Robert Osserman, Min Ru
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 ${\mathbb P}^{m-1}({\mathbb C})$.
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The Gauss map of hypersurfaces with constant weighted mean curvature in the Gaussian space
本文研究 Gauss 空间中常加权平均曲率超曲面，依据其 Gauss 映射像所在区域证明超曲面必为超平面或广义柱面，推广了自收缩子的既有结论。
We construct complete nonorientable minimal surfaces whose Gauss map omits two points of the projective plane. This result proves that Fujimoto's theorem is sharp in nonorientable case.
Klainerman, Luk and Rodnianski derived an anisotropic criterion for formation of trapped surfaces in vacuum, extending the original trapped surface formation theorem of Christodoulou. The effort to understand their result led us to study the intersection of a hyperplane with a lightcone in the Minkowski spacetime. For the intrinsic geometry of the intersection, depending on the hyperplane being spacelike, null or timelike, it has the constant positive, zero or negative Gaussian curvature. For the extrinsic geometry of the intersection, we find that it is a noncompact marginal trapped surface w…
In [15] Robert Osserman proved that the image of the Gauss map of a complete, non flat minimal surface in R^3 with finite total curvature miss at most 3 points. In this paper we prove that the Gauss map of such a minimal immersions omit at most 2 points. This is a sharp result since the Gauss map of the catenoid omits exactly two points. In fact we prove this result for a wider class of isometric immersions, that share the basic differential topological properties of the complete minimal surfaces of finite total curvature.
Given a positive function $F$ on $\mathbb S^n$ which satisfies a convexity condition, for $1\leq r\leq n$, we define for hypersurfaces in $\mathbb{R}^{n+1}$ the $r$-th anisotropic mean curvature function $H_{r; F}$, a generalization of the usual $r$-th mean curvature function. We call a hypersurface is anisotropic minimal if $H_F=H_{1; F}=0$, and anisotropic $r$-minimal if $H_{r+1; F}=0$. Let $W$ be the set of points which are omitted by the hyperplanes tangent to $M$. We will prove that if an oriented hypersurface $M$ is anisotropic minimal, and the set $W$ is open and non-empty, then $x(M)$ …
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.
In a recent paper Jorge and Mercuri proved that the image of Gauss map of a complete non flat minimal surfaces in R3 with finite total curvature omits at most 2 points. In this work we follow their idea and prove 3a similar result for CMC-1 with finite total curvature in H and CMC-1 faces with finite type and regular ends in S31.
The Gauss images of complete minimal surfaces of genus zero of finite total curvature
本文研究欧氏 3、4 维空间中有限全曲率零亏格完备极小曲面的 Gauss 像，构造出含 2 个省略值、总权重 5/2 的完全分歧值的新例子并提出开放问题。
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 $S^{k+1}$.
Let $M$ be an open Riemann surface and let $Λ\subset M$ be a closed discrete subset. In this paper, we prove the existence of complete conformal minimal immersions $M\to\mathbb{R}^n$, $n\ge 3$, with prescribed values on $Λ$ and whose generalized Gauss map $M\to\mathbb{CP}^{n-1}$, $n\ge 3$, avoids $n$ hyperplanes of $\mathbb{CP}^{n-1}$ located in general position. In case $n=3$, we obtain complete nonflat conformal minimal immersions whose Gauss map $M\to\mathbb{S}^2$ omits two (antipodal) values of the sphere. This result is deduced as a consequence of an interpolation theorem for conformal mi…
On the Gauss maps of complete minimal surfaces in $\mathbb{R}^n$
本文证明非平坦完备极小曲面在 $\mathbb{R}^n$ 中的 Gauss 映射可省略次数不超过 $n^{n+2}(n+1)^{n+2}$ 的一般超曲面。
The purpose of this paper is to reveal the relationship between the total curvature and the global behavior of the Gauss map of a complete minimal Lagrangian surface in the complex two-space. To achieve this purpose, we show the precise maximal number of exceptional values of the Gauss map for a complete minimal Lagrangian surface with finite total curvature in the complex two-space. Moreover, we prove that if the Gauss map of a complete minimal Lagrangian surface which is not a Lagrangian plane omits three values, then it takes all other values infinitely many times.
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. …
Surfaces of finite geometric type are complete, immersed into the tree-dimensional Euclidean space with finite total curvature and Gauss map extending to an oriented compact surface as a smooth branched covering map over the unit sphere of the Euclidean three dimensional space. In a recent preprint J. Jorge and F. Mercuri gave a geometric proof that the Gauss map can not omit three or more points if the immersion is minimal and no flat. Here we give a topological proof of this result in the class of no flat finite geometric type surfaces and also give a topological classification when the Gaus…
Modified defect relation for Gauss maps of minimal surfaces with hypersurfaces of projective varieties in subgeneral position
本文建立了完整极小曲面 Gauss 映射与射影簇中次一般位置超曲面的修正亏量关系，给出了相交次数有限且非退化条件下超曲面数量的上界，推广了先前关于超平面的结果并应用于唯一性研究。
We prove a version of the classical Runge and Mergelyan uniform approximation theorems for non-orientable minimal surfaces in Euclidean 3-space R3. Then, we obtain some geometric applications. Among them, we emphasize the following ones: 1. A Gunning-Narasimhan type theorem for non-orientable conformal surfaces. 2. An existence theorem for non-orientable minimal surfaces in R3, with arbitrary conformal structure, properly projecting into a plane. 3. An existence result for non-orientable minimal surfaces in R3 with arbitrary conformal structure and Gauss map omitting one projective direction.
