arXiv research

The study establishes conditions for orientability in spaces with lower Ricci curvature bounds.

The study explores positive scalar curvature metrics on non-orientable manifolds and their covers.

Study curvatures of diffeomorphisms on non-orientable surfaces.

Modeling curvature-sensitive cells in visual cortex using manifold geometry.

Consider an orientable compact surface in three dimensional Euclidean space with minimum total absolute curvature. If the Gaussian curvature changes sign to finite order and satisfies a nondegeneracy condition along closed asymptotic curves, we show that any other isometric surface differs by at most a Euclidean motion…

Let $M^{n+1}$ be an orientable compact Riemannian manifold with positive Ricci curvature. We prove that the Almgren-Pitts width of $M^{n+1}$ is achieved by an orientable index $1$ minimal hypersurface with multiplicity $1$ and optimal regularity. This extends to dimensions $n+1\geq 8$ the results of Ketover-Marques-Nev…

Study on minimal surfaces with closed curvature lines in 3D space.

The paper studies singularities in mean curvature flow and bounds on minimal surface total curvature.

Study on 4-manifolds with positive scalar curvature.

Study reveals flatness of Hessian metrics with non-negative Ricci curvature on foliation leaves.

Compactness results for hypersurfaces with mean curvature prescribed by an ambient function.

We study a positivity condition for the curvature of oriented Riemannian 4-manifolds: The half-$PIC$ condition. It is a slight weakening of the positive isotropic curvature ($PIC$) condition introduced by M. Micallef and J. Moore. We observe that the half-$PIC$ condition is preserved by the Ricci flow and satisfies a m…

Under mean radius of curvature flow, a closed convex surface in Euclidean space is known to expand exponentially to infinity. In the 3-dimensional case we prove that the oriented normals to the flowing surface converge to the oriented normals of a round sphere whose centre is determined by the initial surface. To prove…

We decompose the twisted index obstruction $θ(M)$ against positive scalar curvature metrics for oriented manifolds with spin universal cover into a pairing of a twisted $K$-homology with a twisted $K$-theory class and prove that $θ(M)$ does not vanish if $M$ is an orientable enlargeable manifold with spin universal cov…

The main result of this paper is: Given any constant C, there is $(ε,k,L)$ such that if a complete, orientable, noncompact odd-dimensional manifold with bounded positive sectional curvature contains a $(ε,k,L)$-neck, then the asymptotic scalar curvature ratio is bigger or equal to C. As a application we proved that the…

Applying the general theory about complete spacelike stationary (i.e. zero mean curvature) surfaces in 4-dimensional Lorentz space $\mathbb{R}^4_1$, we classify those regular algebraic ones with total Gaussian curvature $-\int K\mathrm{d}M=4π$. Such surfaces must be oriented and be congruent to either the generalized c…

We prove a general fusion theorem for complete orientable minimal surfaces in $\mathbb{R}^3$ with finite total curvature. As a consequence, complete orientable minimal surfaces of weak finite total curvature with exotic geometry are produced. More specifically, universal surfaces (i.e., surfaces from which all minimal …

In this paper we describe the topology of 4-dimensional closed orientable Riemannian manifolds with a uniform lower bound of sectional curvature and with a uniform upper bound of diameter which collapse to metric spaces of lower dimensions. This enables us to understand the set of homeomorphism classes of closed orient…

Estimates the first eigenvalue for embedded hypersurfaces in manifolds with Ricci curvature bounds.

Study pinched self-dual Weyl curvature in compact 4-manifolds.

Formal manifolds with non-negative Ricci curvature have formal covers.

In this paper, the general formulation for inextensible flows of curves on oriented surface in $\mathbb{R}^3$ is investigated. The necessary and sufficient conditions for inextensible curve flow lying an oriented surface are expressed as a partial differential equation involving the geodesic curvature and the geodesic…

New rigidity result for non-orientable manifolds with scalar curvature constraints.

