Normalized Mean Curvature Research Articles

Codimension two mean curvature flow of entire graphs

AbstractWe consider the graphical mean curvature flow of maps , , and derive estimates on the growth rates of the evolved graphs, based on a new version of the maximum principle for properly immersed submanifolds that extends the well‐known maximum principle of Ecker and Huisken derived in their seminal paper [Ann. of Math. (2) 130:3(1989), 453–471]. In the case of uniformly area decreasing maps , , we use this maximum principle to show that the graphicality and the area decreasing property are preserved. Moreover, if the initial graph is asymptotically conical at infinity, we prove that the normalized mean curvature flow smoothly converges to a self‐expander.

Measurement of outer diameter parameters of manual grinding cable joints based on 3D point cloud processing

For the problems associated with low measurement accuracy caused by surface shrinkage, uneven grinding, and irregular shapes in manual grinding cable joints, as well as the limitations of existing machine vision measurement techniques for the outer diameter parameters, this paper proposes a non-contact outer diameter parameters measurement method of manual grinding cable joints based on three-dimensional (3D) point cloud processing. Firstly, statistical filtering is used to remove discrete noise, and coordinate transformation preprocessing is conducted to ordered the point cloud. Secondly, a bidirectional point cloud slicing method is proposed based on the cable joint structure and point cloud distribution to preserve the original features for effective measurement. Subsequently, the normalized mean curvature is utilized as the weight of point cloud coordinate averaging to construct key points for surface slices, which reduces the amount of data in the sliced point cloud and improves the matching efficiency. Finally, the key points are matched according to the position of the surface slices, and the distance of point pairs is calculated to complete the measurement of the outer diameter parameters. Experiments and error analysis are conducted on multiple manual grinding 110 kV and 500 kV real cable joints. The absolute error in the measurement of the outer diameter is less than 0.2 mm, with a relative error within 1 %. The absolute error of the XY-axis outer diameter deviation is less than 0.1 mm. The experiments demonstrate that the proposed method exhibits superior performance in cable joint outer diameter measurement.

On Curvature Pinching for Submanifolds with Parallel Normalized Mean Curvature Vector

In this note, we investigate the pinching problem for oriented compact submanifolds of dimension n with parallel normalized mean curvature vector in a space form Fn+p(c). We first prove a codimension reduction theorem for submanifolds under lower Ricci curvature bounds. Moreover, if the submanifolds have constant normalized scalar curvature R≥c, we obtain a classification theorem for submanifolds under lower Ricci curvature bounds. It should be emphasized that our Ricci pinching conditions are sharp for even n and p=2.

Biconservative Riemannian Submanifolds in Minkowski 5-Space

In this study, we focus biconservative Riemannian submanifolds with parallel normalized mean curvature vector field (PNMCV) in $\mathbb{E}^5_1$. We obtain explicit classifications for these submanifolds with exactly two distinct principal curvatures of the shape operator along the mean curvature vector field. In particular, we investigate these submanifolds which have time-like and space-like mean curvature vector field in $\mathbb{E}^5_1$.

Timelike surfaces with parallel normalized mean curvature vector field in the Minkowski 4-space

Timelike surfaces with parallel normalized mean curvature vector field in the Minkowski 4-space

Remarks on the normal hyperbolic mean curvature flow

In this paper, we investigate the various aspects of normal hyperbolic mean curvature flow by LeFloch and Smoczyk. It is remarkable that the equation admits the null condition in three-dimensional case and only satisfies the first null condition when [Formula: see text]. Based on the interesting findings, we can obtain the results of global existence of smooth solutions, as well as the stability of hyperplanes under this flow when [Formula: see text], which relates to the famous Bernstein theorem. Some explicit solutions for this flow have been also derived. It should be emphasized that the null structures of this hyperbolic mean curvature flow have not been found before.

