On Bernstein-type theorems

Generalized Bernstein Theorem for Stable Minimal Plateau Surfaces

In this paper, we consider a Generalized Bernstein Theorem for a type of generalized minimal surfaces, namely minimal Plateau surfaces. We show that if an orientable minimal Plateau surface is stable and has quadratic area growth in $\mathbb{R}^3 $, then it is flat.

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.

A spherical Bernstein theorem for minimal submanifolds of higher codimension

Combining the tools of geometric analysis with properties of Jordan angles and angle space distributions, we derive a spherical and a Euclidean Bernstein theorem for minimal submanifolds of arbitrary dimension and codimension, under the condition that the Gauss image is contained in some geometrically defined closed region of a Grassmannian manifold. The proof depends on the subharmoncity of an auxiliary function, the Codazzi equations and geometric measure theory.

三维光锥上的常曲率曲面的Bernstein型定理

Bernstein-type Theorems for constant mean curvature surfaces in the three-dimensional light cone

本文在Gauss曲率下界条件下，证明了三维光锥上整个常均曲率图是horosphere或光锥上的球面。

Bernstein-type theorem for zero mean curvature hypersurfaces without time-like points in Lorentz-Minkowski space

Calabi and Cheng-Yau's Bernstein-type theorem asserts that an entire zero mean curvature graph in Lorentz-Minkowski $(n+1)$-space $\boldsymbol R^{n+1}_1$ which admits only space-like points is a hyperplane. Recently, the third and fourth authors proved a line theorem for hypersurfaces at their degenerate light-like points. Using this, we give an improvement of the Bernstein-type theorem, and we show that an entire zero mean curvature graph in $\boldsymbol R^{n+1}_1$ consisting only of space-like or light-like points is a hyperplane. This is a generalization of the first, third and fourth autho…

Improvement of the Bernstein-type theorem for space-like zero mean curvature graphs in Lorentz-Minkowski space using fluid mechanical duality

Calabi's Bernstein-type theorem asserts that a zero mean curvature entire graph in Lorentz-Minkowski space $\boldsymbol L^3$ which admits only space-like points is a space-like plane. Using the fluid mechanical duality between minimal surfaces in Euclidean 3-space $\boldsymbol E^3$ and maximal surfaces in Lorentz-Minkowski space $\boldsymbol L^3$, we give an improvement of this Bernstein-type theorem. More precisely, we show that a zero mean curvature entire graph in $\boldsymbol L^3$ which does not admit time-like points (namely, a graph consists of only space-like and light-like points) is a…

A Bernstein type theorem for minimal hypersurfaces via Gauss maps

Let $M$ be an $n$-dimensional smooth oriented complete embedded minimal hypersurface in $\mathbb{R}^{n+1}$ with Euclidean volume growth. We show that if the image under the Gauss map of $M$ avoids some neighborhood of a half-equator, then $M$ must be an affine hyperplane.

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.

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。

A Bernstein theorem for special Lagrangian graphs

We obtain a Bernstein theorem for special Lagrangian graphs in n-dimensional complex space for arbitrary n only assuming bounded slope, but no quantitative restriction.

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 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.

Spherical Bernstein theorems for codimension 1 and 2

A result of B.Solomon (On the Gauss map of an area-minimizing hypersurface. 1984. Journal of Differential Geometry, 19(1), 221-232.) says that a compact minimal hypersurface $M^k$ of the sphere $S^{k+1}$ with $H^1(M)=0$, whose Gauss map omits a neighborhood of an $S^{k-1}$ equator, is totally geodesic in $S^{k+1}$. We develop a new proof strategy which can also obtain an analogous result for codimension 2 compact minimal submanifolds of $S^{k+1}$.

