arXiv research

Study Kähler metrics on complex tori with almost non-negative scalar curvature.

Survey on rigidity and almost rigidity of Green functions in non-negative Ricci curvature spaces.

Study shows non-negative curvature on 4-manifolds with torus symmetry.

Paper connects fibre bundles to curvature and homotopy theory.

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

We find a local solution to the Ricci flow equation under a negative lower bound for many known curvature conditions. The flow exists for a uniform amount of time, during which the curvature stays bounded below by a controllable negative number. The curvature conditions we consider include 2-non-negative and weakly $\t…

We generalize most of the known Ricci flow invariant non-negative curvature conditions to less restrictive negative bounds that remain sufficiently controlled for a short time. As an illustration of the contents of the paper, we prove that metrics whose curvature operator has eigenvalues greater than $-1$ can be evolve…

Study almost rigidity of super Ricci flow with non-negative Muller quantity.

Study shows tori metrics converging to flat under specific conditions.

Constructs ε-splitting maps for geodesic balls with non-negative Ricci curvature.

In this paper we study the evolution of almost non-negatively curved (possibly singular) three dimensional metric spaces by Ricci flow. The non-negatively curved metric spaces which we consider arise as limits of smooth Riemannian manifolds (M_i,g_i), i \in N, whose Ricci curvature is not less than -c^2(i), where c^2(i…

In this paper, we derive a mean curvature estimate for eternal solutions (including translating solutions) of almost-calibrated Lagrangian mean curvature flow in complex Euclidean space. As a consequence, we show a non-existence result for eternal solutions of almost-calibrated Lagrangian mean curvature flow.

We provide a somewhat geometric proof of a rigidity theorem by M. Ledoux and C. Xia concerning complete manifolds with non-negative Ricci curvature supporting an Euclidean-type Sobolev inequality with (almost) best Sobolev constant. Using the same technique we also generalize Ledoux-Xia result to complete manifolds wit…

We extend the equivariant classification results of Escher and Searle for closed, simply connected, non-negatively curved Riemannian $n$-manifolds admitting isometric isotropy-maximal torus actions to the class of such manifolds admitting isometric strictly almost isotropy-maximal torus actions. In particular, we prove…

We study an odd-dimensional analogue of the Goldberg conjecture for compact Einstein almost Kähler manifolds. We give an explicit non-compact example of an Einstein almost cokähler manifold that is not cokähler. We prove that compact Einstein almost cokähler manifolds with non-negative $*$-scalar curvature are cokähler…

Study improves understanding of Ricci curvature in manifolds.

New proof linking scalar curvature to volume growth on 3-manifolds.

We show how to lift positive Ricci and almost non-negative curvatures from an orbit space $M/G$ to the corresponding $G$-manifold, $M$. We apply the results to get new examples of Riemannian manifolds that satisfy both curvature conditions simultaneously.

Proves effective linear volume growth for 3-manifolds with positive scalar curvature.

The study examines manifolds with specific curvature properties and finds topological and metric constraints.

We show that a closed almost Kähler 4-manifold of globally constant holomorphic sectional curvature $k\geq 0$ with respect to the canonical Hermitian connection is automatically Kähler. The same result holds for $k<0$ if we require in addition that the Ricci curvature is J-invariant. The proofs are based on the observa…

New bounds on scalar curvature for metric sequences.

As a means to better understanding manifolds with positive curvature, there has been much recent interest in the study of non-negatively curved manifolds which contain points at which all 2-planes have positive curvature. We show that there are generalisations of the well-known Eschenburg spaces and quotients of $\sph^…

We classify closed, simply-connected, non-negatively curved 6-manifolds of almost maximal symmetry rank up to equivariant diffeomorphism.

Developing a non-symmetric strainer theory for spaces with non-negative curvature beyond Alexandrov geometry.

New proof for curved 3-cohom manifold rational ellipticity.

For sequences of warped product metrics on a $3$-torus satisfying the scalar curvature bound $R_j \geq -\frac{1}{j}$, uniform upper volume and diameter bounds, and a uniform lower area bound on the smallest minimal surface, we find a subsequence which converges in both the Gromov-Hausdorff and the Sormani-Wenger Intrin…

The paper proves a gap theorem for almost non-negatively curved manifolds.

Let $\mathcal{M}_{0}^n$ be the class of closed, simply-connected, non-negatively curved Riemannian manifolds admitting an isometric, effective, isotropy-maximal torus action. We prove that if $M\in \mathcal{M}_{0}^n$, then $M$ is equivariantly diffeomorphic to the free linear quotient by a torus of a product of spheres…

We provide a topological procedure to obtain geometric realizations of both classical and `exotic' $G$-manifolds, such as spheres, bundles over spheres and Kervaire manifolds. As an application, we apply the process known as Cheeger deformations to produce new metrics of both positive Ricci and almost non-negative curv…

In this article we show that any finite cover of the moduli space of closed Riemann surfaces of $g$ genus with $g\geq 2$ does not admit any complete finite-volume Hermitian metric of non-negative scalar curvature. Moreover, we also show that the total mass of the scalar curvature of any almost Hermitian metric, which i…

Study of gauge-theoretic functionals leading to constant scalar curvature almost-Kahler 4-manifolds.

The paper constructs solutions to the Kähler-Ricci flow and studies complex structures.

The study examines symmetries in spaces with positive or non-negative curvature.

In this talk, I will discuss the use of harmonic functions to study the geometry and topology of complete manifolds. In my previous joint work with Luen-fai Tam, we discovered that the number of infinities of a complete manifold can be estimated by the dimension of a certain space of harmonic functions. Applying this t…

Graphs with non-negative Ollivier curvature have constant bounded harmonic functions.

Sharp inequality in spaces with non-negative Ricci curvature.

Odd GKM-manifolds with non-negative curvature split cohomology.

Study on rigidity with non-negative intermediate curvature on low-dimensional manifolds.

We prove that the renormalized volume of almost-Fuchsian hyperbolic $3$-manifolds is non-negative, with equality only for Fuchsian manifolds.

Optimal diameter estimates for 3D spaces with non-negative Ricci curvature.

The study extends curvature bounds to non-smooth spaces and proves stability of mean curvature.

Here we develop some basic analytic tools to study compactness properties of $J$-curves (i.e. pseudo-holomorphic curves) when regarded as submanifolds. Incorporating techniques from the theory of minimal surfaces, we derive an inhomogeneous mean curvature equation for such curves, we establish an extrinsic monotonicity…

Lower bound on minimum vertex degree for non-negative Lin-Lu-Yau curvature on graphs.

Sharp inequality for submanifolds in manifolds with non-negative Ricci curvature.

Study on Kähler manifolds with non-negative mixed curvature, proving splitting and structure theorems.

Our main goal in this work is to deal with results concern to the $σ_2$-curvature. First we find a symmetric 2-tensor canonically associated to the $σ_2$-curvature and we present an Almost Schur Type Lemma. Using this tensor we introduce the notion of $σ_2$-singular space and under a certain hypothesis we prove a rigid…

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