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We make a systematic investigation of quadrature properties for quadrics, namely integration of holomorphic functions over planar domains bounded by second degree curves. A full understanding requires extending traditional settings by…

A canal surface is the envelope of a moving sphere with varying radius, defined by the trajectory C(t) (spine curve) of its center and a radius function r(t). In this paper, we investigate when parameter curves of the canal surface are also…

We study surfaces with a constant ratio of principal curvatures in Euclidean and simply isotropic geometries and characterize rotational, channel, ruled, helical, and translational surfaces of this kind under some technical restrictions…

We study focal surfaces of (wave) fronts associated to unbounded principal curvatures near non-degenerate singular points of initial fronts. We give characterizations of singularities of those focal surfaces in terms of types of…

We study surfaces in Euclidean space constructed by the sum of two curves or that are graphs of the product of two functions. We consider the problem to determine all these surfaces with constant Gauss curvature. We extend the results to…

This paper investigates circle patterns with obtuse exterior intersection angles on surfaces of finite topological type. We characterise the images of the curvature maps and establish several equivalent conditions regarding long time…

In this paper we establish conditions on the length of the traceless part of the second fundamental form of a complete constant mean curvature hypersurface immersed in a space of constant sectional curvature in order to show that it is…

We classify the singular loci of real surfaces in three-space that contain two circles through each point. We characterize how a circle in such a surface meets this loci as it moves in its pencil and as such provide insight into the…

The main objective of this paper is to survey some recent results on the Chern--Moser question concerning existence of umbilical points on three dimensional CR submanifolds in $\mathbb C^2$.

We have studied the collective motion of polar active particles confined to ellipsoidal surfaces. The geometric constraints lead to the formation of vortices that encircle surface points of constant curvature (umbilics). We have found that…

PhD dissertation consists in three lines of investigation involving rational elliptic surfaces, namely 1) a study of conic bundles on these surfaces; 2) an investigation of the possible intersection numbers of two sections and 3) a theorem…

An involution on a surface induces involutions on the cohomology, the Chow group and the Brauer group of the surface. We give a detailed study of those actions. We show that the odd part of these groups can be used to describe the geometry…

We describe all local Riemannian metrics on surfaces whose geodesic flows are superintegrable with one integral linear in momenta and one integral cubic in momenta. We also show that some of these metrics can be extended to the 2-sphere.…

At any point of a surface in the four-dimensional Euclidean space we consider the geometric configuration consisting of two figures: the tangent indicatrix, which is a conic in the tangent plane, and the normal curvature ellipse. We show…

This work provides a curve-based approach to Ulrich bundles on surfaces, establishing a correspondence that characterizes their existence, with a focus on applications to surfaces in $\mathbb{P}^3$.

We find all analytic surfaces in space R^3 such that through each point of the surface one can draw two circular arcs fully contained in the surface. The proof uses a new decomposition technique for quaternionic matrices.

Here we study the shapes of droplets captured between chemically distinct parallel plates. This work is a preliminary step toward characterizing the influence of second-phase bridging between biomolecular surfaces on their solution…

We obtain the bifurcation of some special curves on generic 1-parameter families of surfaces in the Minkowski 3-space. The curves treated here are the locus of points where the induced pseudo metric is degenerate, the discriminant of the…

Optical surfaces represented by second-degree polynomials (quadratic or conics) are ubiquitous in optics. We revisit the equations of the conic shapes in the context of grazing incidence optics, gathering together the curves commonly used…

We consider the number of configurations of a surface in two dimensions that has a prescribed length and encloses a prescribed perimeter with respect to a baseline. An approximate analytical treatment in a semi--continuum compares…