Planar Pseudo-geodesics and Totally Umbilic Submanifolds

We study totally umbilic isometric immersions between Riemannian manifolds. First, we provide a novel characterization of the totally umbilic isometric immersions with parallel normalized mean curvature vector, i.e., those having nonzero mean curvature vector and such that the unit vector in the direction of the mean curvature vector is parallel in the normal bundle. Such characterization is based on a family of curves, called planar pseudo-geodesics, representing a natural extrinsic generalization of both geodesics and Riemannian circles: being planar, their Cartan development in the tangent space is planar in the ordinary sense; being pseudo-geodesics, their geodesic and normal curvatures satisfy a linear relation. We study these curves in detail and, in particular, establish their local existence and uniqueness. Moreover, in the case of codimension-one immersions, we prove the following statement: an isometric immersion ι:M↪Q\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\iota :M \\hookrightarrow Q$$\\end{document} is totally umbilic if and only if the extrinsic shape of every geodesic of M is planar. This extends a well-known result about surfaces in R3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {R}^{3}$$\\end{document}.

Biharmonic PNMCV submanifolds in Euclidean 5-space

In this article, we study 3-dimensional biconservative and biharmonic submanifolds of $\mathbb{E}^5$ with parallel normalized mean curvature vector (PNMCV). First, we prove that the principal curvartures and principal directions of biconservative PNMCV isometric immersions into $\mathbb{E}^5$ can be determined intrinsically. Then, we complete the proof of Chen's biharmonic conjecture for PNMCV submanifolds of $\mathbb{E}^5$.

Basic classes of timelike general rotational surfaces in the four-dimensional Minkowski space

In the present paper, we consider timelike general rotational surfaces in the Minkowski 4- space which are analogous to the general rotational surfaces in the Euclidean 4-space introduced by C. Moore. We study two types of such surfaces (with timelike and spacelike meridian curve, respectively) and describe analytically some of their basic geometric classes: flat timelike general rotational surfaces, timelike general rotational surfaces with flat normal connection, and timelike general rotational surfaces with non-zero constant mean curvature. We give explicitly all minimal timelike general rotational surfaces and all timelike general rotational surfaces with parallel normalized mean curvature vector field.

$L$-parabolic linear Weingarten spacelike submanifolds immersed in an Einstein manifold

In this paper, we deal with complete linear Weingarten spacelike submanifolds immersed with parallel normalized mean curvature vector and flat normal bundle in an Einstein manifold \smash{\mathcal E^{n+p}_p} of index p . Under some curvature constraints on the ambient space, we establish an L -parabolicity criterion related to a suitable modified Cheng–Yau's operator L and we apply it to obtain sufficient conditions which guarantee that such a spacelike submanifold must be an isoparametric hypersurface of \smash{\mathcal E^{n+1}_1} with two distinct principal curvatures one of which is simple.

Biconservative Submanifolds with Parallel Normalized Mean Curvature Vector Field in Euclidean Spaces

Biconservative Submanifolds with Parallel Normalized Mean Curvature Vector Field in Euclidean Spaces

On complete submanifolds with parallel normalized mean curvature in product spaces

A Simons type formula for submanifolds with parallel normalized mean curvature vector field (pnmc submanifolds) in the product spaces $M^{n}(c)\times \mathbb {R}$ , where $M^{n}(c)$ is a space form with constant sectional curvature $c\in \{-1,1\}$ , it is shown. As an application is obtained rigidity results for submanifolds with constant second mean curvature.

Classification of f-biharmonic submanifolds in Lorentz space forms

Abstract In this paper, f-biharmonic submanifolds with parallel normalized mean curvature vector field in Lorentz space forms are discussed. When f f is a constant, we prove that such submanifolds have parallel mean curvature vector field with the minimal polynomial of the shape operator of degree ≤ 2 \le 2 . When f f is a function, we completely classify such pseudo-umbilical submanifolds.

Harmonic Evolute Surface of Tubular Surfaces via B -Darboux Frame in Euclidean 3-Space

In this article, we look at a surface associated with real-valued functions. The surface is known as a harmonic surface, and its unit normal vector and mean curvature have been used to characterize it. We use the Bishop-Darboux frame ( B -Darboux frame) in Euclidean 3-space E 3 to study and explain the geometric characteristics of the harmonic evolute surfaces of tubular surfaces. The characterizations of the harmonic evolute surface’s ϱ and ς parameter curves are evaluated, and then, they are compared. Finally, an example of a tubular surface’s harmonic evolute surface is presented, along with visuals of these surfaces.

