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In this study, we investigate constant angle ruled surfaces whose surface normals are parallel to the rulings, central normal vectors, central tangent vectors or unit Darboux vectors of slant ruled surfaces in the Euclidean 3-space. We show that such ruled surfaces are developable and unit director curve of such surfaces coincide with the rulings, Darboux vectors, central tangent vectors or central normal vectors of slant ruled surfaces. Furthermore, we calculate the position vectors of those ruled surfaces along with some examples. [ABSTRACT FROM AUTHOR]

In Palmer and Pámpano (Calc Var Partial Differ Equ 61:79, 2022), the authors studied a particular class of equilibrium solutions of the Helfrich energy which satisfy a second order condition called the reduced membrane equation. In this paper we develop and apply a second variation formula for the Helfrich energy for this class of surfaces. The reduced membrane equation also arises as the Euler–Lagrange equation for the area of surfaces under the action of gravity in the three dimensional hyperbolic space. We study the second variation of this functional for a particular example. [ABSTRACT FROM AUTHOR]

The normal curve is a space curve that plays an important role in the field of differential geometry. This research focuses on analyzing the properties of normal curves on smooth immersed surfaces, considering their invariance under isometric transformations. The primary contribution of this article is to explore the requirements for the image of a normal curve that preserves its invariance under isometric transformations. In this article, we investigate the invariant condition for the component of the position vector of the normal curves under isometry and compute the expression for the normal and geodesic curvature of such curves. Moreover, it has been investigated that the geodesic curvature and Christoffel symbols remain unchanged under the isometry of surfaces in R3{{\mathbb{R}}}^{3}.

The connections between parameter surfaces enable the development of various geometric designs. However, these surfaces are generally connected along their boundaries in a rectangular domain. This study investigated the methods for connecting surfaces along a curve. To this end, we introduced two-variable degenerate functions and utilized their algebraic properties to characterize the form of the parameter surfaces for practical surface construction. The results were used to deform the surfaces along the curve. For application, we presented the examples of deformations using Bézier surfaces and extended them to general surfaces. [ABSTRACT FROM AUTHOR]

This paper aims to present a systematic study on the Gauss images of complete minimal surfaces of genus 0 of finite total curvature in Euclidean 3-space and Euclidean 4-space. We focus on the number of omitted values and the total weight of the totally ramified values of their Gauss maps. In particular, we construct new complete minimal surfaces of finite total curvature whose Gauss maps have 2 omitted values and 1 totally ramified value of order 2, that is, the total weight of the totally ramified values of their Gauss maps are 5 / 2 (= 2.5) in Euclidean 3-space and Euclidean 4-space, respectively. Moreover we discuss several outstanding problems in this study. [ABSTRACT FROM AUTHOR]

Surfaces with concentric K-contours and parallel K-contours in Euclidean 3-space are defined. Crucial examples are presented and characterization of them are given. [ABSTRACT FROM AUTHOR]

The main goal of this paper is to show that helix surfaces and the Enneper surface are the only surfaces in the 3-dimensional Euclidean space R 3 whose isogonal lines are generalized helices and pseudo-geodesic lines. [ABSTRACT FROM AUTHOR]

On the two-sphere Σ , we consider the problem of minimising among suitable immersions f : Σ → R 3 the weighted L ∞ norm of the mean curvature H, with weighting given by a prescribed ambient function ξ , subject to a fixed surface area constraint. We show that, under a low-energy assumption which prevents topological issues from arising, solutions of this problem and also a more general set of "pseudo-minimiser" surfaces must satisfy a second-order PDE system obtained as the limit as p → ∞ of the Euler–Lagrange equations for the approximating L p problems. This system gives some information about the geometric behaviour of the surfaces, and in particular implies that their mean curvature takes on at most three values: H ∈ { ± ‖ ξ H ‖ L ∞ } away from the nodal set of the PDE system, and H = 0 on the nodal set (if it is non-empty). [ABSTRACT FROM AUTHOR]