Application of the Bloch--Ros principle to ramification theorem
本文应用Bloch-Ros原理，统一处理值分布理论、正规族理论以及几类曲面（包括极小曲面、Lorentz-Minkowski 3-空间中的maxfaces和仿射3-空间中的非正常仿射前沿）的Gauss映射理论中关于完全分歧值的现象。
An estimation of the Gauss curvature and the modified defect relation for the Gauss map of immersed harmonic surfaces in $\mathbb{R}^n$
本文针对 $\u211d^3$ 中 $K$-拟共形调和曲面，在法映射遗漏满足特定几何条件的 7 个方向时，证明了高斯曲率的精确上界估计，并进一步验证了 $\u211d^n$ 中浸入调和曲面广义高斯映射的修正亏量关系。
We construct a special class of Lorentz surfaces in the pseudo-Euclidean 4-space with neutral metric which are one-parameter systems of meridians of rotational hypersurfaces with lightlike axis and call them meridian surfaces. We give the complete classification of the meridian surfaces with constant Gauss curvature and prove that there are no meridian surfaces with parallel mean curvature vector field other than CMC surfaces lying in a hyperplane. We also classify the meridian surfaces with parallel normalized mean curvature vector field. We show that in the family of the meridian surfaces th…
We prove that for any open Riemann surface $N,$ natural number $n\geq 3,$ non-constant harmonic map $h:N\to \mathbb{R}^{n-2}$ and holomorphic 2-form $H$ on $N,$ there exists a weakly complete harmonic map $X=(X_j)_{j=1,\ldots,n}:N \to \mathbb{R}^n$ with Hopf differential $H$ and $(X_j)_{j=3,\ldots,n}=h.$ In particular, there exists a complete conformal minimal immersion $Y=(Y_j)_{j=1,\ldots,n}:N \to \mathbb{R}^n$ such that $(Y_j)_{j=3,\ldots,n}=h.$ As a consequence of these results, complete full non-decomposable minimal surfaces with arbitrary conformal structure and whose generalized Gauss m…
Grauert showed that the existence of a complete Kähler metric does not characterize domains of holomorphy by constructing such metrics on the complements of complex analytic sets in a domain of holomorphy. In this note, we study the holomorphic sectional curvatures of such metrics in two prototype cases namely, $\mathbb{C}^n \setminus \{0\}, n \ge 2$ and $\mathbb{B}^N \setminus A$, $N \ge 2$ and $A \subset \mathbb{B}^N$ is a hyperplane of codimension at least two. This is done by computing the Gaussian curvature of its restriction to the leaves of a suitable holomorphic foliation of these two …
The aim of this work is to adapt the complex analytic methods originating in modern Oka theory to the study of non-orientable conformal minimal surfaces in $\mathbb{R}^n$ for any $n\ge 3$. These methods, which we develop essentially from the first principles, enable us to prove that the space of conformal minimal immersions of a given bordered non-orientable surface to $\mathbb{R}^n$ is a real analytic Banach manifold, obtain approximation results of Runge-Mergelyan type for conformal minimal immersions from non-orientable surfaces, and show general position theorems for non-orientable conform…
Timelike meridian surfaces of hyperbolic type in the Minkowski 4-space
研究 Minkowski 4-空间中一类特殊的类时曲面（双曲型类时子午面），证明其具有平坦法向连接，并分类了常 Gauss 曲率、极小、非零常平均曲率以及具有平行归一化平均曲率向量场的曲面。
Modified defect relation of Gauss maps on annular ends of minimal surfaces for hypersurfaces of projective varieties in subgeneral position
本文建立了完备极小曲面环状端上广义高斯映射关于射影簇子一般位置超曲面的修正亏量关系，证明了非退化映射的像不能省略过多超曲面，并首次提出相关的乘积求和不等式，为高斯映射唯一性研究提供了基础。
Complete minimal surfaces of finite total curvature on punctured spheres with totally ramified value number greater than $2$
本文系统构造了满足ν_g>2的极小曲面，证明了Miyaoka-Sato例子的唯一性，并构造了新的极小曲面例子。
Generalizing the Cauchy-Riemann equations, we construct the Osserman system of the first order for a pair $\left(f(x, y), g(x,y) \right)$ of two ${\mathbb{R}}$-valued functions on the domain $Ω\subset {\mathbb{R}}^{2}$. The graph $\left\{\, \left(x, y, f(x, y), g(x,y) \right) \in {\mathbb{R}}^{4} \, \vert \, (x,y) \in Ω\, \right\}$ becomes a minimal surface in ${\mathbb{R}}^{4}$, whose generalized Gauss map lies on the intersection of a hyperplane of the complex projective space ${\mathbb{CP}}^{3}$ and the complex cone ${z_1}^{2}+{z_2}^{2}+{z_3}^{2}+{z_4}^{2}=0$. We present two applications of…
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