Compact Surfaces with Constant Gaussian Curvature in Product Spaces

We prove that the only compact surfaces of positive constant Gaussian curvature in \mathbbH²×\mathbbR{\mathbb{H}^{2}\times\mathbb{R}} (resp. positive constant Gaussian curvature greater than 1 in \mathbbS²×\mathbbR{\mathbb{S}^{2}\times\mathbb{R}}) whose boundary Γ is contained in a slice of the ambient space and such that the surface intersects this slice at a constant angle along Γ, are the pieces of a rotational complete surface. We also obtain some area estimates for surfaces of positive constant Gaussian curvature in \mathbbH²×\mathbbR{\mathbb{H}^{2}\times\mathbb{R}} and positive constant Gaussian curvature greater than 1 in \mathbbS²×\mathbbR{\mathbb{S}^{2}\times\mathbb{R}} whose boundary is contained in a slice of the ambient space. These estimates are optimal in the sense that if the bounds are attained, the surface is again a piece of a rotational complete surface.     

Constant Gaussian Curvature Surfaces with Parallel Mean Curvature Vector in Two-Dimensional Complex Space Forms

We classify constant Gaussian curvature surfaces with nonzero parallel mean curvature vector in two-dimensional complex space forms. As a result, we find new examples of such surfaces.     

On Discrete Constant Mean Curvature Surfaces

Recently, a curvature theory for polyhedral surfaces has been established that associates with each face a mean curvature value computed from areas and mixed areas of that face and its corresponding Gaussian image face. Therefore, a study of constant mean curvature (cmc) surfaces requires studying pairs of polygons with some constant nonvanishing value of the discrete mean curvature for all faces. We focus on meshes where all faces are planar quadrilaterals or planar hexagons. We show an incidence geometric characterization of a pair of parallel quadrilaterals having a discrete mean curvature value of ?1. This characterization yields an integrability condition for a mesh being a Gaussian image mesh of a discrete cmc surface. Thus, we can use these geometric results for the construction of discrete cmc surfaces. In the special case where all faces have a circumcircle, we establish a discrete Weierstrass-type representation for discrete cmc surfaces.     

A Version of the Ruh--Vilms Theorem for Surfaces of Constant Mean Curvature in S 3

We study a version of the Gauss map for a surface immersed in and prove an analog of the Ruh--Vilms theorem which states that this map is harmonic iff has a constant mean curvature. As a corollary, we conclude that an embedded flat torus with constant mean curvature is a spherical Delonay surface.     

Gaussian Curvature and Gyroscopes

We relate Gaussian curvature to the gyroscopic force, thus giving a mechanical interpretation of the former and a geometrical interpretation of the latter. We do so by considering the motion of a spinning disk constrained to be tangent to a curved surface. It is shown that the spin gives rise to a force on the disk that is equal to the magnetic force on a point charge moving in a magnetic field normal to the surface, of magnitude equal to the Gaussian curvature, and of charge equal to the disk's axial spin. In a special case, this demonstrates that the precession of Lagrange's top is due to the curvature of a sphere determined by the parameters of the top. © 2017 Wiley Periodicals, Inc.     

A Theorem on Infinitesimal I-isometry of Surfaces Immersed in a Space with Constant Curvature

1.IntroductionTheproblemsofinfinitesimalrigidityandisometricdeformationhavebeenextensivelyin-vestigatedforsurfacesinthreedimensionalEuclideanspaceE'.See,forexample,A.SvecL1]'E.Kann[2j,ZhouJiazu[3Jandthepapers[4j,[5J,[6JbyYangWenmao3also,KobayashiandNomizu[7].ThefirstauthorandZhangGaoyonggeneralizedthetheoryofinfinitesimali-sometriesofsurfacesinE3tothehypersurfacesimmersedinaspaceofconstantcurvature[8j.K.Tenenblat[9]andR.A.GoldsteinandP.J.Ryan[1o]studiedtheinfinitesimalI-isometryfors…     

On Two-Dimensional Polynomially Integrable Billiards on Surfaces of Constant Curvature

Doklady Mathematics - The algebraic version of the Birkhoff conjecture is solved completely for billiards with a piecewise C2-smooth boundary on surfaces of constant curvature: Euclidean plane,...     

Inequalities for the Curvature of Curves and Surfaces

In this paper we bound the difference between the total mean curvatures of two closed surfaces in in terms of their total absolute curvatures and the Frechet distance between the volumes they enclose. The proof relies on a combination of methods from algebraic topology and integral geometry. We also bound the difference between the lengths of two curves using the same methods.     

Bonnet Surfaces with Constant Curvature

We classify Bonnet surfaces with constant curvature in 3-dimensional space forms and show that they are parametrized by curves in 2-dimensional space forms with specific geodesic curvature.     

Baillon's Theorem on Maximal Regularity

The aim of this note is to give a proof of Baillon's Theorem on Maximal Regularity. Though it is in some sense a negative result (it states that for abstract Cauchy problems maximal regularity can occur only in very special cases), it is commonly accepted that it is important. Many people believe that its proof is very complicated. This might be due to the fact that Baillon's note in the Comptes Rendus is rather short and sometimes difficult to understand. The proof outlined here follows basically Baillon's lines. However it is simplified and (hopefully) easier to understand.     

全平面上的高斯曲率方程

本文用Ｓｃｈａｕｄｅｒ不动点定理证明了一维Ｋ≥０的解、二维Ｋ≥０的径向解的存在性，同时证明了当Ｋ≤０时，在无穷远处有不同渐近性的Ｋ所对应的极大解的渐近性，并给出了径向解的刻画，推广了前人结果．     

A Combinatorial Application of the Maximal Ergodic Theorem

Let X be a compact space,µ a Borel probability measureon X, T: X X a measure preserving continuous transformationand g: X R a continuous function. Then for some yX, This Lemma is used to give an alternative proof of a resultby Ruzsa [6], which implies the following extension of a resultof Bergelson [1]. If E N satisfies then there exists a set N such that n–1|[1,n]| (E) for all, n 1, and any finite subset{₁, ... _k} satisfies Ø. 7 Moria St., Ramat Hasharon, Israel     

A Splitting Theorem for Scalar Curvature

We show that a Riemannian 3-manifold with nonnegative scalar curvature is flat if it contains an area-minimizing cylinder. This scalar-curvature analogue of the classical splitting theorem of J. Cheeger and D. Gromoll (1971) was conjectured by D. Fischer-Colbrie and R. Schoen (1980) and by M. Cai and G. Galloway (2000). © 2018 the Authors. Communications on Pure and Applied Mathematics is published by the Courant Institute of Mathematical Sciences and Wiley Periodicals, Inc.     

Kirszbraun's Theorem and Metric Spaces of Bounded Curvature

We generalize Kirszbraun's extension theorem for Lipschitz maps between (subsets of) euclidean spaces to metric spaces with upper or lower curvature bounds in the sense of A.D. Alexandrov. As a by-product we develop new tools in the theory of tangent cones of these spaces and obtain new characterization results which may be of independent interest. Submitted: June 1996, final version: November 1996     

New proof of the Calabi-Bernstein theorem

By means of a local integral formula on a maximal surface, which involves the hyperbolic angle between the unit normal vector field and a fixed timelike direction, we give a new proof of the parametric Calabi-Bernstein theorem. A suitable modification of the integral formula gives an independent proof to the non-parametric case of this result.     

The Self-conjugate Maximal Surfaces in L~(3*)

in this note we study the maximal surfaces which are congruent with theirconjugate ones in L3, and give a criterion for this kind of surfaces and some examples.