We consider the problem of producing a ruled Möbius band by subjecting an unstretchable, homogeneous, isotropic, elastic material surface material surface in a circular helicoidal reference configuration to a deformation that is isometric and chirality preserving. We find that such a Möbius band is completely determined by the unit binormal of the Frenet frame of its midline, which must be a geodesic and must have uniform torsion inversely proportional to the pitch of the helicoidal reference configuration. Granted that the energy density of the material surface depends quadratically on the mean curvature of its deformed configuration, we show that the total energy stored in producing a ruled Möbius band as described reduces, in closed form and without approximation, to an integral over the midline of the Möbius band. We formulate and numerically solve a constrained variational problem for finding relative minima of the dimensionally reduced bending energy and construct corresponding stable Möbius bands. The only input parameter entering our variational problem is the number ν of turns in a helicoidal reference configuration. We only find solutions if ν exceeds a certain threshold, which we compute to machine precision. Above that threshold, an interplay between the operative constraints leads to a multiplicity of coexisting stable solutions with n ≥ 3 half twists. For each n ≥ 3 , we construct an energetically optimal Möbius band which exhibits n -fold rotational symmetry. All other energy minima yield Möbius bands which lack symmetry. To our knowledge, this study contains the first examples of stable Möbius bands produced by isometrically deforming reference configurations that are not flat. [ABSTRACT FROM AUTHOR]

The aim of this paper is to give an upper bound for the intrinsic diameter of a surface with boundary immersed in a conformally flat three dimensional Riemannian manifold in terms of the integral of the mean curvature and of the length of its boundary. Of particular interest is the application of the inequality to minimal surfaces in the three-sphere and in the hyperbolic space. Here the result implies an a priori estimate for connected solutions of Plateau's problem, as well as a necessary condition on the boundary data for the existence of such solutions. The proof follows a construction of Miura and uses a diameter bound for closed surfaces obtained by Topping and Wu–Zheng. [ABSTRACT FROM AUTHOR]

An osculating curve is a type of curve in space that holds significance in the study of differential geometry. In this article, we investigate certain geometric invariants of osculating curves on smooth and regularly immersed surfaces under conformal transformations in Euclidean space R3{{\mathbb{R}}}^{3}. The primary objective of this article is to explore conditions sufficient for the conformal invariance of the osculating curve under both conformal transformations and isometries. We also compute the tangential and normal components of the osculating curves, demonstrating that they remain invariant under the isometry of the surfaces in R3{{\mathbb{R}}}^{3}.

The Gauss map of a generic immersion of a smooth, oriented surface into R 4 is an immersion. But this map takes values on the Grassmanian of oriented 2-planes in R 4 . Since this manifold has a structure of a product of two spheres, the Gauss map has two components that take values on the sphere. We study the singularities of the components of the Gauss map and relate them to the geometric properties of the generic immersion. Moreover, we prove that the singularities are generically stable, and we connect them to the contact type of the surface and J -holomorphic curves with respect to an orthogonal complex structure J on R 4 . Finally, we get some formulas of Gauss–Bonnet type involving the geometry of the singularities of the components with the geometry and topology of the surface. [ABSTRACT FROM AUTHOR]

We prove that every 1-Lipschitz map from a closed metric surface onto a closed Riemannian surface that has the same area is an isometry. If we replace the target space with a non-smooth surface, then the statement is not true and we study the regularity properties of such a map under different geometric assumptions. Our proof relies on a coarea inequality for continuous Sobolev functions on metric surfaces that we establish, and which generalizes a recent result of Esmayli–Ikonen–Rajala. [ABSTRACT FROM AUTHOR]

It is well known that the total torsion of a closed spherical curve is zero. Furthermore, if the total torsion of any closed curve on the surface is zero, then it is part of a plane or a sphere. In this paper, we examine the total torsion of a spherical curve during infinitesimal bending. We find the appropriate bending fields and show that the variation of the total torsion of a closed spherical curve is equal to zero. Some examples are considered both analytically and using our own software tool. For figures, we use colors to represent the value of torsion at different points of the curve, together with a colour-value scale. [ABSTRACT FROM AUTHOR]

Conformable fractional calculus is a relatively new branch of mathematics that seeks to extend traditional calculus to include non-integer order derivatives and integrals. This new form of calculus allows for a more precise description of physical phenomena, such as those related to fractal geometry and chaos theory. It also offers new tools for solving problems in physics, engineering, finance, and other areas. By using conformable fractional calculus, researchers can gain insight into the behavior of systems that would otherwise be difficult or impossible to analyze. In this article, we will discuss the fundamentals of conformable fractional calculus and its applications in various fields. For this purpose ruled surfaces are studied with conformable fractional calculus. The ruled surface is rearranged concerning the conformable surface definition and geometric properties are investigated. [ABSTRACT FROM AUTHOR]

We consider a general class of non-homogeneous contracting flows of convex hypersurfaces in Rn+1 ${\mathbb{R}}^{n+1}$ , and prove the existence and regularity of the flow before extincting to a point in finite time.

