Yuxin Dong, Yingbo Han, Qingchun Ji
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 $C^n$
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We show the regularity of, and derive a-priori estimates for (weakly) harmonic maps from a Riemannian manifold into a Euclidean sphere under the assumption that the image avoids some neighborhood of a half-equator. The proofs combine constructions of strictly convex functions and the regularity theory of quasi-linear elliptic systems. We apply these results to the spherical and Euclidean Bernstein problems for minimal hypersurfaces, obtaining new conditions under which compact minimal hypersurfaces in spheres or complete minimal hypersurfaces in Euclidean spaces are trivial.
The authors obtained the classification theorem for Lagrangian H-umbilical submanifolds in quaternion Euclidean spaces using the idea of quaternion extensors.
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。
In this paper, we introduce the quaternionic similar curves in 4-dimensional Euclidean space. We show that the families of quaternionic curves with vanishing curvatures form the families of quaternionic similar curves.
Using the idea of special Legendre curves, the authors obtained the explicit description of flat Lagrangian H-umbilical submanifolds in quaternion Euclidean spaces.
In [18], L. R. Pears proved that Bertrand curves in E-n(n > 3) are degenerate curves. This result restate in [16] by Matsuda and Yorozu. They proved that there is no special Bertrand curves in E-n(n > 3) and they define new kind of Bertrand curves called (1, 3)-type Bertrand curves in 4-dimensional Euclidean space. In this study, we define a quaternionic Bertrand curve ?(4) in Euclidean space E4 and investigate its properties for two cases. In the first case; we consider quaternionic Bertrand curve in the Euclidean space E4 for r-K = 0 where r is the torsion of the spatial quaternionic curve ?…
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。
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.
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.
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 filling volume functions of the n-th quaternionic Heisenberg group grow, up to dimension n, as fast as the ones of the Euclidean space. We identify the growth rate of the filling volume function in dimension n+1, which is strictly faster than the growth rate of the (n+1)-dimensional filling volume function of the Euclidean space.
In [5], Matsuda and Yorozo obtained that Mannheim curves in 4-dimensional Euclidean space. In this study, we define quaternionic Mannheim curves and we give some characterizations of them in Euclidean 3-space and 4-space.
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。
Bernstein problem for affine maximal type equation has been a core problem in affine geometry. A conjecture proposed firstly by Chern for entire graph and then extended by Trudinger-Wang to its fully generality asserts that any Euclidean complete, affine maximal type, locally uniofrmly convex C^4-hypersurface in R^{N+1} must be an elliptic paraboloid. At the same time, this conjecture was solved completely by Trudinger-Wang for dimension N=2 and θ=3/4, and later extended by Jia-Li to N=2, θ\in(3/4,1] (see also [Zhou]). On the past twenty years, much efforts were done toward higher dimensional …
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.
We prove that any complete, uniformly elliptic Weingarten surface in Euclidean $3$-space whose Gauss map image omits an open hemisphere is a cylinder or a plane. This generalizes a classical theorem by Hoffman, Osserman and Schoen for constant mean curvature surfaces. In particular, this proves that planes are the only complete, uniformly elliptic Weingarten multigraphs. We also show that this result holds for a large class of non-uniformly elliptic Weingarten equations. In particular, this solves in the affirmative the Bernstein problem for entire graphs for that class of elliptic equations. …
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.
We proved that any complete hypersurface in the Euclidean space $\mathbb{R}^{n+1}$ whose Gauss image is contained in an open hemisphere has to be proper. As applications, we derive a counterpart of Hoffman-Osserman-Schoen's result for $λ$-hypersurfaces, which gives an affirmative answer to the Bernstein type problem proposed by Cheng-Wei \cite{CW14b}.
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.
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.
This paper gives, in generic situations, a complete classification of ruled minimal surfaces in pseudo-Euclidean space with arbitrary index. In addition, we discuss the condition for ruled minimal surfaces to exist, and give a counter-example on the problem of Bernstein type.
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…
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。
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…
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.
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.
We prove that in the Heisenberg group $\mathbb{H}^1$ with a sub-Finsler structure, an $(X,Y)$-Lipschitz surface which is complete, oriented, connected and stable must be a vertical plane. In particular, the result holds for entire intrinsic graphs of Euclidean Lipschitz functions.
In this note we consider the problem of local classification of Bonnet pairs of surfaces in 3-dimensional Minkowski space. We use split quaternions in a way similar to the use of quaternions in, arXiv:dg-ga/9610006, for the solution in Euclidean 3-Space.
