arXiv research

The paper estimates curvature for a specific type of equations.

Constructs two types of Eguchi-Hanson metrics with negative scalar curvature.

Ricci flows with bounded scalar curvature cannot develop Type I singular points.

New examples of mixed-type zero-curvature graphs found.

Extends Ricci flow theory to Kato-type curvature bounds, proving manifold properties.

The study classifies timelike meridian surfaces in Minkowski 4-space.

We extend the definition of curvature homogeneity of type (1,3) to include the possibility that there is a homothety between any two points of a manifold preserving the first r covariant derivatives of the curvature operator simultaneously; we call this strong curvature homogeneity of type (1,3) up to order r. We chara…

The paper studies curvature tensors and hypersurfaces in Kenmotsu type manifolds.

In this paper, we show that if the mean curvature of a closed smooth embedded mean curvature flow in R^3 is of type-I, then the rescaled flow at the first finite singular time converges smoothly to a self-shrinker flow with multiplicity one. This result confirms Ilmanen's multiplicity-one conjecture under the assumptio…

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…

The projective curvature tensor $P$ is invariant under a geodesic preserving transformation on a semi-Riemannian manifold. It is well known that $P$ is not a generalized curvature tensor and hence it possesses different geometric properties than other generalized curvature tensors. The main object of the present paper …

Local singularity analysis for Ricci flows with applications to bounded scalar curvature.

A mixed type surface is a connected regular surface in a Lorentzian 3-manifold with non-empty spacelike and timelike point sets. The induced metric of a mixed type surface is a signature-changing metric, and their lightlike points may be regarded as singular points of such metrics. In this paper, we investigate the beh…

The paper proves uniformization for specific curvature types on manifolds.

The paper proves properties of Kähler surfaces with zero scalar curvature.

Paper constructs hyperbolic metrics using circle packings and curvature parameters.

Study rigidifies Einstein-type manifolds with boundary and constant curvature.

In this paper we study the Type IIb mean curvature flow. We first prove that if the convex entire graph $(y,u(|y|))$ over $\mathbb{R}^n$, $n\geq 2$, satisfying there exist positive constants $ε$, $c$ and $N$ such that $u'(r)\geq c r^ε$ for $r\geq N$, the longtime solution to mean curvature flow with initial data $(y,…

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…

Study precise asymptotics of noncompact Type-IIb solutions to mean curvature flow.

The paper proves finite topological type theorems for open manifolds with non-negative Ricci curvature and almost maximal local rewinding volume.

Study shows different behaviors of noncompact hypersurfaces under mean curvature flow.

Generalizes Ricci flow starting from small curvature concentration with a Morrey-type condition.

Willmore-type inequalities for bounded domains in manifolds with curvature bounds.

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…

Maximal diameter theorem for graphs with positive Ricci curvature.

In this paper, we obtain a Cartan type identity for curvature-adapted isoparametric hypersurfaces in symmetric spaces of compact type or non-compact type. This identity is a generalization of Cartan-D'Atri's identity for curvature-adapted(=amenable) isoparametric hypersurfaces in rank one symmetric spaces. Furthermore,…

Constructs Kähler metrics with constant scalar curvature on blown-up manifolds.

It is classically known that the only zero mean curvature entire graphs in the Euclidean 3-space are planes, by Bernstein's theorem. A surface in Lorentz-Minkowski 3-space $\boldsymbol{R}^3_1$ is called of mixed type if it changes causal type from space-like to time-like. In $\boldsymbol{R}^3_1$, Osamu Kobayashi found …

Solves Calabi-Yau equation on complex manifolds with non-positive astheno-Ricci curvature.

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

Study timelike meridian surfaces in Minkowski 4-space with specific properties.

The paper proves rigidity theorems for Type II singularities in Lagrangian flows.

Paper shows how solutions to Allen-Cahn converge to multiphase mean curvature flow.

New type of ruled surfaces studied with properties and examples.

The study explores metrics with constant curvature on compact manifolds.

Paper solves curvature equations in Minkowski space for non-convex domains.

A zero mean curvature surface in the Lorentz-Minkowski 3-space is said to be of Riemann-type if it is foliated by circles and at most countably many straight lines in parallel planes. We classify all zero mean curvature surfaces of Riemann-type according to their causal characters, and as a corollary, we prove that if …

The paper derives inequalities on Finsler manifolds, influenced by their curvatures.

We discuss a question by Felix, Oprea, and Tanre concerning nonnegative curvature and (rational) homotopy type.

The paper examines geometric properties of a unique spacetime model.

We obtain several rigidity results for biharmonic submanifolds in $\mathbb{S}^{n}$ with parallel normalized mean curvature vector field. We classify biharmonic submanifolds in $\mathbb{S}^{n}$ with parallel normalized mean curvature vector field and with at most two distinct principal curvatures. In particular, we dete…

The study analyzes weighted manifolds with curvature bounds, proving eigenvalue estimates and inequalities.

Constructs metrics with negative constant scalar curvature.

Develops Llarull type theorems for 3D and 4D bands with spectral scalar curvature bounds.

Ancient solutions of Ricci flow with Type I growth are classified.

In this article, we will use the harmonic mean curvature flow to prove a new class of Alexandrov-Fenchel type inequalities for strictly convex hypersurfaces in hyperbolic space in terms of total curvature, which is the integral of Gaussian curvature on the hypersurface. We will also use the harmonic mean curvature flow…

Study classifies zero mean curvature surfaces with planar curvature lines.