arXiv research

The paper proves conditions for compact Kähler manifolds to be projective or rationally connected.

The Ricci tensor (Ric) is fundamental to Einstein's geometric theory of gravitation. The 3-dimensional Ric of a spacelike surface vanishes at the moment of time symmetry for vacuum spacetimes. The 4-dimensional Ric is the Einstein tensor for such spacetimes. More recently the Ric was used by Hamilton to define a non-li…

In this paper we study the class of compact Kähler manifolds with positive orthogonal Ricci curvature: $Ric^\perp>0$. First we illustrate examples of Kähler manifolds with $Ric^\perp>0$ on Kähler C-spaces, and construct ones on certain projectivized vector bundles. These examples show the abundance of Kähler manifolds …

We establish metrics of positive $2^\mathrm{nd}$-intermediate Ricci curvature, i.e. $\mathrm{Ric}_2>0$, on products of positively curved homogeneous spaces. Using these examples, we demonstrate that the Hopf conjectures, Petersen-Wilhelm conjecture, Berger fixed point theorem, and Hsiang-Kleiner theorem for positively …

In this paper, we use a Killing form on a Riemannian manifold to construct a class of Finsler metrics. We find equations that characterize Einstein metrics among this class. In particular, we construct a family of Einstein metrics on $S^3$ with ${\rm Ric} = 2 F^2$, ${\rm Ric}=0$ and ${\rm Ric}=- 2 F^2$, respectively. T…

The paper describes a fine representation of the Ricci tensor and Hessian on RCD spaces.

The paper explores numerical characteristics of compact Riemannian manifolds and proves inequalities.

The study constructs metrics with positive 2nd Ricci curvature on various manifolds.

Let G be a three-dimensional unimodular Lie group, and let T be a left-invariant symmetric (0, 2)-tensor field on G. We provide the necessary and sufficient conditions on T for the existence of a pair (g, c) consisting of a left-invariant Riemannian metric g and a positive constant c such that Ric(g) = cT, where Ric(g)…

This article provides a new toolbox to derive sparse recovery guarantees from small deviations on extreme singular values or extreme eigenvalues obtained in Random Matrix Theory. This work is based on Restricted Isometry Constants (RICs) which are a pivotal notion in Compressed Sensing and High-Dimensional Statistics a…

Study shows Gromov's Betti number bound fails for certain intermediate Ricci curvatures.

In this paper, we investigate complete Riemannian manifolds satisfying the lower weighted Ricci curvature bound $\mathrm{Ric}_{N} \geq K$ with $K>0$ for the negative effective dimension $N<0$. We analyze two $1$-dimensional examples of constant curvature $\mathrm{Ric}_N \equiv K$ with finite and infinite total volumes.…

Let $(\M^n, g_{ij})$ be a complete Riemammnian manifold. For some constants $p,\ r>0$, define $\displaystyle k(p,r)=\sup_{x\in M}r^2\left(\oint_{B(x,r)}|Ric^-|^p dV\right)^{1/p}$, where $Ric^-$ denotes the negative part of the Ricci curvature tensor. We prove that for any $p>\frac{n}{2}$, when $k(p,1)$ is small enough,…

Let $P_t$ be the diffusion semigroup generated by $L:=Δ+\nabla V$ on a complete connected Riemannian manifold with $\operatorname {Ric}\ge-(σ^2ρ_o^2+c)$ for some constants $σ, c>0$ and $ρ_o$ the Riemannian distance to a fixed point. It is shown that $P_t$ is hypercontractive, or the log-Sobolev inequality holds for the…

Let (M.F) be a complete Finsler manifold and P be a minimal and compact submanifold of M. Ric_k(x), x in M is a differential invariant that interpolates between the flag curvature and the Ricci curvature. We prove that if on any geodesic c(t) emanating orthogonally from P we have \int_{0}^{\infty}\mathbf{Ric}_{k}(t)>0,…

The paper finds many manifolds with intermediate Ricci curvature for small k.

