Francisco J. Lopez, Francisco Martin
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
Remarks on the Gauss images of complete minimal surfaces in Euclidean four-space
系统研究欧氏四维空间中完备极小曲面的高斯映射像，给出例外值最大数量的几何解释，并为拉格朗日曲面及不可定向曲面提供最优结果。
It was proved that the fundamental group of the space of harmonic polynomials of degree $n(n \geq 2)$, with the same Gaussian curvature is not trivial. Furthermore, we give an example of topologically nonequivalent conjugate harmonic functions having the same Gaussian curvature.
We give a short proof of the following fact. Let $Σ$ be a connected, finitely connected, noncompact manifold without boundary. If $g$ is a complete Riemannian metric on $Σ$ whose Gaussian curvature $K$ is nonnegative at infinity, then $K$ must be integrable. In particular, we obtain a new short proof of the fact that if $Σ$ admits a complete metric whose Gaussian curvature is nonnegative and positive at one point, then $Σ$ is diffeomorphic to $\mathbb{R}^2$.
Using polar convex bodies and the $C_0$-bounds from Guan and Ni \cite{PL}, we obtain a uniform lower bound on the Gauss curvature of the normalized solution of the Gauss curvature flow without using Chow's Harnack inequality \cite{Ch2}.
We use a Simons type equation in order to characterize complete non-minimal pmc surfaces with non-negative Gaussian curvature.
The classical result of Nevanlinna states that two nonconstant meromorphic functions on the complex plane having the same images for five distinct values must be identically equal to each other. In this paper, we give a similar uniqueness theorem for the Gauss maps of complete minimal surfaces in Euclidean four-space.
We describe the gauge-theoretic approach to transformations in integrable geometry through discussion of two classical examples: surfaces of constant negative Gauss curvature and isothermic surfaces. These are purely expository notes written to accompany some lectures I gave in Fukuoka in May 2015.
Lecture notes for a minicourse to given in the XVII Brazilian School of Geometry, UFAM (Amazonas), Brazil, July 2012.
In this paper, we prove a local index theorem for the DeRham Hodge-laplacian which is defined by the connection compatible with metric. This connection need not be the Levi-Civita connection. When the connection is Levi-Civita connection, this is the classical local Gauss-Bonnet-Chern theorem.
A note on parallel mean curvature surfaces and Codazzi operators
本文提出用Codazzi算子刻画的曲面内蕴Klotz-Osserman型定理，并利用Simons公式研究乘积空间中具非正Gauss曲率的平行平均曲率曲面。
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.
The skew mean curvature flow (SMCF) is a natural generalization of the famous vortex filament equation. In this note, we show that the Gauss map of the SMCF satisfies a Schrödinger flow equation. In this regard, we explore the geometry of the oriented Grassmannian manifold explicitly by embedding it into the exterior product space.
Let $M^4$ be a closed immersed minimal hypersurface with constant squared length of the second fundamental form $S$ and constant 3-mean curvature $H_3$ in $\mathbb{S}^{5}$. If $H_3^2\leq \frac{1}{2}.$ and Gauss-Kronecker curvature $K_M$ satisfies $K_M\leq1$ $($or $ K_M\leq\frac{S^2}{144}$$)$, then $M^4$ is isoparametric.
In this short note we prove the convexity of minimizers of some variational problem in the Gauss space. This proof is based on a geometric version of an older argument due to Korevaar.
The aim of this note is to provide an intrinsic proof of the Gauss--Bonnet theorem without invoking triangulations, which is achieved by exploiting complex structures.
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).
In this note, we prove that the abstract gradient flow introduced by Baird-Fardoun-Regbaoui \cite{BFR}is well-posed on a closed Riemann surface with conical singularity. Long time existence and convergence of the flow are proved under certain assumptions. As an application, the prescribed Gaussian curvature problem is solved when the singular Euler characteristic of the conical surface is non-positive.
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。