arXiv research

The paper examines complete Yamabe solitons with finite total scalar curvature.

Proves strong finite total curvature hypersurfaces are proper and diffeomorphic to a compact manifold minus finitely many points.

Ancient flows converge fast with finite curvature and convexity.

Study minimal surfaces with finite curvature in a specific manifold.

An approximation theorem for minimal surfaces by complete minimal surfaces of finite total curvature in $\mathbb{R}^3$ is obtained. This Mergelyan type result can be extended to the family of complete minimal surfaces of weak finite total curvature, that is to say, having finite total curvature on proper regions of fin…

Ancient solutions to curve shortening with finite total curvature created by gluing Grim Reapers.

Study on Gauss images of specific minimal surfaces with finite curvature.

The study classifies flows of finite curvature in 3D space.

Finite totally geodesic hypersurfaces in curved manifolds proven.

The paper studies finite total $Q$-curvature on locally conformally flat manifolds.

Study on minimal submanifolds with finite curvature in Euclidean space.

In this paper we prove that a complete minimal surface immersed in H^2xR, with finite total curvature and two ends, each one asymptotic to a vertical geodesic plane, must be a horizontal catenoid. Moreover, we give a geometric description of minimal ends of finite total curvature in H^2xR. We also prove that a minimal …

The paper studies harmonic forms on submanifolds with finite total curvature.

Totally umbilic surfaces in hyperbolic 3-manifolds are constructed and characterized.

Characterizes minimal surfaces with finite curvature in specific spaces.

Complete minimal surfaces with finite curvature can be extended and hit optimally.

The study shows conditions for complete minimal surfaces to have finite total curvature.

Revisits conformal metrics with finite Q-curvature, providing necessary and sufficient conditions.

Characterizes metrics with finite total Q-curvature and introduces new volume entropy.

In this paper, we obtain the isoperimetric inequality on conformally flat manifold with finite total $Q$-curvature. This is a higher dimensional analogue of Li and Tam's result \cite{L-T} on surfaces with finite total Gaussian curvature. The main step in the proof is based on the construction of a quasiconformal map wh…

This paper improves Osserman's result on complete minimal surfaces with finite curvature.

Extends theory of minimal surfaces to elliptic special Weingarten surfaces.

Paper proves fractional Poincaré inequalities on manifolds with finite Q-curvature.

The plane and catenoid are the only capillary minimal surfaces outside a unit ball with one end and finite total curvature.

The paper proves properties of surfaces with finite curvature in 3D space.

Given a definable function f, enough differentiable, we study the continuity of the total curvature function t --> K(t), total curvature of the level {f=t}, and the total absolute curvature function t-->|K| (t), total absolute curvature of the level {f=t}. We show they admits at most finitely many discontinuities.

The paper classifies hypersurfaces in a specific product space with finite curvature.

Knot theory is the study of isotopy classes of embeddings of the circle $S^1$ into a 3-manifold, specifically $R^3$. The Fáry-Milnor Theorem says that any curve in $R^3$ of total curvature less than $4π$ is unknotted. More generally, a (finite) graph consists of a finite number of edges and vertices. Given a topologica…

We construct the first examples of complete, properly embedded minimal surfaces in $\mathbb{H}^2 \times \mathbb{R}$ with finite total curvature and positive genus. These are constructed by gluing copies of horizontal catenoids or other nondegenerate summands. We also establish that every horizontal catenoid is nondegen…

In this paper, we consider minimal hypersurfaces in the product space $\mathbb{H}^n \times \mathbb{R}$. We begin by studying examples of rotation hypersurfaces and hypersurfaces invariant under hyperbolic translations. We then consider minimal hypersurfaces with finite total curvature. This assumption implies that the …

Paper uses Gromov-Hausdorff convergence to re-examine surface classification.

Minimal surfaces with two ends have least total absolute curvature.

Properly immersed surfaces in hyperbolic 3-manifolds have bounded curvature and finite topology.

Study soap bubbles with almost constant higher-order mean curvature, proving unique limits under certain conditions.

Study non-integrated defect relations for complete minimal surfaces in higher dimensions.

Proves conditions for Willmore surfaces to have finite ends or finite total curvature.

Corrects a mistake in Ricci Flow's curve-shrinking flow equations.

We construct three kinds of complete embedded minimal surfaces in $\Bbb H^2\times \Bbb R$. The first is a simply connected, singly periodic, infinite total curvature surface. The second is an annular finite total curvature surface. These two are conjugate surfaces just as the helicoid and the catenoid are in $\mathbb R…

Laurent Hauswirth and Harold Rosenberg developed the theory of minimal surfaces with finite total curvature in $\H^2\times\R$. They showed that the total curvature of one such a surface must be a non-negative integer multiple of $-2π$. The first examples appearing in this context are vertical geodesic planes and Scherk…

Minimal hypersurfaces with specific properties are shown to be catenoids.

Embedded minimal surfaces of finite total curvature in $\mathbb{R}^3$ are reasonably well understood: From far away, they look like intersecting catenoids and planes, suitably desingularized. We consider the larger class of harmonic embeddings in $\mathbb{R}^{3}$ of compact Riemann surfaces with finitely many punctures…

We prove a global smooth isometric immersion for negatively curved surfaces with finite total curvature.

We consider surfaces with parallel mean curvature vector field and finite total curvature in product spaces of type $\mathbb{M}^n(c)\times\mathbb{R}$, where $\mathbb{M}^n(c)$ is a space form, and characterize certain of these surfaces. When $n=2$, our results are similar to those obtained in \cite{bds} for surfaces wit…

Study provides obstructions for Q-curvature on complete metrics in n-space.

We use the solution space of a pair of ODEs of at least second order to construct a smooth surface in Euclidean space. We describe when this surface is a proper embedding which is geodesically complete with finite total Gauss curvature. If the associated roots of the ODEs are real and distinct, we give a universal uppe…

Study weak Frenet frame for non-smooth curves with finite curvature and torsion.

We will construct surfaces of revolution with finite total curvature whose Gauss curvatures are not bounded. Such a surface of revolution is employed as a reference surface of comparison theorems in radial curvature geometry. Moreover, we will prove that a complete non-compact Riemannian manifold M is homeomorphic to t…

Timelike curves in Lorentzian length spaces have a total curvature notion that agrees with smooth curves.