A convergent evolving finite element algorithm for Willmore flow of closed surfaces

A proof of convergence is given for a novel evolving surface finite element semi-discretization of Willmore flow of closed two-dimensional surfaces, and also of surface diffusion flow. The numerical method proposed and studied here discretizes fourth-order evolution equations for the normal vector and mean curvature, reformulated as a system of second-order equations, and uses these evolving geometric quantities in the velocity law interpolated to the finite element space. This numerical method admits a convergence analysis in the case of continuous finite elements of polynomial degree at least two. The error analysis combines stability estimates and consistency estimates to yield optimal-order \(H^1\)-norm error bounds for the computed surface position, velocity, normal vector and mean curvature. The stability analysis is based on the matrix–vector formulation of the finite element method and does not use geometric arguments. The geometry enters only into the consistency estimates. Numerical experiments illustrate and complement the theoretical results.

On the parabolicity of linear Weingarten spacelike submanifolds in the de Sitter space

In this paper, we deal with complete linear Weingarten spacelike sub-manifold immersed with parallel normalized mean curvature vector in de Sitter space of index p. Imposing appropriate restrictions on the values of the mean curvature function H, we establish a parabolicity criterion related to a suitable Cheng-Yau’s modified operator and we use it to obtain sufficient conditions which guarantee that such a spacelike submanifold must be either totally umbilical or isometric to certain hyperbolic cylinders of .

A Simons type integral inequality for closed submanifolds in the product space [formula omitted

A Simons type formula for submanifolds with parallel normalized mean curvature vector field (pnmc submanifolds) in the product spaces Sn×R, where Sn is the unit Euclidean sphere, is proved. As an application, an integral inequality for pnmc closed submanifolds in Sn×R is obtained. By this, it is shown that the sharpness in this inequality is attained, in the totally umbilical hypersurfaces, and in a certain family of standard tori of the form S1(1−r2)×Sn−1(r) with 0<r<1.

A sharp integral inequality for closed spacelike submanifolds immersed in the de Sitter space

In this paper, we establish a sharp integral inequality for n-dimensional closed spacelike submanifolds with constant scalar curvature immersed with parallel normalized mean curvature vector field in the de Sitter space $$\mathbb S_p^{n+p}$$ of index p, and we use it to characterize totally umbilical round spheres $$\mathbb S^n(r)$$ , with $$r>1$$ , of $$\mathbb S_1^{n+1}\hookrightarrow \mathbb S_p^{n+p}$$ . Our approach is based on a suitable lower estimate of the Cheng-Yau operator acting on the square norm of the traceless second fundamental form of such a spacelike submanifold.

Yoğunluklu Öklidyen 3-Uzayında Helis Yüzeyleri

The differential geometry of helix curves and helix hypersurfaces in different spaces has important application areas in many disciplines. Also, the notion of weighted manifold is become to be a very popular topic for scientists in recent years. In this context, after defining the notions of weighted mean curvature (or φ-mean curvature) and weighted Gaussian curvature (or φ-Gaussian curvature) of an n-dimensional hypersurface on manifolds with density, lots of studies have been done by differential geometers in different spaces with different densities. So, in the present study, firstly we give the normal vector field, mean curvature and Gaussian curvature of a helix surface in three dimensional Euclidean space and after that, we obtain the weighted mean curvature and weighted Gaussian curvature of a helix surface generated by a unit speed planar curve in three dimensional Euclidean space with different three densities by stating the parametric equation of this surface. However, we know that a hypersurface is weighted minimal and weighted flat in Eucilidean 3-space with density if the weighted mean curvature and the weighted Gaussian curvature vanish, respectively. So, by using these definitions, we obtain the weighted minimal helix surfaces for these different densities and give some results for weighted flatness of the helix surfaces in Euclidean 3-space. We hope that, this study will bring a new viewpoint to differential geometers who are dealing with constant angle surfaces and in near future, one can handle these surfaces in different spaces with another densities.

Complete linear Weingarten spacelike submanifolds immersed in the anti-de Sitter space

We deal with $n$-dimensional complete linear Weingarten spacelike submanifolds having nonnegative sectional curvature and immersed in the anti-de Sitter space $\mathbb H_p^{n+p}$ of index $p$ with parallel normalized mean curvature vector field. We recall