Surfaces In Euclidean Space Research Articles

A New Fractional Curvature of Curves and Surfaces in Euclidean Space Using the Caputo’s Fractional Derivative

In this paper, the authors generalize the fractional curvature of plane curves introduced by Rubio et al. in 2023, to regular curves in the Euclidean space R3, and study the geometric properties of the curve using Caputo’s fractional derivative. Furthermore, we introduce a new definition of fractional curvature and fractional mean curvature of a regular surface, using fractional principal curvatures; and prove that such concepts are invariant under isometries; i.e., they belong to the intrinsic geometry of the regular surface. Also, a geometric interpretation is given to Caputo’s fractional derivative of algebraic polynomials.

Umbilics of Surfaces in the Lorentz–Minkowski 3-Space

In this paper, we prove several fundamental properties on umbilics of a space-like or time-like surface in the Lorentz–Minkowski space {{mathbb {L}}}^3. In particular, we show that the local behavior of the curvature line flows of the germ of a space-like surface in {{mathbb {L}}}^3 is essentially the same as that of a surface in Euclidean space. As a consequence, for each positive integer m, there exists a germ of a space-like surface with an isolated C^{infty }-umbilic (resp. C^1-umbilic) of index (3-m)/2 (resp. 1+m/2). We also show that the indices of isolated umbilics of time-like surfaces in {{mathbb {L}}}^3 that are not the accumulation points of quasi-umbilics are always equal to zero. On the other hand, when quasi-umbilics accumulate, there exist countably many germs of time-like surfaces which admit an isolated umbilic with non-zero indices.

On the Index of Willmore spheres

We consider unbranched Willmore surfaces in Euclidean space that arise as inverted complete minimal surfaces with embedded planar ends. Several statements are proven about upper and lower bounds on the Morse Index—the number of linearly independent variational directions that locally decrease the Willmore energy. We in particular compute the Index of a Willmore sphere in the three‑space. This Index is $m-d$, where $m$ is the number of ends of the corresponding complete minimal surface and $d$ is the dimension of the span of the normals at the $m$-fold point. The dimension $d$ is either two or three. For $m=4$, we prove that $d=3$. In general, we show that there is a strong connection of the Morse Index to the number of logarithmically growing Jacobi fields on the corresponding minimal surface.

Surfaces of Osculating Circles in Euclidean Space

The aim of this paper is to define a new class of surfaces in Euclidean space using the concept of osculating circle. Given a regular curve C, the surface of osculating circles generated by C is the set of all osculating circles at all points of C. It is proved that these surfaces contain a one-parametric family of planar lines of curvature. A classification of surfaces of osculating circles is given in the family of canal surfaces, Weingarten surfaces, surfaces with constant Gauss curvature and surfaces with constant mean curvature.

Mean Curvature Flow in Null Hypersurfaces and the Detection of MOTS

We study the mean curvature flow in 3-dimensional null hypersurfaces. In a spacetime a hypersurface is called null, if its induced metric is degenerate. The speed of the mean curvature flow of spacelike surfaces in a null hypersurface is the projection of the codimension-two mean curvature vector onto the null hypersurface. We impose fairly mild conditions on the null hypersurface. Then for an outer un-trapped initial surface, a condition which resembles the mean-convexity of a surface in Euclidean space, we prove that the mean curvature flow exists for all times and converges smoothly to a marginally outer trapped surface (MOTS). As an application we obtain the existence of a global foliation of the past of an outermost MOTS, provided the null hypersurface admits an un-trapped foliation asymptotically.

Singular Minimal Translation Surfaces in Euclidean Spaces Endowed with Semi-symmetric Connections

In this paper, we study and classify singular minimal translation surfaces in a Euclidean space of dimension $3$ endowed with a certain semi-symmetric (non-)metric connection.

Holomorphic representation of minimal surfaces in simply isotropic space

It is known that minimal surfaces in Euclidean space can be represented in terms of holomorphic functions. For example, we have the well-known Weierstrass representation, where part of the holomorphic data is chosen to be the stereographic projection of the normal of the corresponding surface, and also the Bj\"orling representation, where it is prescribed a curve on the surface and the unit normal on this curve. In this work, we are interested in the holomorphic representation of minimal surfaces in simply isotropic space, a three-dimensional space equipped with a rank 2 metric of index zero. Since the isotropic metric is degenerate, a surface normal cannot be unequivocally defined based on metric properties only, which leads to distinct definitions of an isotropic normal. As a consequence, this may also lead to distinct forms of a Weierstrass and of a Bj\"orling representation. Here, we show how to represent simply isotropic minimal surfaces in accordance with the choice of an isotropic surface normal.

딥러닝 성능 향상을 위한 3D 의료 영상 증강법

This paper proposes a 3D image augmentation method for improving the generalization performance of deep neural networks. It allows us to enrich the diversity of training data samples that is essential in medical image segmentation tasks, thus reducing the data overfitting problem caused by the fact the scale of medical image dataset is typically smaller. It also enables us to predict medical segmentation surfaces in Euclidean space without additional labeled datasets. This method includes image transformation functions, which are comprised of a spatial deformation and image intensity change, enabling the synthesis of complex effects such as variations in anatomy and image acquisition procedures. Our numerical experiments demonstrate that the proposed approach provides significant improvements over state-of-the-art methods for 3D medical image segmentation.

