Surfaces In Euclidean Three-space Research Articles

Geometric inequalities involving mean curvature for closed surfaces

In this paper we prove some geometric inequalities for closed surfaces in Euclidean three-space. Motivated by Gage's inequality for convex curves, we first verify that for convex surfaces the Willmore energy is bounded below by some scale-invariant quantities. In particular, we obtain an optimal scaling law between the Willmore energy and the isoperimetric ratio under convexity. In addition, we also address Topping's conjecture relating diameter and mean curvature for connected closed surfaces. We prove this conjecture in the class of simply-connected axisymmetric surfaces, and moreover obtain a sharp remainder term which ensures the first evidence that optimal shapes are necessarily straight even without convexity.

GAUSSIAN CURVATURE AND UNICITY PROBLEM OF GAUSS MAPS OF VARIOUS CLASSES OF SURFACES

In this article, we establish a new estimate for the Gaussian curvature of open Riemann surfaces in Euclidean three-space with a specified conformal metric regarding the uniqueness of the holomorphic maps of these surfaces. As its applications, we give new proofs on the unicity problems for the Gauss maps of various classes of surfaces, in particular, minimal surfaces in Euclidean three-space, constant mean curvature one surfaces in the hyperbolic three-space, maximal surfaces in the Lorentz–Minkowski three-space, improper affine spheres in the affine three-space and flat surfaces in the hyperbolic three-space.

QA-deformation of surface of negative Gaussian curvature

An infinitesimal deformation with the given law of changing the element of area of a surface in Euclidean three-space was considered in this article. Such deformation in the article was called the quasiareal infinitesimal deformation or, briefly, the QA-deformation. The problem of finding the QA-deformation, under which the unit normal vector to the surface is preserved, was reduced to the study of one nonhomogeneous partial differential equation of the second order with respect to one unknown function. The initial conditions, under which the only one QA-deformation with the stationary unit normal vector exists, were defined for the surfaces of a negative Gaussian curvature. In this case, for the above equation, the Cauchy and Goursat problems were applied. The initial conditions of these tasks were expressed through the deforming vector.

Semi-Discrete Constant Mean Curvature Surfaces of Revolution in Minkowski Space

In this paper we describe semi-discrete isothermic constant mean curvature surfaces of revolution with smooth profile curves in Minkowski three-space. Unlike the case of semi-discrete constant mean curvature surfaces in Euclidean three-space, they might have certain types of singularities in a sense defined by the second author in a previous work. We analyze the singularities of such surfaces.

Function-theoretic Properties for the Gauss Maps of Various Classes of Surfaces

AbstractWe elucidate the geometric background of function-theoretic properties for the Gauss maps of several classes of immersed surfaces in three-dimensional space forms, for example, minimal surfaces in Euclidean three-space, improper affine spheres in the affine three-space, and constant mean curvature one surfaces and flat surfaces in hyperbolic three-space. To achieve this purpose, we prove an optimal curvature bound for a specified conformal metric on an open Riemann surface and give some applications. We also provide unicity theorems for the Gauss maps of these classes of surfaces.

Generalized surfaces with constant H/K in Euclidean three-space

Surfaces in Euclidean three-space with constant ratio of mean curvature to Gauss curvature arise naturally as the parallel surfaces to minimal surfaces. They might possess singularities which occur naturally as focal points of minimal surfaces. We study geometric properties and the singularities of such surfaces, prove some global results about them, and provide a Bjorling formula to construct such surfaces with prescribed point or curve singularities.

Fluctuation of Sectional Curvature for Closed Hypersurfaces

Liebmann proved in 1899 that the only closed surfaces in Euclidean three-space that have constant Gauss curvature are round spheres. Thus, if a closed surface in three-space is not a topological sphere, its Gauss curvature must fluctuate. We consider quantitative formulations of this fact, also in higher dimensions.

Willmore two-spheres in the four-sphere

Genus zero Willmore surfaces immersed in the three-sphereS3\mathbb {S}^3correspond via the stereographic projection to minimal surfaces in Euclidean three-space with finite total curvature and embedded planar ends. The critical values of the Willmore functional are4πk4\pi k, wherek∈N∗k\in \mathbb {N}^*, withk≠2,3,5,7k\ne 2,3,5,7. When the ambient space is the four-sphereS4\mathbb {S}^4, the regular homotopy class of immersions of the two-sphereS2\mathbb {S}^2is determined by the self-intersection numberq∈Zq\in \mathbb {Z}; here we shall prove that the possible critical values are4π(|q|+k+1)4\pi (|q|+k+1), wherek∈Nk\in \mathbb {N}. Moreover, ifk=0k=0, the corresponding immersion, or its antipodal, is obtained, via the twistor Penrose fibrationP3→S4\mathbb {P}^3\rightarrow \mathbb {S}^4, from a rational curve inP3\mathbb {P}^3and, ifk≠0k\ne 0, via stereographic projection, from a minimal surface inR4\mathbb {R}^4with finite total curvature and embedded planar ends. An immersion lies in both families when the rational curve is contained in someP2⊂P3\mathbb {P}^2\subset \mathbb {P}^3or (equivalently) when the minimal surface ofR4\mathbb {R}^4is complex with respect to a suitable complex structure ofR4\mathbb {R}^4.

Developable surfaces in Euclidean space

AbstractThe classical notion of a two-dimensional develpable surface in Euclidean three-space is extended to the case of arbitrary dimension and codimension. A collection of characteristic properties is presented. The theorems are stated with the minimal possible integer smoothness. The main tool of the investigation is Cartan's moving frame method.