$\u211d^7$与$\u211d^8$中校准子流形的Bernstein定理

Bernstein Theorems for Calibrated Submanifolds in $\mathbb{R}^7$ and $\mathbb{R}^8$

本文研究$\u211d^7$中余结合子流形对应的$\u211d^4$到$\u211d^3$光滑映射的Bernstein问题，给出由斜率第二初等对称多项式表述的仿射性条件，并得到$\u211d^8$中Cayley子流形的对应定理。

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。

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…

Remarks on Bernstein theorems for translating solitons

In this paper, we study the properties of potential function of the translating soliton $M$ in $R^{n+1}$ and the volume growth of the intersection of Euclidean balls with $M$. We give a condition to obtain the Bernstein theorem for the translating solitons. We also give an outline of a simple proof of the Bernstein theorem due to Bombieri-De Giorgi-Miranda.

高维仿射Bernstein问题和Plateau问题

The higher dimensional affine Bernstein and Plateau problems

本文证明了所有维数下仿射Plateau问题的内部正则性，给出了维数十及以上仿射Bernstein问题的光滑反例，并在增长条件下证明了所有维数的仿射Bernstein定理。

Bernstein Type Results for Lagrangian Graphs with Partially Harmonic Gauss Map

We establish Bernstein Theorems for Lagrangian graphs which are Hamiltonian minimal or have conformal Maslov form. Some known results of minimal (Lagrangian) submanifolds are generalized.

Bernstein Type Theorems For Minimal Lagrangian Graphs of Quaternion Euclidean space

In this paper, we prove some Bernstein type results for $n$-dimensional minimal Lagrangian graphs in quaternion Euclidean space $H^n\cong R^{4n}$. In particular, we also get a new Bernstein Theorem for special Lagrangian graphs in $C^n$

A Bernstein type result of translating solitons

By using a new test function and the gradient estimate technique, we obtain a better Bernstein type result of translating solitons.

The Geometry of Maximum Principles and a Bernstein Theorem in Codimension 2

Moser's Bernstein theorem \cite{moser61} says that an entire minimal graph of codimension 1 with bounded slope must be a hyperplane. An analogous result for arbitrary codimension is not true, by an example of Lawson-Osserman. Here, we show that Moser's theorem nevertheless extends to codimension 2, i.e., a minimal $p$-submanifold in $R^{p+2}$, which is the graph of a smooth function defined on the entire $R^p$ with bounded slope, must be a $p$-plane. Our method depends on convexity properties of Grassmannians which come into play as the targets of the (harmonic) Gauss maps of our minimal subma…

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, +∞)。

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…

A Bernstein type theorem for graphic self-shrinkers with flat normal bundle

In this note we will prove that an $n$ dimensional graphic self-shrinker in $R^{n+m}$ with flat normal bundle is a linear subspace. This result is a generalization of the corresponding result of Lu Wang in codimension one case.

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.

$\mathbb{R}^{7}$ 中的稳定 Bernstein 定理

The stable Bernstein theorem in $\mathbb{R}^{7}$

本文通过 Green 函数方法证明了 $\mathbb{R}^{7}$ 中稳定极小超曲面的 Bernstein 定理，解决了该问题的最后情形。

The Gauss image of $λ$-hypersurfaces and a Bernstein type problem

In this paper, our purpose is to study rigidity theorems for $λ$-hypersurfaces in Euclidean space under Gauss map. As a Bernstein type problem for $λ$-hypersurfaces, we prove that an entirely graphic $λ$-hypersurface in Euclidean space is a hyperplane.

Bernstein theorem for translating solitons of hypersurfaces

In this paper, we prove a monotonicity formula and some Bernstein type results for translating solitons of hypersurfaces in $\re^{n+1}$, giving some conditions under which a trantranslating soliton is a hyperplane. We also show a gap theorem for the translating soliton of hypersurfaces in $R^{n+k}$, namely, if the $L^n$ norm of the second fundamental form of the soliton is small enough, then it is a hyperplane.