General rotational ξ−surfaces in Euclidean spaces

General rotational ξ−surfaces in Euclidean spaces

Competency approach to the study of the second-order surface interface as the geometric location of points in space

The necessity of the formation of students' functional literacy as a competency approach to the training of future Mathematics teachers is substantiated on the example of studying of one of the topics of analytical geometry. It has been established that a prerequisite for the development of any competency prescribed in the standards of secondary education is the initial existence of a sufficiently new concept of functional literacy for a student of a certain level. The basic literacy comes down to the ability to read, write and express of one's thoughts correctly. Let us consider the issue of functional literacy from the point of view of the pedagogic specialty. Acquaintance with the well-known textbooks of analytic geometry allows us to say that 2nd order algebraic surfaces in Euclidean space are determined in most cases algebraically by means of equations. A constructive approach is also of use – surfaces are obtained by rotating 2nd degree curves around their symmetry axes and by deformation of the resulting surfaces by compression. The metric approach, as it used for 2nd order curves, is restricted only by the formulation of problems to find the certain locus of points in space. The exception is the article Dmitriy Perepyolkin which was published in 1936. In this paper the locus of points in space with the following characteristic property is studied – the ratio of the distance to a given point to the distance to a given straight line is constant. The strait line is assumed not to contain the point. The study is held out in pure geometrical manner – it is done using the method of sections and known loci of points on the surface. In the present article we study the locus of points in space defined by metric relation to a certain set of pairs of points, lines and planes. It is shown that any non-degenerate 2nd order surface can be considered as a certain locus of points of space and this interpretation is not unique.

Offset Ruled Surface in Euclidean Space with Density

Abstract In this paper, offset ruled surfaces in these spaces are defined by using the geometry of ruled surfaces in Euclidean space with density. The mean curvature and Gaussian curvature of these surfaces are studied. In addition, the relationships between the mean curvature and mean curvature with density, and the Gaussian curvature and the Gaussian curvature with density of the offset ruled surfaces in E 3 with density e z and e − x 2− y 2 are given.

Complete $$\lambda $$-surfaces in $${\mathbb {R}}^3$$

The purpose of this paper is to study complete \(\lambda \)-surfaces in Euclidean space \({\mathbb {R}}^3\). A complete classification for 2-dimensional complete \(\lambda \)-surfaces in Euclidean space \(\mathbb R^3\) with constant squared norm of the second fundamental form is given, which confirms a conjecture of Guang (Self-shrinkers and translating solitons of mean curvature flow, 2016, p 74).

Translation surfaces in Euclidean space with constant Gaussian curvature

We prove that the only surfaces in $3$-dimensional Euclidean space $\R^3$ with constant Gaussian curvature $K$ and constructed by the sum of two space curves are cylindrical surfaces, in particular, $K=0$.

A characteristic property of Delaunay surfaces

We prove that Delaunay surfaces, except the plane and the catenoid, are the only surfaces in Euclidean space with nonzero constant mean curvature that can be expressed as an implicit equation of type $f(x)+g(y)+h(z)=0$, where $f$, $g$ and $h$ are smooth real functions of one variable.

The Lorentzian Version of a Theorem of Krust

In Lorentz–Minkowski space, we prove that the conjugate surface of a maximal graph over a convex domain is also a graph. We provide three proofs of this result that show a suitable correspondence between maximal surfaces in Lorentz–Minkowski space and minimal surfaces in Euclidean space.

Semmes surfaces and intrinsic Lipschitz graphs in the Heisenberg group

A Semmes surface in the Heisenberg group is a closed set S S that is upper Ahlfors-regular with codimension one and satisfies the following condition, referred to as Condition B. Every ball B ( x , r ) B(x,r) with x ∈ S x \in S and 0 > r > diam ⁡ S 0 > r > \operatorname {diam} S contains two balls with radii comparable to r r which are contained in different connected components of the complement of S S . Analogous sets in Euclidean spaces were introduced by Semmes in the late 1980s. We prove that Semmes surfaces in the Heisenberg group are lower Ahlfors-regular with codimension one and have big pieces of intrinsic Lipschitz graphs. In particular, our result applies to the boundary of chord-arc domains and of reduced isoperimetric sets. The proof of the main result uses the concept of quantitative non-monotonicity developed by Cheeger, Kleiner, Naor, and Young. The approach also yields a new proof for the big pieces of Lipschitz graphs property of Semmes surfaces in Euclidean spaces.

Gluing Delaunay ends to minimal n-noids using the DPW method

we construct constant mean curvature surfaces in euclidean space by gluing n half Delaunay surfaces to a non-degenerate minimal n-noid, using the DPW method.

Osculating surfaces of curves on surfaces

Oosculating sphere have been studied in classical differential geometry [1]. In this article the osculating surfaces of higher order of space curves on surfaces in Euclidean space is considered. We study the intrinsic differential geometry of curves on surfaces by analyzing their contact with surfaces of higher order.

Intrinsic Differentiability and Intrinsic Regular Surfaces in Carnot Groups

A Carnot group $\mathbb {G}$ is a connected, simply connected, nilpotent Lie group with stratified Lie algebra. Intrinsic regular surfaces in Carnot groups play the same role as $\mathbb {C}^{1}$ surfaces in Euclidean spaces. As in Euclidean spaces, intrinsic regular surfaces can be locally defined in different ways: e.g. as non critical level sets or as continuously intrinsic differentiable graphs. The equivalence of these natural definitions is the problem that we are studying. Precisely our aim is to generalize the results on Ambrosio et al. (J. Geom. Anal., 16, 187–232, 2006) valid in Heisenberg groups to the more general setting of Carnot groups.

Classification of rotational surfaces in Euclidean space satisfying a linear relation between their principal curvatures

AbstractWe classify all rotational surfaces in Euclidean space whose principal curvatures κ1 and κ2 satisfy the linear relation , where a and b are two constants. As a consequence of this classification, we find closed (embedded and not embedded) surfaces and periodic (embedded and not embedded) surfaces with a geometric behaviour similar to Delaunay surfaces. Finally, we give a variational characterization of the generating curves of these surfaces.