Curvature-adapted hypersurfaces of 2-type in non-flat quaternionic space forms
本文分类了非平坦四元数空间形式中在四元-Hermitian矩阵空间内的Chen 2型曲率适应实超曲面，确定了射影与双曲空间中的具体几何类型，并判别了其质量对称性与极小性。
A conformal map from a Riemann surface to a Euclidean space of dimension greater than or equal to three is explained by using the Clifford algebra, in a similar fashion to quaternionic holomorphic geometry of surfaces in the Euclidean three- or four-space. The Weierstrass representation, the spin transform, the Darboux transforms, surfaces of parallel mean curvature vector, families of flat connections associated with a harmonic map from a Riemann surface to a sphere are explained. The degree of the spinor bundle associated with a conformal immersion is calculated. Analogues of a polar surface…
A dichotomy for minimal submanifolds
本文综述最新成果，提出极小子流形的二分性，通过体积加倍定理等机制推导其受限性带来的几何与函数论限制，将$\bf{R}^3$嵌入极小圆盘理论弱推广至所有维数。
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.
Using the stress energy tensor, we establish some monotonicity formulae for vector bundle-valued p-forms satisfying the conservation law, provided that the base Riemannian (resp. Kähler) manifolds poss some real (resp. complex) p-exhaustion functions. Vanishing theorems follow immediately from the monotonicity formulae under suitable growth conditions on the energy of the p-forms. As an application, we establish a monotonicity formula for the Ricci form of a Kähler manifold of constant scalar curvature and then get a growth condition to derive the Ricci flatness of the Kähler manifold. In part…
We introduce the complete lifts of maps between (real and complex) Euclidean spaces and study their properties concerning holomorphicity, harmonicity and horizontal weakly conformality. As applications, we are able to use this concept to characterize holomorphic maps $ϕ:{\Bbb C}^{m}\supset U\longrightarrow {\Bbb C}^{n}$ (Proposition 2.3) and to construct many new examples of harmonic morphisms (Theorem 3.3). Finally we show that the complete lift of the quaternion product followed by the complex product is a simple and explicit example of a harmonic morphism which does not arise (see Definitio…
Stable anisotropic capillary hypersurfaces in a half-space
本文研究了半空间中各向异性毛细超曲面的稳定性，证明了紧致常平均曲率超曲面弱稳定当且仅当为截断Wulff形状，并给出了三维情形下的Bernstein型定理。
Some estimates on stable minimal hypersurfaces in Euclidean space
本文推导了 $R^{n+1}$ 中稳定极小超曲面的若干估计，揭示了现有 Bernstein 定理证明方法在 $n=6$ 时的局限性，并指出该推导方法有望应用于其他问题。
On potentials for sub-Laplacians and geometric applications
本文提出了一种在同构卡拉尼集团中利用几何完备性估计极集哈代维度的新方法，并应用于CR几何等领域。
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.
We introduce the beta function of a knot in euclidean three-space. This is a meromorphic function of a complex variable which we prove admits a Bernstein type functional equation. We determine the first residues.
New non-quadratic Euclidean complete affine maximal type hypersurfaces via Calabi affine geometry
本文从子流形理论与Calabi仿射几何视角研究仿射极大型曲面，刻画了已有显式欧氏完备反例的几何特征，并构造新的非二次欧氏完备仿射极大型超曲面，拓展了对应参数的取值范围。
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…
We prove a positive mass theorem for spaces which asymptotically approach a flat Euclidean space times a Calabi-Yau manifold (or any special honolomy manifold except the quaternionic Kähler). This is motivated by the very recent work of Hertog-Horowitz-Maeda.
A super-conformal map and a minimal surface are factored into a product of two maps by modeling the Euclidean four-space and the complex Euclidean plane on the set of all quaternions. One of these two maps is a holomorphic map or a meromorphic map. These conformal maps adopt properties of a holomorphic function or a meromorphic function. Analogs of the Liouville theorem, the Schwarz lemma, the Schwarz-Pick theorem, the Weierstrass factorization theorem, the Abel-Jacobi theorem, and a relation between zeros of a minimal surface and branch points of a super-conformal map are obtained.
Complex Quaternions and Superminimal Surfaces in Four-Space
利用复四元数代数，将共形超极小曲面表示为全纯零曲线因子分解，并证明超极小性等价于广义高斯映射的一个分量恒定。
This paper shows that the Grassmann Manifolds $G_{\bf F}(n,N)$ can all be imbedded in an Euclidean space $M_{\bf F}(N)$ naturally and the imbedding can be realized by the eigenfunctions of Laplacian $\triangle$ on $G_{\bf F}(n,N)$. They are all minimal submanifolds in some spheres of $M_{\bf F}(N)$ respectively. Using these imbeddings, we construct some degenerate Morse functions on Grassmann Manifolds, show that the homology of the complex and quaternion Grassmann Manifolds can be computed easily.
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