arXiv research

For Legendre curves, we consider surfaces of revolution of frontals. The surface of revolution of a frontal can be considered as a framed base surface. We give the curvatures and basic invariants for surfaces of revolution by using the curvatures of Legendre curves. Moreover, we give properties of surfaces of revolutio…

In this paper we consider the conformal type (parabolicity or non-parabolicity) of complete ends of revolution immersed in simply connected space forms of constant sectional curvature. We show that any complete end of revolution in the $3$-dimensional Euclidean space or in the $3$-dimensional sphere is parabolic. In th…

Sharp upper bound found for Steklov spectrum on revolution submanifolds.

The study defines new surfaces with specific cut locus properties and provides conditions for their existence.

We prove here that when all planes transverse and nearly perpendicular to the axis of a surface of revolution intersect it in loops having central symmetry, the surface must be quadric. It follows that the quadrics are the only surfaces of revolution without skewloops. Similar statements hold for hypersurfaces of revol…

Upper bound found for Steklov eigenvalue of a surface of revolution.

Study examines noncompact cases of Gauss Curvature Flow on revolution surfaces.

Characterizes rotational solitons for curve shortening flow on revolution surfaces.

In this paper we study geodesic mappings of $n$-dimensional surfaces of revolution. From the general theory of geodesic mappings of equidistant spaces we specialize to surfaces of revolution and apply the obtained formulas to the case of rotational ellipsoids. We prove that such $n$-dimensional ellipsoids admit non tri…

Smooth maps preserve distances on specific revolution surfaces.

This paper establishes an interesting connection between the family of CMC surfaces of revolution in $\mathbb E_1^3$ and some specific families of elliptic curves. As a consequence of this connection, we show in the class of spacelike CMC surfaces of revolution in the $\mathbb E_1^3$, only spacelike cylinders and stand…

Fermat constants fail to fully identify Clairaut constants for certain geodesics on a surface of revolution.

New 2-spheres of revolution with simple cut locus structures.

Study of Ricci flow on discrete surfaces of revolution with constant Gaussian curvature.

We consider surfaces of revolution in the three-dimensional Euclidean space which are of coordinate finite type with respect to the third fundamental form. We show that a surface of revolution satisfying the preceding relation is a catenoid or part of a sphere.

The study finds conditions for certain surfaces to have a specific type of metric.

Proves inequality for special 3D shapes, generalizing to non-symmetric ones.

Using the theory of geodesics on surfaces of revolution, we introduce the period function. We use this as our main tool in showing that any two-dimensional orbifold of revolution homeomorphic to S^2 must contain an infinite number of geometrically distinct closed geodesics. Since any such orbifold of revolution can be …

In the previous paper, the structure of the cut locus was determined for a class of surfaces of revolution homeomorphic to a cylinder. In this paper, we prove the structure theorem of the cut locus for a wider class of surfaces of revolution homeomorphic to a cylinder.

We present numerical visualizations of Ricci Flow of surfaces and 3-dimensional manifolds of revolution. Ricci_rot is an educational tool which visualizes surfaces of revolution moving under Ricci flow. That these surfaces tend to remain embedded in R3 is what makes direct visualization possible. The numerical lessons …

We show that the surface energy introduced by Auckly and Sadun attains the minimum value at the Clifford torus among tori of revolution.

The study finds parametrizations for surfaces of revolution with a linear curvature ratio.

Study of tori of revolution under Willmore flow converges to Clifford Torus.

An extrinsic representation of a Ricci flow on a differentiable n-manifold M is a family of submanifolds S(t), each smoothly embedded in R^{n+k}, evolving as a function of time t such that the metrics induced on the submanifolds S(t) by the ambient Euclidean metric yield the Ricci flow on M. When does such a representa…

Economics does not need a scientific revolution. Economics needs accurate measurements according to high standards of natural sciences and meticulous work on revealing empirical relationships between measured variables.

Study properties of surfaces with nonvanishing third fundamental form.

We find sharp upper bounds for the multiplicities and the numerical values of all the distinct eigenvalues on a surface of revolution diffeomorphic to the sphere.

This paper classifies symmetries of biharmonic heat equations on surfaces of revolution.

