arXiv research

Classifies surfaces with special curvature properties.

New relation between curvature bounds and spacetime inextendibility.

New relation between curvature bounds and spacetime inextendibility.

The study classifies polynomial relation tubular surfaces in 3-spaces.

Lectures on mean curvature flow and its related equations.

Study relates Gaussian curvature signs to cuspidal edge types and geometric invariants.

In this paper, we consider orthogonal Ricci curvature $Ric^{\perp}$ for Kähler manifolds, which is a curvature condition closely related to Ricci curvature and holomorphic sectional curvature. We prove comparison theorems and a vanishing theorem related to these curvature conditions, and construct various examples to i…

Paper establishes a relation between Berwald scalar curvature and S-curvature.

We define a Weyl-type curvature tensor of $(1,2)$-type to provide a characterization for Finsler metrics of constant flag curvature. This Weyl-type curvature tensor is projective invariant only to projective factors that are Hamel functions. Based on this aspect we construct new families of projectively related Finsler…

We explore the relation among volume, curvature and properness of a $m$-dimensional isometric immersion in a Riemannian manifold. We show that, when the $L^p$-norm of the mean curvature vector is bounded for some $m \leq p\leq \infty$, and the ambient manifold is a Riemannian manifold with bounded geometry, properness …

Study Clairaut maps on Kähler manifolds with Ricci solitons, finding curvature and scalar relations.

The study finds parametrizations for surfaces of revolution with a linear curvature ratio.

For singular corank 1 surfaces in $\mathbb R^3$ we introduce a distinguished normal vector called the axial vector. Using this vector and the curvature parabola we define a new type of curvature called the axial curvature, which generalizes the singular curvature for frontal type singularities. We then study contact pr…

For Einstein four-manifolds with positive scalar curvature, we derive relations among various positivity conditions on the curvature tensor, some of which are of great importance in the study of the Ricci flow. These relations suggest possible new ideas to study the well-known rigidity conjecture for positively curved …

Study on curvature measures in non-Euclidean spaces linked to Euclidean geometry.

The paper explores gaps in curvature-related metrics and rigidity.

This paper connects graph curvature to community structure.

Integrable nets described with curvature relations to pseudospherical surfaces.

Study of a flow related to the Orlicz-Minkowski problem for convex hypersurfaces.

The study connects surface geometry in 5D to 4D projections and umbilic curvatures.

Study connects isoperimetric sets, mass concepts, and nonnegative scalar curvature.

In this article, we propose some conditions on the modified defect relations of the Gauss map of a complete minimal surface $M$ to show that $M$ has finite total curvature.

We discuss notions of Gauss curvature and mean curvature for polyhedral surfaces. The discretizations are guided by the principle of preserving integral relations for curvatures, like the Gauss/Bonnet theorem and the mean-curvature force balance equation.

We present several problems and results relating the scalar curvatures of manifolds with mean curvatures of their boundaries

Sharp inequality linking interior and boundary Yamabe invariants on specific manifolds.

By establishing two general quadratic inequalities, we obtain some inequalities related to Ricci curvatures for Lagrangian submanifolds of K$\ddot{\mathrm{a}}$hler QCH-manifolds, which generalize some results for Lagrangian submanifolds of complex space forms.

We examine relations between geometry and the associated curvature decompositions in Weyl geometry.

Study finds multiple periodic solutions to ODEs related to curvature problems.

Detect spacetime curvature with event causality measurements.

Study on properties and transformations of Weingarten surfaces in 3D space.

We shall introduce the singular curvature function on cuspidal edges of surfaces, which is related to the Gauss-Bonnet formula and which characterizes the shape of cuspidal edges. Moreover, it is closely related to the behavior of the Gaussian curvature of a surface near cuspidal edges and swallowtails.

The study finds the minimum average area ratio on hyperbolic manifolds and its relation to scalar curvature.

In this paper, we prove the following two results: First, we study a class of conformally invariant operators $P$ and their related conformally invariant curvatures $Q$ on even-dimensional Riemannian manifolds. When the manifold is locally conformally flat(LCF) and compact without boundary, $Q$-curvature is naturally r…

Study Einstein warped-product manifolds with specific curvature conditions.

Inspired by a formula of Stern that relates scalar curvature to harmonic functions, we evaluate the mass of an asymptotically flat $3$-manifold along faces and edges of a large coordinate cube. In terms of the mean curvature and dihedral angle, the resulting mass formula relates to Gromov's scalar curvature comparison …

We define a Weyl-type curvature tensor that provides a characterisation for Finsler metrics of constant flag curvature. When the Finsler metric reduces to a Riemannian metric, the Weyl-type curvature tensor reduces to the classic projective Weyl tensor. In the general case, the Weyl-type curvature tensor differs from t…

We show that any space with a positive upper curvature bound has in a small neighborhood of any point a closely related metric with a negative upper curvature bound.

Defines a new tensor related to special geometric spaces.

At each point in an immersed surface in $\mathbb R^4$ there is a curvature ellipse in the normal plane which codifies all the local second order geometry of the surface. More recently, at the singular point of a corank 1 singular surface in $\mathbb R^3$, a curvature parabola in the normal plane which codifies all the …

The Riemann curvature tensor is a central mathematical tool in Einstein's theory of general relativity. Its related eigenproblem plays an important role in mathematics and physics. We extend M-eigenvalues for the elasticity tensor to the Riemann curvature tensor. The definition of M-eigenproblem of the Riemann curvatur…

Study on Einstein manifolds linking stability and rigidity.

The study explores surfaces with curvature satisfying a specific relation, leading to isometric immersions and topological obstructions.

In this paper we introduce the hyperbolic mean curvature flow and prove that the corresponding system of partial differential equations are strictly hyperbolic, and based on this, we show that this flow admits a unique short-time smooth solution and possesses the nonlinear stability defined on the Euclidean space with …

Since Li and Yau obtained the gradient estimate for the heat equation, related estimates have been extensively studied. With additional curvature assumptions, matrix estimates that generalize such estimates have been discovered for various time-dependent settings, including the heat equation on a Kähler manifold, Ricci…

The study examines connections and their curvatures on different types of bundles.

Study curvature of direct image bundles in deformations of maps.

Paper solves Orlicz-Aleksandrov problem using Gauss curvature flow.

We generalize the classical Bochner formula for the heat flow on M to martingales on the path space PM, and develop a formalism to compute evolution equations for martingales on path space. We see that our Bochner formula on PM is related to two sided bounds on Ricci curvature in much the same manner that the classical…