Narrow Results

In this paper, we consider translating solitons in ${ℝ}^{n+1}$ which is foliated by spheres. In three-dimensional Euclidean space, we show that such a translating soliton is a surface of revolution and the axis of revolution is parallel to the translating direction of the translating soliton. We also show that the same result holds for a higher dimension case with a hypersurface foliated by spheres in parallel hyperplanes that are perpendicular to the translating direction.

In this paper, we explore the zeta function arising from a small perturbation on a surface of revolution and the effect of this on the functional determinant and on the change of the Casimir energy associated with the surface.

A ruled surface of revolution with moving axes and angles is a rational tensor product surface generated from a line and a rational space curve by rotating the line (the directrix) around vectors and angles generated by the rational space curve (the director). Only right circular cylinders and right circular cones are ruled surfaces that are also surfaces of revolution, but we show that a rich collection of other ruled surfaces such as hyperboloids of one sheet, 2-fold Whitney umbrellas, and a wide variety other interesting ruled shapes are ruled surfaces of revolution with moving axes and angles. We present a fast way to compute the implicit equation of a ruled surface of revolution with moving axes and angles from two linearly independent vectors that are perpendicular to the directrix of the surface. We also provide an algorithm for determining whether or not a given rational ruled surface is a ruled surface of revolution with moving axes and angles.

In this paper, we show that the constant property of the Gaussian curvature of surfaces of revolution in both ℝ⁴ and depend only on the radius of rotation. We then give necessary and sufficient conditions for the Gaussian curvature of the general rotational surfaces whose meridians lie in two-dimensional planes in ℝ⁴ to be constant, and define the parametrization of the meridians when both the Gaussian curvature is constant and the rates of rotation are equal.