Let Σbe a compact oriented surface immersed in a four dimensional Kähler-Einstein manifold M. We consider the evolution of Σin the direction of its mean curvature vector. It is proved that being symplectic is preserved along the flow and the flow does not develop type I singularity. When M has two parallel Kähler forms…

In this paper we prove that stable, compact without boundary, oriented, nonzero constant mean curvature surfaces in the de Sitter-Schwarzschild and Reissner-Nordstrom manifolds are the slices, provided its mean curvature satisfies some positive lower bound. More generally, we prove that stable, compact without boundary…

In this paper we will show the following result: Let $\mathcal{N}$ be a complete (noncompact) connected orientable Riemannian three-manifold with nonnegative scalar curvature $S \geq 0$ and bounded sectional curvature $K_{s} \leq K$. Suposse that $Σ\subset \mathcal{N}$ is a complete orientable connected area-minimi…

We introduce the notion of translational Riemannian manifolds and define a Gauss map for orientable immersed hypersurfaces lying in these ambients, an associated translational curvature and prove a Gauss-Bonnet theorem. We also use this Gauss map to prove that if $M^{n}$ is a compact, connected and oriented immersed hy…

Proves cobordism of CP^2 bundles generating oriented ring.

The paper examines curvature properties of twistor spaces.

The paper bounds the $L^2$-norm of Euler class for foliations on 3-manifolds.

Classifies 3-manifolds with uniformly positive scalar curvature.

The paper reduces normal curvature and enhances homology recovery via embedded submanifolds.

We introduce and study co-dimension one area-minimizing locally rectifiable currents $T$ with $C^{1,α}$ tangentially immersed boundary: $\partial T$ is locally a finite sum of orientable co-dimension two submanifolds which only intersect tangentially with equal orientation. We show that any such $T$ is supported in a s…

A Finsler space is said to be geodesically reversible if each oriented geodesic can be reparametrized as a geodesic with the reverse orientation. A reversible Finsler space is geodesically reversible, but the converse need not be true. In this note, building on recent work of LeBrun and Mason, it is shown that a geodes…

New manifolds with negative curvature limit to one with negative curvature.

In this paper we determine the topology of three-dimensional complete orientable Riemannian manifolds with a uniform lower bound of sectional curvature whose volume is sufficiently small.

Let $M$ be a closed oriented $4$-manifold admitting a rank-$2$ oriented foliation with a metric of leafwise positive scalar curvature. If $b^+>1$, then we will show that the Seiberg-Witten invariant vanishes for all \spinc structures.

We present conditions on the Ricci curvature for complete, oriented, minimal submanifolds of Euclidean space, as well as the standard unit sphere, when the Gauss maps are bounded embeddings.

The paper studies scalar curvature and harmonic forms on 3-manifolds with boundaries.

Proves Kähler-Einstein property for certain Einstein 4-manifolds.

In this paper, we derive the evolution equation for the first eigenvalue of the Witten-Laplace operator acting on the space of functions along the mean curvature flow on a closed oriented manifold. We show some interesting monotonic quantities under the mean curvature flow.

We give a curvature identity derived from the generalized Gauss-Bonnet formula for 4-dimensional compact oriented Riemannian manifolds. We prove that the curvature identity holds on any 4-dimensional Riemannian manifold which is not necessarily compact. We also provide some applications of the identity.

We prove that every closed, smooth $n$-manifold $X$ admits a Riemannian metric together with a smooth, transversely oriented CMC foliation if and only if its Euler characteristic is zero, where by CMC foliation we mean a codimension-one, transversely oriented foliation with leaves of constant mean curvature and where t…

The abstract finds conditions for creating curves of constant curvature.

We study $n$-dimensional area-minimizing currents $T$ in $\mathbb{R}^{n+1},$ with boundary $\partial T$ satisfying two properties: $\partial T$ is locally a finite sum of $(n-1)$-dimensional $C^{1,α}$ orientable submanifolds which only meet tangentially and with same orientation, for some $α\in (0,1]$; $\partial T$ has…

We show that a closed orientable Riemannian $n$-manifold, $n \ge 5$, with positive isotropic curvature and free fundamental group is homeomorphic to the connected sum of copies of $S^{n-1} \times S^1$.

The goal of this article is to study the pinching problem proposed by S.-T. Yau in 1990 replacing sectional curvature by one weaker condition on biorthogonal curvature. Moreover, we classify 4-dimensional compact oriented Riemannian manifolds with nonnegative biorthogonal curvature. In particular, we obtain a partial a…

The study preserves positive Ricci curvature on connected sums of fibre bundles.