arXiv research

Ricci flow preserves positive sectional curvature on homogeneous spheres

Paper shows $L^\infty$-positivity and stochastic completeness are equivalent.

This paper proposes grid cells encode position via a conformal isometric embedding of 2D physical space.

The paper proves positivity preservation and self-adjointness for Schrödinger operators on incomplete Riemannian manifolds.

The study preserves Ricci curvature positivity on homogeneous spaces under the normalized Ricci flow.

The paper proves a conjecture about positivity preserving in Riemannian manifolds.

Solves Lempert's question on Nakano semi-positivity preservation.

The paper studies Ricci flow on manifolds with boundary, proving existence, uniqueness, and boundary conditions preservation.

Preserves positive intermediate curvature on manifolds.

The purpose of this paper is to prove that the Hermitian Curvature Flow (HCF) on an Hermitian manifold $(M,g,J)$ preserves many natural curvature positivity conditions. Following Wilking, for an $Ad\,{GL(T^{1,0}M)}$-invariant subset $S\subset End(T^{1,0}M)$ and a ncie function $F\colon End(T^{1,0}M)\to\mathbb R$ we con…

Study Bismut-Griffiths-positivity in non-Kähler manifolds under Hermitian curvature flows.

Maps preserving mass and injective on boundary are isometries.

In this paper we consider a Ricci de Turck flow of spaces with isolated conical singularities, which preserves the conical structure along the flow. We establish that a given initial regularity of Ricci curvature is preserved along the flow. Moreover under additional assumptions, positivity of scalar curvature is prese…

It was proved by H. Chen earlier that the property of the sum of any two eigenvalues of the curvature operator is positive is preserved under the ricci flow in all dimensional. By a recent result of Phong-Sturm, a similar notion of positive 2-traceless bisectional curvature positive is preserved on complex surface. We …

Curvature flows in hyperbolic space preserve positive sectional curvature and contract to a point.

In this paper, we study any Kähler manifold where the positive orthogonal bisectional curvature is preserved on the Kähler Ricci flow. Naturally, we always assume that the first Chern class $C_1$ is positive. In particular, we prove that any irreducible Kähler manifold with such property must be biholomorphic to $\math…

Study preserves metrics with positive Bakry-Émry Ricci curvature via surgery.

Riemannian submersions can preserve positive intermediate Ricci curvature, but not necessarily.

We provide explicit examples which show that mean convexity (i.e. positivity of the mean curvature) and positivity of the scalar curvature are non-preserved curvature conditions for hypersurfaces of the Euclidean space evolving under either the volume- or the area preserving mean curvature flow. The relevance of our ex…

In this paper we further investigate the geometry of monads of order-preserving functionals and of positively homogeneous functionals. We prove that for any compactum X with $w(X) = τ$ the map $μ_F X$, where $F\in\{O,OH\}$, is homeomorphic to trivial $I^τ$-fibration if and only if $X$ is openly generated $χ$-homogeneou…

We begin a systematic study of a curvature condition (strongly positive curvature) which lies strictly between positive curvature operator and positive sectional curvature, and stems from the work of Thorpe in the 1970s. We prove that this condition is preserved under Riemannian submersions and Cheeger deformations, an…

Study on positivity properties of vector bundle Monge-Ampère equation.

Let (M,g) be a complete 3-dimensional asymptotically flat manifold with everywhere positive scalar curvature. We prove that, given a compact subset K of M, all volume preserving stable constant mean curvature surfaces of sufficiently large area will avoid K. This complements the work of G. Huisken and S.-T. Yau and J. …

Our aim in this note is to extend the semi discrete technique by combine it with the split step method. We apply our new method to the Ait-Sahalia model and propose an explicit and positivity preserving numerical scheme.

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

Study variations of Riemannian submersions to maintain geodesic fibers and positive curvatures.

The paper studies totally nonnegative parts of flag varieties and their topologies.

Proves rigidity of certain transformations on specific geometric manifolds.

If $π:M\rightarrow B$ is a Riemannian Submersion and $M$ has positive sectional curvature, O'Neill's Horizontal Curvature Equation shows that $B$ must also have positive curvature. We show there are Riemannian submersions from compact manifolds with positive Ricci curvature to manifolds that have small neighborhoods of…

Convex hypersurfaces in hyperbolic space evolve to geodesic spheres.

In this note we review some results regarding higher order elliptic differential operators on manifolds without boundary.

In this paper, we introduce a flow over the projective bundle $p:P(E^*)\to M$, which is a natural generalization of both Hermitian-Yang-Mills flow and Kähler-Ricci flow. We prove that the semipositivity of curvature of the hyperplane line bundle $\mathcal{O}_{P(E^*)}(1)$ is preserved along this flow under the null eige…

We consider the quermassintegral preserving flow of closed \emph{h-convex} hypersurfaces in hyperbolic space with the speed given by any positive power of a smooth symmetric, strictly increasing, and homogeneous of degree one function $f$ of the principal curvatures which is inverse concave and has dual $f_*$ approachi…

Nearly spherical, positively curved surfaces are mapped from a sphere.

The paper proves that under certain conditions, solutions to a specific differential inequality are nonnegative.

Survey on Ricci flow on spaces with conical singularities.

In this paper, we mainly study the mean curvature flow in Kähler surfaces with positive holomorphic sectional curvatures. We prove that if the ratio of the maximum and the minimum of the holomorphic sectional curvatures is less than 2, then there exists a positive constant $δ$ depending on the ratio such that $\cosα\ge…

Ricci flow can change metrics with positive curvature to those without.

The Ricci flow preserves positivity on Stiefel manifolds.

Ricci flow deforms metrics with positive curvature to include negative curvature.

Study shows limits of metrics with positive scalar curvature on spheres.

We show that the reduced sl(n) homology defined by Khovanov and Rozansky is invariant under component-preserving positive mutation when n is odd.

Given a 3-holed sphere decomposition of an orientable closed surface, it is shown that each orientation preserving homeomorphism of the surface is isotopic to a composition AB where A is a product of positive Dehn twists and B is a product of negative Dehn twists on the decomposition curves.

We study the interplay between sequential decision making and avoiding discrimination against protected groups, when examples arrive online and do not follow distributional assumptions. We consider the most basic extension of classical online learning: "Given a class of predictors that are individually non-discriminato…

We prove that manifolds with complicated enough fundamental group admit measure-preserving homeomorphisms which have positive stable fragmentation norm with respect to balls of bounded measure.

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…

Normalized Ricci flow preserves positivity of Ricci curvature on some generalized Wallach spaces.

We prove that Stein surfaces with boundary coincide up to orientation preserving diffeomorphisms with simple branched coverings of $\B^4$ whose branch set is a positive braided surface. As a consequence, we have that a smooth oriented 3-manifold is Stein fillable iff it has a positive open-book decomposition.