal bundle is flat. We also prove minimal graphs of dimension no greater than six and any codimension is flat if the the normal bundle is flat and the density at infinity is finite. Such a Bernstein type theorem holds in any dimension if we assume additionally growth conditions on the volume element.

On maximal hypersurfaces in Lorentz manifolds admitting a parallel lightlike vector field

We study constant mean curvature spacelike hypersurfaces and in particular maximal hypersurfaces immersed in pp-wave spacetimes satisfying the timelike convergence condition. We prove the non-existence of compact spacelike hypersurfaces whose constant mean curvature is non-zero and also that every compact maximal hypersurface is totally geodesic. Moreover, we give an extension of the classical Calabi-Bernstein theorem to this class of pp-wave spacetimes.

至多七维空间形式中有限指标常平均曲率超曲面

Finite index constant mean curvature hypersurfaces in space forms of dimension at most seven

研究至多七维空间形式中有限指标常平均曲率超曲面，在球面、欧氏及双曲空间中分别获得紧致性或极小性结论，并给出维数依赖的显式阈值。

Complete Foliations of Space Forms by Hypersurfaces

We study foliations of space forms by complete hypersurfaces, under some mild conditions on its higher order mean curvatures. In particular, in Euclidean space we obtain a Bernstein-type theorem for graphs whose mean and scalar curvature do not change sign but may otherwise be nonconstant. We also establish the nonexistence of foliations of the standard sphere whose leaves are complete and have constant scalar curvature, thus extending a theorem of Barbosa, Kenmotsu and Oshikiri. For the more general case of {\em r-}minimal foliations of the Euclidean space, possibly with a singular set, we ar…

On the fundamental tone of minimal submanifolds with controlled extrinsic curvature

The aim of this paper is to obtain the fundamental tone for minimal submanifolds of the Euclidean or hyperbolic space under certain restrictions on the extrinsic curvature. We show some sufficient conditions on the norm of the second fundamental form that allow us to obtain the same upper and lower bound for the fundamental tone of minimal submanifolds in a Cartan-Hadamard ambient manifold. As an intrinsic result, we obtain a sufficient condition on the volume growth of a Cartan-Hadamard manifold to achieve the lowest bound for the fundamental tone given by McKean.

欧几里得空间中稳定极小超曲面的格林核梯度估计及$\mathbb{R}^4$中的稳定伯恩斯坦定理

Gradient estimates for the Green kernel under spectral Ricci bounds, and the stable Bernstein theorem in $\mathbb{R}^4$

本文通过新积分不等式证明了稳定极小超曲面的性质，并给出$\mathbb{R}^4$中完整双侧稳定极小超曲面是平面的简单证明，核心方法基于非负曲率空间中稳定极小超曲面具有谱里奇曲率下界。

A Spectral Bernstein Theorem

We study the spectrum of the Laplace operator of a complete minimal properly immersed hypersurface $M$ in $\R^{n+1}$. (1) Under a volume growth condition on extrinsic balls and a condition on the unit normal at infinity, we prove that $M$ has only essential spectrum consisting of the half line $[0, +\infty)$. This is the case when $\lim_{\tilde{r}\to +\infty}\tilde{r}κ_i=0$, where $\tilde{r}$ is the extrinsic distance to a point of $M$ and $κ_i$ are the principal curvatures. (2) If the $κ_i$ satisfy the decay conditions $|κ_i|\leq 1/\tilde{r}$, and strict inequality is achieved at some point $…

Minimal spacelike surfaces and the graphic equations in R^4_1

In this paper we study an extension of the Bernstein Theorem for minimal spacelike surfaces of the four dimensional Minkowski vector space form and we obtain the class of those surfaces which are also graphics and have non-zero Gauss curvature. That is the class of entire solutions of a system of two elliptic non-linear equations that is an extension of the equation of minimal graphic of $\mathbb R^3$. Therefore, we prove that the so-called Bernstein property does not hold in general for the case of graphic spacelike surfaces in $\mathbb R^4_1$. In addition, we also obtain explicitly the conju…

Compact stable surfaces with constant mean curvature in Killing submersions

A Killing submersion is a Riemannian submersion from a 3-manifold to a surface, both connected and orientable, whose fibres are the integral curves of a Killing vector field, not necessarily unitary. The first part of this paper deals with the classification of all Killing submersions in terms of two geometric functions, namely the bundle curvature and the length of the Killing vector field, which can be prescribed arbitrarily. In a second part, we show that if the base is compact and the submersion admits a global section, then it also admits a global minimal section. These turn out to be the…

Lagrangian Mean Curvature flow for entire Lipschitz graphs

We consider the mean curvature flow of entire Lagrangian graphs with Lipschitz continuous initial data. Assuming only a certain bound on the Lipschitz norm of an initial entire Lagrangian graph in $\R^{2n}$, we show that the parabolic equation \eqref{PMA} for the Lagrangian potential has a longtime solution which is smooth for all positive time and satisfies uniform estimates away from time $t=0$. In particular, under the mean curvature flow the graph immediately becomes smooth and the solution exists for all time such that the second fundamental form decays uniformly to 0 on the graph as $t\t…

收缩子的强弗兰克尔定理

A strong Frankel Theorem for shrinkers

本文证明了任意维数下平均曲率流收缩子的强弗兰克尔定理，即大球内任意两个收缩子必相交且收缩子自身连通，其证明核心是建立不完全稳定高斯曲面的强伯恩斯坦定理。

On the Geometry of Conformally Stationary Lorentz Spaces

In this paper we study several aspects of the geometry of conformally stationary Lorentz manifolds, and particularly of GRW spaces, due to the presence of a closed conformal vector field. More precisely, we begin by extending to these spaces a result of J. Simons on the minimality of cones in Euclidean space, and apply it to the construction of complete, noncompact maximal submanifolds of both de Sitter and anti-de Sitter spaces. Then we state and prove very general Bernstein-type theorems for spacelike hypersurfaces in conformally stationary Lorentz manifolds, one of which not assuming the hy…

Dual quadratic differentials and entire minimal graphs in Heisenberg space

We define holomorphic quadratic differentials for spacelike surfaces with constant mean curvature in the Lorentzian homogeneous spaces $\mathbb{L}(κ,τ)$ with isometry group of dimension 4, which are dual to the Abresch-Rosenberg differentials in the Riemannian counterparts $\mathbb{E}(κ,τ)$, and obtain some consequences. On the one hand, we give a very short proof of the Bernstein problem in Heisenberg space, and provide a geometric description of the family of entire graphs sharing the same differential in terms of a 2-parameter conformal deformation. On the other hand, we prove that entire m…