We study the Gauss map $G$ of surfaces of revolution in the 3-dimensional Euclidean space ${\mathbb{E}^3}$ with respect to the so called Cheng-Yau operator $\square$ acting on the functions defined on the surfaces. As a result, we establish the classification theorem that the only surfaces of revolution with Gauss map …

Spinor fields on surfaces of revolution conformally immersed into 3-dimensional space are considered in the framework of the spinor representations of surfaces. It is shown that a linear problem (a 2-dimensional Dirac equation) related with a modified Veselov- Novikov hierarchy in the case of the surface of revolution …

We give an explicit formula for singular surfaces of revolution with prescribed unbounded mean curvature. Using it, we give conditions for singularities of that surfaces. Periodicity of that surface is also discussed.

In this paper, we will show that the limit of the Brown-York mass of a family of convex revolution surfaces in an asymptotically Schwarzschild manifold is the ADM mass.

In this work, we study spacelike and timelike surfaces of revolution in Minkowski space $\e_{1}^{3}$ that satisfy $aH+bK=c$, where $H$ and $K$ denote the mean curvature and the Gauss curvature of the surface and $a$, $b$ and $c$ are constants. The classification depends on the causal character of the axis of revolution…

We consider the Kepler problem on surfaces of revolution that are homeomorphic to $S^2$ and have constant Gaussian curvature. We show that the system is maximally superintegrable, finding constants of motion that generalize the Runge-Lentz vector. Then, using such first integrals, we determine the class of surfaces tha…

We prove that the systolic ratio of a sphere of revolution $S$ does not exceed $π$ and equals $π$ if and only if $S$ is Zoll. More generally, we consider the rotationally symmetric Finsler metrics on a sphere of revolution which are defined by shifting the tangent unit circles by a Killing vector field. We prove that i…

Dedicated to Professor K. Shiohama on the occasion of his seventieth birthday: This article is the third in a series of our investigation on a complete non-compact connected Riemannian manifold $M$. In the first series [arXiv:0901.4010], we showed that all Busemann functions on an $M$ which is not less curved than a vo…

We prove that among all constant width bodies of revolution, the minimum of the ratio of the volume to the cubed width is attained by the constant width body obtained by rotation of the Reuleaux triangle about an axis of symmetry.

The aim of this paper is to determine the structure of the cut locus for a class of surfaces of revolution homeomorphic to a cylinder. Let $M$ denote a cylinder of revolution which admits a reflective symmetry fixing a parallel called the equator of $M.$ It will be proved that the cut locus of a point $p$ of $M$ is a s…

We study the behavior of geodesics on a Randers surface of revolution. The main tool is the extension of Clairaut relation from Riemannian case to the Randers case. Moreover, we show that our Randers surface of revolution can be embedded in a Minkowski space as hypersurface.

Study of Randers metrics on spheres with simple cut loci.

For all positive integers n we construct a 1-parameter family of conformal tori of revolution in the 3-sphere with n bulges. These tori arise by Darboux transformations of constant mean curvature tori but do not have constant mean curvature in the 3-sphere.

In this work we give a new lower bound on the Morse index for constant mean curvature tori of revolution immersed in the three-sphere $\mathbb{S}^3$, by computing some explicit negative eigenvalues for the corresponding Jacobi operator.

Fix two parallel circles in $\mathbb{R}^3$ centered about a common axis. Among surfaces of revolution immersed in $\mathbb{R}^3$ whose boundary is given by these circles, there is one which maximizes the first Dirichlet eigenvalue. If the circles are sufficiently close together, then this surface is unique.

In an ambient space with rotational symmetry around an axis (which include the Hyperbolic and Euclidean spaces), we study the evolution under the volume-preserving mean curvature flow of a revolution hypersurface M generated by a graph over the axis of revolution and with boundary in two totally geodesic hypersurfaces …

In this paper, by the studying of the Gauss map, Laplacian operator, curvatures of surfaces in $\mathbb{R}_{1}^{3}$ and Bour's theorem, we are going to identify surfaces of revolution with pointwise 1-type Gauss map property in $3-$dimensional Minkowski space.

Any given surface of revolution embedded in Euclidean three-space can always be perturbed by arbitrarily small ambient isotopies as to admit highly nontrivial vector fields inducing infinitesimal deformations. For this matter Morse Theory is used, clarifying and giving a general response of a problem started with an id…

A 2D Riemannian space has only 2 injective geodesics.

This study classifies quadric surfaces in 3-sphere as Weingarten surfaces.