arXiv research

Survey Bernstein-type theorems for graphical surfaces in Euclidean and Lorentz-Minkowski spaces.

Ancient solutions to mean curvature flow have unique shapes.

Ancient symplectic solutions to mean curvature flow are flat.

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 spa…

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 $C^n$

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 the…

Paper proves stable minimal surfaces in 3D are flat.

We summarize results concerning the Bernstein property of differential equations.

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.

Explains Bernstein theorems for various geometric PDEs.

Paper proves a theorem about constant mean curvature surfaces in isotropic 3-space.

Based on a calibration argument, we prove a Bernstein type theorem for entire minimal graphs over Gauss space $\mathbb{G}^n$ by a simple proof.

The study proves that certain constant mean curvature surfaces in a specific cone are either spheres or horospheres.

Bernstein theorem proven for 2-valued minimal graphs in 4D.

Strong Frankel theorem for shrinkers in all dimensions.

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 hy…

Classifies surfaces with no Gaussian curvature.

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.

New theorems prove uniqueness of solutions to geometric PDEs.

The paper proves conditions for zero Gaussian curvature convex hypersurfaces to be hyperplanes.

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 poi…

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…

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.

Study on uniqueness of hypersurfaces in hyperbolic space with constant mean curvature.

The paper extends Bernstein Theorem for minimal spacelike surfaces in 4D Minkowski space.

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-Minko…

Minimal graph theorem proven for convex domains.

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-Liouvil…

The paper solves Bernstein problems for specific submanifolds in high-dimensional spaces.

Anisotropic minimal graphs over half-spaces are flat.

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 re…

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…

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.

Paper proves generalizations of Bernstein's theorem in higher dimensions.

The paper extends stable minimal hypersurface results to δ-stable hypersurfaces in R^(n+1).

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.

Proof that certain complex equations have only simple solutions.

Solves an old problem by showing round spheres are the only compact surfaces with specific curvature properties.

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.

Paper studies stability of curved surfaces in a half-space.

Minimal surface equation results in constant solutions on RCD spaces.

Formula proves almost monotonicity for H-minimal surfaces in Heisenberg group.

We study the problem of finding a minimal graph with prescribed boundary data in arbitrary dimension and codimension. Existence, uniqueness, stability and regularity are treated. We first present the well-known results for codimension one: Jenkins-Serrin's existence theorem, convexity properties of the area which give …

Paper studies curvature of stable surfaces meeting at a common boundary.

The study proves that certain minimal surfaces are flat under specific conditions.

We prove some Bernstein theorems for entire space-like submanifolds in pseudo-Euclidean spaces and, as a corollary, we obtain a new proof of the Calabi-Pogorelov theorem on global solutions of Monge-Ampere equations.

The paper proves that certain stationary hypersurfaces in high dimensions are essentially flat.

A celebrated result of S. Bernstein states that every solution of the minimal surface equation over the entire plane has to be an affine linear function. Since the paper of Bernstein appeared in 1927, many different proofs and generalizations of this beautiful theorem were given, namely to higher dimensions and to more…