Related papers: Torus Curves With Vanishing Curvature

We give a full classification of complete rotationally invariant surfaces with constant Gauss curvature in Berger spheres: they are either Clifford tori, which are flat, or spheres of Gauss curvature $K \geq K_0$ for a positive constant…

A triangulation of a punctured or pinched surface is irreducible if no edge can be shrunk without producing multiple edges or changing the topological type of the surface. The finiteness of the set of (non-isomorphic) irreducible…

We establish the nonexistence of extremizers for a local Fourier restriction inequality on a certain class of planar convex curves whose curvature satisfies a natural assumption. We accomplish this by studying the local behavior of the…

We consider a family of embedded, mean convex hypersurfaces in a Riemannian manifold which evolve by the mean curvature flow. We show that, given any number $T>0$ and any $\delta>0$, we can find a constant $C_0$ with the following property:…

In this paper, we consider soliton solutions of the mean curvature flow in the unit sphere $S^{2n+1}$ moving along the integral curves of the Hopf unit vector field. While such solitons must necessarily be minimal if compact, we produce a…

In an ambient space with rotational symmetry around an axis (which include the Hyperbolic and Euclidean spaces), we study the evolution under the volume-preserving mean curvature flow of a revolution hypersurface M generated by a graph over…

In this paper, we show that, by applying some results on modular symbols, for a family of certain elliptic curves defined over $\mathbb Q$, there is a large class of explicit quadratic twists whose complex $L$-series does not vanish at…

We prove that the total curvature of a planar graph with nonnegative combinatorial curvature is at least $\frac{1}{12}$ if it is positive. Moreover, we classify the metric structures of ambient polygonal surfaces for planar graphs attaining…

We study complete non-compact manifolds of positive scalar curvature, with a focus on how curvature decay is constrained by topology at infinity. Our first main result shows that topological linking at infinity forces polynomial decay of…

The dynamics of torus vortex configurations $V_{n,p,q}$ in a superfluid liquid at zero temperature ($n$ is the number of quantum vortices, $p$ is the number of turns of each filament around the symmetry axis of the torus, and $q$ is the…

In this paper we prove that some well known K\"ahler surfaces with zero scalar curvature are QCH K\"ahler. We prove that family of generalized Taub-Nut K\"ahler surfaces parametrized by $k\in[-1,1]$ is of orthotoric type for $k\in(-1,1)$…

We consider surfaces in Euclidean space parametrized on an annular domain such that the first fundamental form and the principal curvatures are rotationally invariant, and the principal curvature directions only depend on the angle of…

There is no 5,7-triangulation of the torus, that is, no triangulation with exactly two exceptional vertices, of degree 5 and 7. Similarly, there is no 3,5-quadrangulation. The vertices of a 2,4-hexangulation of the torus cannot be…

Asymptotically flat, time-symmetric, axially symmetric and conformally flat initial data for vacuum general relativity are studied numerically on $R^3$ with the interior of a standard torus cut out. By the choice of boundary condition the…

We consider analytic maps and vector fields defined in $\mathbb{R}^2 \times \mathbb{T}^d$, having a $d$-dimensional invariant torus $\mathcal{T}$. The map (resp. vector field) restricted to $\mathcal{T}$ defines a rotation of frequency…

We introduce the notion of curvature parameters for singular plane curves with finite multiplicities and define the notion of curvatures for them. We then provide criteria to determine their singularity types for A-simple singularities. As…

We give a complete classification of homogeneous $(\alpha,\beta)$-metrics with positive flag curvature and vanishing S-curvature.

We study the relationship between free curves and periodic points for torus homeomorphisms in the homotopy class of the identity. By free curve we mean a homotopically nontrivial simple closed curve that is disjoint from its image. We prove…

Let $(X,g)$ be a compact $n$-dimensional smooth Riemannian manifold with a lower bound on the average of the lowest $n-p$ eigenvalues of the curvature operator and the diameter of $X$ is bounded above by $D>0$. In this article, we…

In this paper we show that a particular extrinsic pointwise hypersurface invariant is always non-positive on minimal hypersurfaces of constant curvature spaces and vanishes identically if and only if the hypersurface is rotational. We show…