We prove existence and nonexistence results for annular type parametric surfaces with prescribed, almost constant mean curvature, characterized as normal graphs of compact portions of unduloids or nodoids in R 3 , and whose boundary consists of two coaxial circles of the same radius. [ABSTRACT FROM AUTHOR]

Let F : M → C P 2 be an isometric immersion of a closed surface in the complex projective plane C P 2 . In this paper, we consider the functional W N (F) = ∫ M ( K ¯ ⊥ - K ⊥) d M , which is a global conformal invariant. The critical surfaces of W N (F) are called normal critical surfaces. We compute the first variation of W N (F) . Moreover, we build an index formula for the normal critical surfaces in the spirits of Webster (J Differ Geom 20:463–470, 1984.), Wolfson (J Diff Geom 29:281–294, 1989), Chen-Tian (Geom Funct Anal 7:873–916, 1997) and Han-Li (J Eur Math Soc 12:505–527, 2010). In particular, when the Kähler angle α satisfies cos α ⩽ - 5 3 or - 5 3 ⩽ cos α ⩽ 5 3 or cos α ⩾ 5 3 , M is the normal critical surface if and only if it is the minimal surface with constant Kähler angle α . [ABSTRACT FROM AUTHOR]

We study the space of Hopf differentials of almost fuchsian minimal immersions of compact Riemann surfaces. We show that the extrinsic curvature of the immersion at any given point is a concave function of the Hopf differential. As a consequence, we show that the set of all such Hopf differentials is a convex subset of the space of holomorphic quadratic differentials of the surface. In addition, we address the non-equivariant case, and obtain lower and upper bounds for the size of this set. [ABSTRACT FROM AUTHOR]

We use Clifford algebras to construct a unified formalism for studying constant extrinsic curvature immersed surfaces in Riemannian and semi-Riemannian 3-manifolds in terms of immersed bilegendrian surfaces in their unitary bundles. As an application, we provide full classifications of both complete and compact immersed bilegendrian surfaces in the unit tangent bundle U S 3 of the 3-sphere. [ABSTRACT FROM AUTHOR]

We study geometric properties of caustics of pseudo-spherical surfaces, that is, surfaces with constant negative Gaussian curvature -1 in the Euclidean 3-space ℝ 3 . We investigate the Gaussian and the mean curvature of caustics of pseudo-spherical surfaces. Moreover, a certain condition required for the caustics to be minimal surfaces is derived. [ABSTRACT FROM AUTHOR]

In this article, we investigate the relationships between the instantaneous invariants of a one-parameter spatial movement and the local invariants of the axodes. Specifically, we provide new proofs for the Euler-Savary and Disteli formulas using the E. Study map in spatial kinematics, showcasing its elegance and efficiency. In addition, we introduce two line congruences and thoroughly analyze their spatial equivalence. Our findings contribute to a deeper understanding of the interplay between spatial movements and axodes, with potential applications in fields such as robotics and mechanical engineering.

In the present research paper, we investigate the motion of surfaces in ℝ3 according to their curvatures. We study the motion of the torus of revolution via the normal velocity. We consider two cases: when the normal velocity is a function of both the time and the coordinates of the torus, and when it is a function of time only. We also study how the torus moves under different types of curvature flows, such as inverse mean curvature flow, inverse Gaussian curvature flow, and harmonic mean curvature flow. Moreover, we present some new applications of these flows.

This paper investigates the kinematic motions of space-like and time-like curves specified by acceleration fields in Minkowski space ℝ2,1. Through the motion, the relationship between the acceleration fields and velocity fields is determined. In this study, we focus on studying the flows of inextensible space-like curves with a space-like principal normal vector specified by a normal acceleration that equals the curvature of the curve. Through the motion of the inextensible space-like curve with normal acceleration, we prove that the position vector of the curve satisfies a one-dimensional wave equation. We present some novel applications and visualize the flows of curves and their curvatures.

We study quotients of quadratic forms and associated polar lines in the projective plane. Our results, applied pointwise to quadratic differential forms, shed some light on classical binary differential equations (BDEs) associated to congruences of lines in Euclidean 3-space and lead us to construct a new one. The new BDE yields a new singular surface in the Euclidean 3-space associated to the congruence of lines. We determine the generic local configurations of the new and the classical BDEs. [ABSTRACT FROM AUTHOR]

We determine all helical surfaces in three-dimensional Euclidean space which possess a constant ratio a : = κ 1 / κ 2 of principal curvatures (CRPC surfaces), thus providing the first explicit CRPC surfaces beyond the known rotational ones. Our approach is based on the involution of conjugate surface tangents and on well chosen generating profiles such that the characterizing differential equation is sufficiently simple to be solved explicitly. We analyze the resulting surfaces, their behavior at singularities that occur for a > 0 , and provide an overview of the possible shapes. [ABSTRACT FROM AUTHOR]

