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We consider a mean curvature flow in a cone, that is, a hypersurface in a cone which moves toward the opening with normal velocity equaling to the mean curvature, and the contact angle between the hypersurface and the cone boundary being…
In this paper, we study the computation of curvatures at the singular points of algebraic curves and surfaces. The idea is to convert the problem to compute the curvatures of the corresponding regular parametric curves and surfaces, which…
We state that any constant curvature Riemannian metric with conical singularities of constant sign curvature on a compact (orientable) surface $S$ can be realized as a convex polyhedron in a Riemannian or Lorentzian) space-form. Moreover…
Topology and geometry are deeply intertwined in the study of surfaces, though their interaction manifests differently in smooth and discrete settings. In the smooth category, a classical result asserts that any closed smooth surface…
We use PDE methods as developed for the Liouville equation to study the existence of conformal metrics with prescribed singularities on surfaces with boundary, the boundary condition being constant geodesic curvature. Our first result shows…
This paper investigates conformal deformations of the scalar curvature and mean curvature on complete Riemannian manifolds with boundary. We establish sufficient conditions for the existence of conformal deformations to complete metrics…
The problem of defining correctly geometric objects such as the curvature is a hard one in discrete geometry. In 2009, Ollivier defined a notion of curvature applicable to a wide category of measured metric spaces, in particular to graphs.…
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…
The singular and regular type of a point on a real hypersurface $\mathcal H$ in $\mathbb C^n$ are shown to agree when the regular type is strictly less than 4. If $\mathcal H$ is pseudoconvex, we show they agree when the regular type is 4.…
We prove that finite area isolated singularities of surfaces with constant positive curvature in R^3 are removable singularities, branch points or immersed conical singularities. We describe the space of immersed conical singularities of…
We investigate orbit spaces of isometric actions on unit spheres and find a universal upper bound for the infimum of their curvatures.
The fine curve graph of a surface is the graph whose vertices are simple closed essential curves in the surface and whose edges connect disjoint curves. In this paper, we prove that the automorphism group of the fine curve graph of a…
Hilbert's fourth problem seeks the classification of metric geometries where straight lines are shortest paths. Its regular case identifies the projectively flat Finsler manifolds. This broader framework breaks the equivalence between…
We study the curvature-dimension inequality in regular graphs. We develop techniques for calculating the curvature of such graphs, and we give characterizations of classes of graphs with positive, zero, and negative curvature. Our main…
We study the geometry of surfaces in $\mathbb R^5$ by relating it to the geometry of regular and singular surfaces in $\mathbb R^4$ obtained by orthogonal projections. In particular, we obtain relations between asymptotic directions, which…
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.
A degree-regular triangulation is one in which each vertex has identical degree. Our main result is that any such triangulation of a (possibly non-compact) surface $S$ is geometric, that is, it is combinatorially equivalent to a geodesic…
The nonlinear dynamics of the free surface of an ideal conducting liquid in a strong external electric field is studied. It is establish that the equations of motion for such a liquid can be solved in the approximation in which the surface…
Flatness of a plate is a parameter has been put under consideration for long time. Factors influencing the accuracy of this parameter have been recognized and examined carefully but placed scatterringly. Beside that those reports have not…
Motivated by recent experiments, we consider theoretically the compression of droplets pinned at the bottom on a surface of finite area. We show that if the droplet is sufficiently compressed at the top by a surface, it will always develop…