Constant Mean Curvature Surfaces with Two Ends in Hyperbolic Space

We invest igate the close relationship between minimal surfaces in Euclidean three-space and surfaces of constant mean curvature 1 in hyperbolic three-space. Just as in the case of minimal surfaces in Euclidean three-space, the only complete connected embedded surfaces of constant mean curvature 1 with two ends in hyperbolic space are well-understood surfaces of revolution: the catenoid cousins. In contrast to this, we show that, unlike the case of minimal surfacesin Euclidean three-space, there do exist complete connected immersed surfaces of constant mean curvature 1 with two ends in hyperbolic space that are not surfaces of revolution: the genus-one catenoid cousins. These surfaces are of interest because they show that, although minimal surfaces in Euclidean three-spaceand surfacesof constant mean curvature 1 in hyperbolic three-space are intimately related, there are essential differences between these two sets of surfaces. The proof we give of existence of the genus-one catenoid cousins is a mathematically rigorous verification that the results of a computer experiment are sufficiently accurate to imply existence.

Nonexistence results for extremals of curvature functionals

The following investigation deals with higher order variational integrals involving mean and Gaussian curvature of surfaces in Euclidean three-space. We prove that the few known examples of particular tori and cylinder surfaces are essentially the only possible solutions of the corresponding Euler equations in the classes of tubes and ruled surfaces.

Surfaces of constant mean curvature with boundary in a sphere

of the mean curvature. Themean curvature in a point is defined by the average of the two principal curvaturesin this point, so, the sphere of radius r > 0 has 1/r as mean curvature in anywhere ifthe Gauss map is chosen to point inside. Remember that a cmc surface is orientableand so, we may choose a globally defined Gauss map TV on Σ. A cmc surface inEuclidean three-space can be viewed as a surface where the exterior pressure andthe surface tension forces are balanced. For this reason they are thought of as

Affine classification of points of multidimensional surfaces

The classification of the points of a regular two-dimensional surface in Euclidean threespace is well known. One distinguishes three nondegenerate classes of points: hyperbolic points, at which the Gaussian curvature is negative; elliptic points, at which the Gaussian curvature is positive; and parabolic points, with Gaussian curvature zero; moreover, there is one degenerate class flat points, at which both principal curvatures vanish. In exactly the same way one can classify points on regular hypersurfaces in a Euclidean space: the class of a point is determined by the signs of the principal curvatures of the surface~

A Characteristic Property of the Sphere

Let $S$ be any connected piece of surface in Euclidean three-space, of class ${C^3}$ and ${g_{ij}}$, ${l_{ij}}$ be the coefficients of the first and second fundamental forms of $S$. If these coefficients satisfy the system of differential equations obtained by interchanging the ${g_{ij}}$ and ${l_{ij}}$ having same indices in the Mainardi-Codazzi equations, $S$ is part of a sphere. Furthermore, if two metrics on $S$ satisfy a similar condition, they are proportional.

Smooth approximation of polyhedral surfaces regarding curvatures

In this paper we prove that every closed polyhedral surface in Euclidean three-space can be approximated (uniformly with respect to the Hausdorff metric) by smooth surfaces of the same topological type such that not only the (Gaussian) curvature but also the absolute curvature and the absolute mean curvature converge in the measure sense. This gives a direct connection between the concepts of total absolute curvature for both smooth and polyhedral surfaces which have been worked out by several authors, particularly N. H. Kuiper and T. F. Banchoff.

A characteristic property of the sphere

Let S S be any connected piece of surface in Euclidean three-space, of class C 3 {C^3} and g i j {g_{ij}} , l i j {l_{ij}} be the coefficients of the first and second fundamental forms of S S . If these coefficients satisfy the system of differential equations obtained by interchanging the g i j {g_{ij}} and l i j {l_{ij}} having same indices in the Mainardi-Codazzi equations, S S is part of a sphere. Furthermore, if two metrics on S S satisfy a similar condition, they are proportional.

Zur inneren Kr�mmung der zweiten Grundform

LetKII denote the inner curvature of the second fundamental form of a surface in euclidean threespace and letH be the usual mean curvature. It is well known that the sphere is the only closed ovaloid satisfyingKII=H. In the present paper it is shown that the surfaces of revolution satisfyingKII=H are locally characterized by constancy of the ratio of the principal curvatures. Using previous results ofH. Hopf this implies that there exist closed analytic surfaces (which are different from the sphere) satisfying the equalityKII=H everywhere except at two flat points whereKII is not defined but limKII=0.

The November meeting in Los Angeles

The seven hundred eighteenth meeting of the American Mathematical Society was held at the University of Southern California in Los Angeles, California, on Saturday, November 23, 1974. There were 76 registrants including 70 members of the Society. By invitation of the Committee to Select Hour Speakers for Far Western Sectional Meetings, there were two invited hour addresses. Professor C. Edmund Burgess of the University of Utah lectured on Embeddings of surfaces in Euclidean three-space. He was introduced by Professor F. Burton Jones. Professor Paul R. Chernoff of the University of California, Berkeley, gave the second address. The title of his lecture was Quasi-analytic vectors and quasi-analytic functions. He was introduced by Professor Theodore W. Gamelin. There were two sessions of ten-minute contributed papers, chaired by Professors John Greever and Frank B. Miles. Professor William A. Harris, Jr. of the University of Southern California organized a special session of twenty-minute papers on Analytic Theory of Ordinary Differential Equations. The speakers were Ivar Bakken, Louis J. Grimm, William A. Harris, Jr., Frederick A. Howes, James S. Muldowney, Robert E. O'Malley, Jr., Yasutaka Sibuya, and Gilbert Stengle. KENNETH A. Ross

Embeddings of surfaces in euclidean three-space

Embeddings of surfaces in euclidean three-space