In this paper, we find a full description of concircular hypersurfaces in space forms as a special family of ruled hypersurfaces. We also characterize concircular helices in 3-dimensional space forms by means of a differential equation involving the concircular factor and their curvature and torsion, and we show that the concircular helices are precisely the geodesics of the concircular surfaces. [ABSTRACT FROM AUTHOR]

Suppose that the surfaces K 0 and K r are the boundaries of two convex, complete, connected C 2 bodies in R 3 . Assume further that the (Euclidean) distance between any point x in K r and K 0 is always r ( r > 0 ). For x in K r , let Π (x) denote the nearest point to x in K 0 . We show that the projection Π preserves geodesics in these surfaces if and only if both surfaces are concentric spheres or co-axial round cylinders. This is optimal in the sense that the main step to establish this result is false for C 1 , 1 surfaces. Finally, we give a non-trivial example of a geodesic preserving projection of two C 2 non-constant distance surfaces. The question whether for any C 2 convex surface S 0 , there is a surface S whose projection to S 0 preserves geodesics is open. [ABSTRACT FROM AUTHOR]

There are many works related to metrics and surfaces of positive and negative curvature. This paper is a survey of results related to locally Euclidean metrics and surfaces with such metrics. There are many problems included in the intersection of geometry, complex analysis, and differential equations that can become a source of new interesting research. [ABSTRACT FROM AUTHOR]

In this paper, we obtain a stochastic criterion for a minimal surface in terms of transition densities of stochastic processes generated by the fundamental forms of the surface. [ABSTRACT FROM AUTHOR]

In the literature various notions of nonlocal curvature can be found. Here we propose a notion of nonlocal curvature tensor. This we do by generalizing an appropriate representation of the classical curvature tensor and by exploiting some analogies with certain fractional differential operators. [ABSTRACT FROM AUTHOR]

Dilational materials, for which the angles between pairs of material fibers are preserved under deformations, are an important class of metamaterials. Although these materials are typically made by assembling discrete elemental building blocks in repeating patterns, continuum mechanics provides a powerful tool for exploring their macroscopic properties and response. We present an analysis of the constraint, the constitutive relation, and the equilibrium equations for homogeneous and isotropic dilational elastic material surfaces. We also describe the possibility of penalizing deviations from local area preservation to yield a framework for approximating isometric deformations of unstretchable elastic material surfaces. [ABSTRACT FROM AUTHOR]

In this paper, we prove that if u is a solution to the Liouville equation 0.1 Δ u + e 2 u = 0 in R 2 , then the diameter of R 2 under the conformal metric g = e 2 u δ is bounded below by π . Here δ is the Euclidean metric in R 2 . Moreover, we explicitly construct a family of solutions to (0.1) such that the corresponding diameters of R 2 range over [ π , 2 π) . We also discuss supersolutions to (0.1). We show that if u is a supersolution to (0.1) and ∫ R 2 e 2 u d x < ∞ , then the diameter of R 2 under the metric e 2 u δ is less than or equal to 2 π . For radial supersolutions to (0.1), we use both analytical and geometric approaches to prove some inequalities involving conformal lengths and areas of disks in R 2 . We also discuss the connection of the above results with the sphere covering inequality in the case of Gaussian curvature bounded below by 1. Higher dimensional generalizations are also discussed. [ABSTRACT FROM AUTHOR]

In this study, some basic concepts related to the surface are examined with the help of conformable fractional analysis. As known, the best thing that makes fractional analysis popular is that it gives numerically more approximate results compared to classical analysis. For this reason, the concepts that enable us to make calculations based on the measurement on the surface have been redefined to give more numerical results with conformable fractional analysis. In addition, with the help of fractional analysis, it is explained which concepts are changed or not. Finally, an example is given to better understand the obtained results and its graph is drawn with the help of Mathematica. The reason for using conformable fractional analysis in this study is that it is more suitable for the algebraic structure of differential geometry. [ABSTRACT FROM AUTHOR]

We compute three-term semiclassical asymptotic expansions of counting functions and Riesz-means of the eigenvalues of the Laplacian on spheres and hemispheres, for both Dirichlet and Neumann boundary conditions. Specifically for Riesz-means we prove upper and lower bounds involving asymptotically sharp shift terms, and we extend them to domains of S d . We also prove a Berezin–Li–Yau inequality for domains contained in the hemisphere S + 2 . [ABSTRACT FROM AUTHOR]