A one-sentence proof of the Bernstein type theorem for special Lagrangian equation in two dimensions

The aim of this expository article is to reveal the equivalence of Jörgens' Theorem on the two dimensional unimodular Hessian equation and Fu's Theorem on the two dimensional special Lagrangian equation.

A Bernstein Type Theorem For Self-similar Shrinkers

In this note, we prove that smooth self-shrinkers in $\Real^{n+1}$, that are entire graphs, are hyperplanes. Previously Ecker and Huisken showed that smooth self-shrinkers, that are entire graphs and have at most polynomial growth, are hyperplanes. The point of this note is that no growth assumption at infinity is needed.

A Bernstein Type Theorem for Minimal Graphs over Convex Domains

Given any $n \geq 2$, we show that if $Ω\subsetneq \mathbb{R}^n$ is an open convex domain (e.g. a half-space), and $u : Ω\to \mathbb{R}$ is a solution to the minimal surface equation which agrees with a linear function on $\partial Ω$, then $u$ must itself be linear.

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。

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。

非线性几何偏微分方程的 Bernstein 定理

Bernstein theorems for nonlinear geometric PDEs

本文围绕极小曲面、Monge–Ampère、特殊 Lagrangian 方程及极小曲面系统，综述几何偏微分方程中的 Bernstein 问题，内容源自 2024 年欧洲微分几何博士学校讲座。

各向异性极小图的半空间Bernstein定理

A half-space Bernstein theorem for anisotropic minimal graphs

该研究证明半空间上带平坦边界的各向异性极小图必为平坦的，将Edelen-Wang的结果推广至各向异性情形，采用最大值原理与完全非线性PDE理论思想完成证明。

Euler-Dierkes-Huisken泛函的稳态超曲面的Bernstein型定理

Bernstein-type theorem for stationary hypersurfaces of the Euler-Dierkes-Huisken functional

本文证明，对于α>0，所有通过原点的光滑、完备、连通、嵌入的α-稳态超曲面必为线性超平面。

New examples of Moser-Bernstein type problems for some nonlinear elliptic partial differential equations arising in geometry

A family of nonlinear partial differential equations of divergence form is considered. Each one is the Euler-Lagrange equation of a natural Riemaniann variational problem of geometric interest. New uniqueness results for the entire solutions of these equations on a parabolic Riemaniann manifold of arbitrary dimension are given. In particular, several Moser-Bernstein type theorems are proved.

A Bernstein theorem for two-valued minimal graphs in dimension four

We prove a Bernstein-type theorem for two-valued minimal graphs in the four-dimensional Euclidean space $\mathbf{R}^4$. This states that two-valued functions defined on the entire $\mathbf{R}^3$, and whose graph is a minimal surface, must necessarily be linear. This is a two-valued analogue of the classical Bernstein theorem, which asserts that in dimensions up to $n+1 \leq 8$, an entire single-valued minimal graph is linear. The main contrast with the single-valued theory is the presence of a large set of singularities in the graphs of two-valued functions. Indeed two-valued minimal graphs ar…

半空间中毛细管图的Bernstein型定理

A Bernstein-type theorem for capillary graphs in a half-space

证明低维或无穷远切锥不分裂垂直线的条件下，半空间中整体毛细极小图必为线性，并证明整体毛细极小超曲面的正则集连通，讨论与单相Bernoulli问题的联系。

带自由边界的$3$维各向异性极小图的Bernstein定理

Bernstein theorem for $3$-dimensional anisotropic minimal graphs with free boundary

本文研究带自由边界的各向异性极小曲面方程，解决混合边值问题，并证明$\mathbb{R}^3_+$上任何各向异性极小图必平坦，推广了Simon的经典结果。

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 $…

伯恩斯坦型梯度估计用于带有势项的加权局部热方程系统

Bernstein type gradient estimate for system of weighted local heat equations with potential term

本文研究了带有势项的加权局部热方程系统，推导了伯恩斯坦型梯度估计，并部分解决了Bhattacharyya等人提出的问题。