arXiv research

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

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 …

Study of Randers metrics on spheres with simple cut loci.

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…

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…

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 …

An upper bound on the first S^1 invariant eigenvalue of the Laplacian for invariant metrics on the 2-sphere is used to find obstructions to the existence of isometric embeddings of such metrics in (R^3,can). As a corollary we prove: If the first four distinct eigenvalues have even multiplicities then the surface of rev…

We give a surface for which the Ricci Flow applied to the metric will increase the topological entropy of the geodesic flow. Specifically, we first adapt the Melnikov method to apply to a Ricci Flow perturbation and then we construct a surface which is closely related to a surface of revolution, but does not quite have…

A semi-isotropic space is a real affine 3-space endowed with the non-degenerate metric dx^{2}-dy^{2}. The main purpose of this paper is to describe the surfaces of revolution in the semi-isotropic space that satisfy some equations in terms of the position vector and the Laplace operators with respect to the first and t…

Sharp upper bounds found for Steklov eigenvalues of warped products.

In this paper, we study the geometry of the manifolds of geodesics of a Zoll surface of positive Gauss curvature, show how these metrics induce Finsler metrics of constant flag curvature and give some explicit constructions.

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…

We will construct surfaces of revolution with finite total curvature whose Gauss curvatures are not bounded. Such a surface of revolution is employed as a reference surface of comparison theorems in radial curvature geometry. Moreover, we will prove that a complete non-compact Riemannian manifold M is homeomorphic to t…

Sharp upper bounds found for Steklov eigenvalues of a specific hypersurface.

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 geometry on a slope of a mountain is the geometry of a Finsler metric, called here the {\it slope metric}. We study the existence of globally defined slope metrics on surfaces of revolution as well as the geodesic's behavior. A comparison between Finslerian and Riemannian areas of a bounded region is also studied.

The detailed analysis of the generalised Weierstrass representation of surfaces of revolution and their deformations induced by the modified Korteweg--de Vries (mKdV) equations is done. In particular, it is shown that these deformations preserve tori. The geometric meaning of the potential of surface is discussed and t…

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.

In the special case of S^1 invariant metrics on S^2, we find necessary and sufficient conditions for the existence of isometric embeddings into the canonical R^3, in other words: a Weyl type theorem with converse.

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 prove optimal systolic inequalities on Finsler Mobius bands relating the systole and the height of the Mobius band to its Holmes-Thompson volume. We also establish an optimal systolic in- equality for Finsler Klein bottles of revolution, which we conjecture to hold true for arbitrary Finsler metrics. Extremal metric…

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.

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

The normalized eigenvalues $Λ_i(M,g)$ of the Laplace-Beltrami operator can be considered as functionals on the space of all Riemannian metrics $g$ on a fixed surface $M$. In recent papers several explicit examples of extremal metrics were provided. These metrics are induced by minimal immersions of surfaces in $\mathbb…

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 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.

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 equip many non compact non simply connected surfaces with smooth Riemannian metrics whose isoperimetric profile is smooth, a highly non generic property. The computation of the profile is based on a calibration argument, a rearrangement argument, the Bol-Fiala curvature dependent inequality, together with new result…

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…