For smooth mappings of the unit disc into the oriented Grassmannian manifold G n , 2 , Hélein (Harmonic Maps Conservation Laws and Moving Frames, Cambridge University Press, Cambridge, 2002) conjectured the global existence of Coulomb frames with bounded conformal factor provided the integral of | A | 2 , the squared-length of the second fundamental form, is less than γ n = 8 π . It has since been shown that the optimal bounds that guarantee this result are: γ 3 = 8 π and γ n = 4 π for n ≥ 4 . For isothermal immersions in R 3 , this hypothesis is equivalent to saying the integral of the sum of the squares of the principal curvatures is less than γ 3 . The goal here is to prove that when n = 3 the same conclusion holds under weaker hypotheses. In particular, it holds for isothermal immersions when | A | 2 is integrable and the integral of | K | , where K is the Gauss curvature, is less than 4 π . Since 2 | K | ≤ | A | 2 this implies the known result for isothermal immersions, but | K | may be small when | A | 2 is large. The method, which is purely analytic, is then developed to examine the case n = 3 when | A | is only square-integrable. The possibility of extending that result in the language of Grassmannian manifolds to the case n > 3 is outlined in an Appendix. [ABSTRACT FROM AUTHOR]

Inspired by previous work of Kusner and Bauer–Kuwert, we prove a strict inequality between the Willmore energies of two surfaces and their connected sum in the context of isoperimetric constraints. Building on previous work by Keller, Mondino and Rivière, our strict inequality leads to existence of minimizers for the isoperimetric constrained Willmore problem in every genus, provided the minimal energy lies strictly below 8 ⁢ π . Besides the geometric interest, such a minimization problem has been studied in the literature as a simplified model in the theory of lipid bilayer cell membranes. [ABSTRACT FROM AUTHOR]

We classify simply-connected, orientable, complete Willmore surfaces with vanishing Gaussian curvature. We also study the Willmore cones in R 3 and give a classification. As an application, we show that for a complete Willmore embedding f : R 2 → R 3 , if its corresponding Gaussian curvature is nonnegative and the image of its Gauss map lies in { θ ∈ S 2 : d S 2 (θ , θ 0) ≤ α < π 2 } , f (R 2) is a plane. [ABSTRACT FROM AUTHOR]

Curves on surface are called osculating Darboux curve and rectifying Darboux curve if its position vectors always lies in the osculating Darboux plane and rectifying Darboux plane which are spanned by Darboux frame. We study the case that osculating Darboux curve and rectifying Darboux curve are geodesic, line of curvature or asymptotic curve. In this case, we determine each curve is related to rectifying curve, spherical curve and planer curve. Using this result, we also give the classification of a special developable surface under the condition of the existence of osculating Darboux curve as a geodesic or a line of curvature. As a result, in this case, we determine these developable surfaces become conical surfaces. [ABSTRACT FROM AUTHOR]

Cone-nets are conjugate nets on a surface such that along each individual curve of one family of parameter curves there is a cone in tangential contact with the surface. The corresponding conjugate curve network is projectively invariant and is characterized by the existence of particular transformations. We study properties of that transformation theory and illustrate how several known surface classes appear within our framework. We present cone-nets in the classical smooth setting of differential geometry as well as in the context of a consistent discretization with counterparts to all relevant statements and notions of the smooth setting. We direct special emphasis towards smooth and discrete tractrix surfaces which are characterized as principal cone-nets with constant geodesic curvature along one family of parameter curves. [ABSTRACT FROM AUTHOR]

In this paper, we first establish the existence of blow-up solutions with two antipodal points to the fourth order mean field equations on S 4 . Moreover, we construct non-axially symmetric solutions with blow-up points at the vertices of regular configurations, i.e., equilateral triangles on a great circle, regular tetrahedrons, cubes, octahedrons, icosahedrons and dodecahedrons. The bubbling rates of these blow-up solutions rely on various bubbling configurations. [ABSTRACT FROM AUTHOR]

Generalising classical result of Müller and Šverák (J. Differ. Geom. 42(2), 229-258, 1995), we obtain a pointwise estimate of the conformal factor of sequences of conformal immersions from the unit disk of the complex plane of uniformly bounded total curvature and converging strongly outside of a concentration point towards a branched immersions for which the quantization of energy holds. We show that the multiplicity associated to the conformal parameter becomes eventually constant to an integer equal to the order of the branch point of the limiting branched immersion. Furthermore, we deduce a C 0 convergence of the normal unit in the neck regions. Finally, we show that these improved energy quantizations hold for Willmore surfaces of uniformly bounded energy and precompact conformal class, and for Willmore spheres arising as solutions of min-max problems in the viscosity method. [ABSTRACT FROM AUTHOR]