Geometry, Volume 3, Issue 1

We obtain improved regularity estimates on solutions of partial differential equations by combining the method of Fuchsian Reduction with geometric transformations. Examples include the meron problem and the regularity of the conformal radius. In eac...

A focal circular cubic is a locus of the foci of conics tangent to a given quadrilateral. In this article, we derive some of the focal curve’s basic properties. We study the curve in the complex plane and prove that the complex coordinates of p...

Given an ellipse rolling along a straight line without slipping and a point P on the ellipse, we will determine the shape of the elliptic cycloid traced by P as the ellipse rolls and compute the area under one arch of the elliptic cycloid. We also in...

A long-standing conjecture of Lapidus asserts that under certain conditions a self-similar fractal set is not Minkowski measurable if and only if it is of lattice-type. For self-similar sets in $\mathbb{R}$, the Lapidus conjecture has been confirmed. However, i...

By using a Morse function and a Witten deformation argument, we obtain an upper bound for the dimensions of the space of divergence-free symmetric Killing p-tensors on a closed Riemannian manifold and explicitly calculate it for $p=2$.

Let $▵ABC$ be a triangle in the plane $\mathcal{E}$, K be its symmedian point, and $\mathcal{C}$ be its circumcircle. Assume that P is a point on $\mathcal{C}$ such that it is not A, B, or C, it does not lie on the medians of $▵ABC$, and the lines ${\ell }_{AP}$, ${\ell }_{BP}$, and $T_{}$...