arXiv research

Square-like quadrilaterals inscribed in space curves proven for finite total curvature.

In this paper, we obtain a basic Chen's inequality for a C-totally real submanifold in a generalized $(κ,μ)$-contact space forms involving intrinsic invariants, namely the scalar curvature and the sectional curvatures of the submanifold on left hand side and the main extrinsic invariant, namely the squared mean curvatu…

We obtain a basic inequality involving the Laplacian of the warping function and the squared mean curvature of any warped product isometrically immersed in a Riemannian manifold without assuming any restriction on the Riemann curvature tensor of the ambient manifold. Applying this general theory, we obtain basic inequa…

This paper classifies all planar p-elasticae and their properties.

Proves curvature bounds for submanifolds in negatively curved spaces.

Let M be a compact pseudo-umbilical submanifold of the unit sphere S. In the present note, it is shown that if the normal curvature, scalar curvature S and square of the length of second fundamental form satisfy certain conditions, then M is totally geodesic.

Estimates manifold dimension using local graph structure.

On a closed 4-dimensional Riemannian manifold, we give a lower bound for the square of the first eigenvalue of the Yamabe operator in terms of the total Branson's Q-curvature. As a consequence, if the manifold is spin, we relate the first eigenvalue of the Dirac operator to the total Branson's Q-curvature. On a closed …

Researchers classify special curved spheres in a complex space.

The paper studies how submanifolds of a sphere evolve over time.

Study minimal surfaces in 4D, find specific tori with total curvature -8π.

Using the notion of vacuum pairs we show how the (square of the) mass matrix of the fermions can be considered geometrically as curvature. This curvature together with the curvature of space-time, defines the total curvature of the Clifford module bundle representing a ``free'' fermion within the geometrical setup of s…

In this paper we establish a general inequality involving the Laplacian of the warping functions and the squared mean curvature of any doubly warped product isometrically immersed in a Riemannian manifold. Moreover, we obtain some geometric inequalities for C-totally real doubly warped product submanifolds of generaliz…

We study curvature restrictions of Levi-flat real hypersurfaces in complex projective planes, whose existence is in question. We focus on its totally real Ricci curvature, the Ricci curvature of the real hypersurface in the direction of the Reeb vector field, and show that it cannot be greater than -4 along a Levi-flat…

In this paper we extend Efimov's Theorem by proving that any complete surface in $\mathbb{R}^3$ with Gauss curvature bounded above by a negative constant outside a compact set has finite total curvature, finite area and is properly immersed. Moreover, its ends must be asymptotic to half-lines. We also give a partial so…

Let $M$ be a compact Riemannian manifold not containing any totally geodesic surface. Our main result shows that then the area of any complete surface immersed into $M$ is bounded by a multiple of its extrinsic curvature energy, i.e. by a multiple of the integral of the squared norm of its second fundamental form.

Through the direct study of the analysis estimator we derive oracle inequalities with fast and slow rates by adapting the arguments involving projections by Dalalyan, Hebiri and Lederer (2017). We then extend the theory to the square root analysis estimator. Finally, we focus on (square root) total variation regularize…

Study on minimal hypersurfaces in a unit sphere, proving specific isometries.

The paper constructs foliations of minimal surfaces in negatively curved 3-manifolds.

In this paper, we prove conformal positive mass theorems for asymptotically flat manifolds with charge. We apply conformal relations to show that if the conformal sum of scalar curvature is not less than the norm square of electric field and electric density, the sum of the mass will not less than the modulus of total …

In this paper, we study locally projectively flat Finsler metrics with constant flag curvature ${\bf K}$. We prove those are totally determined by their behaviors at the origin by solving some nonlinear PDEs. The classifications when ${\bf K}=0$, ${\bf K}=-1$ and ${\bf K} =1$ are given respectively in an algebraic way.…

The paper finds surfaces closest to being flat that span a given contour.

A complete surface of constant mean curvature 1 (CMC-1) in hyperbolic 3-space with constant curvature -1 has two natural notions of "total curvature"-- one is the total absolute curvature which is the integral over the surface of the absolute value of the Gaussian curvature, and the other is the dual total absolute cur…

Study variations of metrics on Riemannian submersions to preserve fiber geometry.

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.

Minimal hypersurfaces in S^5 with specific curvature properties are totally geodesic.

We review recent results on classifying complete constant mean curvature 1 (CMC 1) surfaces in hyperbolic 3-space with low total curvature. There are two natural notions of "total curvature" -- one is the total absolute curvature, which is the integral over the surface of the absolute value of the Gaussian curvature, a…

Totally geodesic hypersurfaces in a sphere have small total curvature.

We survey our recent results on classifying complete constant mean curvature 1 (CMC-1) surfaces in hyperbolic 3-space with low total curvature. There are two natural notions of "total curvature"-- one is the total absolute curvature which is the integral over the surface of the absolute value of the Gaussian curvature,…

Estimates parameters of interconnected linear systems using total variation penalization.

Paper compares total quotient curvature and proves bounds for Einstein metric.

Paper solves degenerated circle pattern metric problem in spherical geometry.

In the present paper, we study hemi-slant submanifolds of a locally product Riemannian manifold. We prove that the anti-invariant distribution which is involved in the definition of hemi-slant submanifold is integrable and give some applications of this result. We get a necessary and sufficient condition for a proper h…

A new approach is proposed for study structure and properties of the total squared mean curvature $W$ of surfaces in ${\bf R}^3$. It is based on the generalized Weierstrass formulae for inducing surfaces. The quantity $W$ (Willmore functional) is shown to be invariant under the modified Novikov--Veselov hierarchy of in…

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

Extensions of the generalized Weierstrass representation to generic surfaces in 4D Euclidean and pseudo-Euclidean spaces are given. Geometric characteristics of surfaces are calculated. It is shown that integrable deformations of such induced surfaces are generated by the Davey -Stewartson hierarchy. Geometrically thes…

Ancient flows converge fast with finite curvature and convexity.

Study shows convergence for mean curvature flow on almost minimal totally real submanifolds.

In this paper, we consider the classical variational problem in the Galilean space. we develop the Euler-Lagrange equations for a elastic line on an oriented surface in the Galilean 3-dimensional space $G_3$. Using the varia- tion method, we will try to give some characterization for the solution curve (the elastic lin…

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

In this paper, we show that steady or shrinking complete gradient Yamabe solitons with finite total scalar curvature and non-positive Ricci curvature are Ricci flat. Moreover, under certain pinching condition for Ricci curvature, we show that steady or shrinking complete gradient Yamabe solitons with finite total scala…

Totally geodesic minimal hypersurfaces in $\mathbb H^5$ with specific curvature properties.

We introduce the notions of pointwise almost h-slant submanifolds and pointwise almost h-semi-slant submanifolds as a generalization of slant submanifolds, pointwise slant submanifolds, semi-slant submanifolds, and pointwise semi-slant submanifolds. We have characterizations and investigate the integrability of distrib…

Paper proves total curvature for convex hypersurfaces in equiaffine space.

The study preserves upper bounds of total scalar curvature in conformal classes.

Paper proves curvature estimate for curved spaces.

Total curvatures of certain hypersurfaces are continuous.

We prove a phenomenon of concentration of total curvature for stable minimal surfaces in the product space H^2xR; where H^2 is the hyperbolic plane. Under some geometric conditions on the asymptotic boundary of an oriented stable minimal surface immersed in H^2xR, it has infinite total curvature. In particular, we infe…