Fifty years ago, Eells and Sampson have proved a famous theorem in which they argued that any harmonic mapping $f:(M,g) \rightarrow (\bar{M},\bar{g})$ is totally geodesic if $(M, g)$ is a compact manifold with the nonnegative Ricci tensor and the section curvature of $(\bar{M},\bar{g})$ is nonpositive. Moreover, other …

First we confirm a conjecture asserting that any compact Kähler manifold $N$ with $\Ric^\perp>0$ must be simply-connected by applying a new viscosity consideration to Whitney's comass of $(p, 0)$-forms. Secondly we prove the projectivity and the rational connectedness of a Kähler manifold of complex dimension $n$ under…

In this paper is considered the differential equation Ric(g)=T, where Ric(g) is the Ricci tensor of the metric g and T is a rotational symmetric tensor on R^n. A new, geometric, proof of the existence of smooth solutions of this equation, based on qualitative theory of implicitdifferential equations, is presented here.…

The paper proves conditions under which critical point metrics are Einstein.

Paper generalizes Schwarz Lemma for VT harmonic maps with conditions.

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

Given an Einstein structure with positive scalar curvature on a four-dimensional Riemannian manifolds, that is $Ric=λg$ for some positive constant $λ$. For convenience, the Ricci curvature is always normalized to $Ric=1$. A basic problem is to classify four-dimensional Einstein manifolds with positive or nonnegative cu…

Paper proves Liouville theorems for harmonic functions under specific curvature bounds.

This is first of series papers on new two-side Gaussian bounds for the heat kernel $H(x,y,t)$ on a complete manifold $(M,g)$. In this paper, on a complete manifold $M$ with $Ric(M)\geq 0$, we obtain new two-side Gaussian bounds for the heat kernel $H(x,y,t)$, which improve the well-known Li-Yau's two-side bounds. As ap…

In this short paper we study $L_f^p$-Liouville property with $0<p<1$ for nonnegative $f$-subharmonic functions on a complete noncompact smooth metric measure space $(M,g,e^{-f}dv)$ with $\mathrm{Ric}_f^m$ bounded below for $0<m\leq\infty$. We prove a sharp $L_f^p$-Liouville theorem when $0<m<\infty$. We also prove an $…

In \cite{LiWang2001complete1,LiWang2001complete2}, Li-Wang proved a splitting theorem for an n-dimensional Riemannian manifold with $Ric\geqslant -(n-1)$ and the bottom of spectrum $λ_0(M)=\frac{(n-1)^2}{4}$. For an n-dimensional compact manifold $M$ with $Ric\geqslant-(n-1)$ with the volume entropy $h(M)=n-1$, Ledrapp…

Classifies 3-manifolds with Killing vector fields, extending results from Riemannian to Lorentzian.

The paper sets limits on the number of ends of certain geometric structures.

5D shrinking Ricci solitons with constant scalar curvature are rigid.

We extend the range of $N$ to negative values in the $(K,N)$-convexity (in the sense of Erbar--Kuwada--Sturm), the weighted Ricci curvature $Ric_N$ and the curvature-dimension condition $CD(K,N)$. We generalize a number of results in the case of $N>0$ to this setting, including Bochner's inequality, the Brunn--Minkowsk…

We show that a complete Riemannian manifold of dimension $n$ with $\Ric\geq n{-}1$ and its $n$-st eigenvalue close to $n$ is both Gromov-Hausdorff close and diffeomorphic to the standard sphere. This extends, in an optimal way, a result of P. Petersen. We also show that a manifold with $\Ric\geq n{-}1$ and volume close…

We use the framework used by Bakry and Emery in their work on logarithmic Sobolev inequalities to define a notion of coarse Ricci curvature on smooth metric measure spaces alternative to the notion proposed by Y. Ollivier. This function can be used to recover the Ricci tensor on smooth Riemannian manifolds by the formu…

The paper finds solutions for specific curvature conditions on 5D Lie groups.

The study classifies 4D manifolds with specific curvature properties.

Let $M$ be a compact Riemannian manifold and $h$ a smooth function on $M$. Let $ρ^h(x)=\inf_{|v|=1}\left(Ric_x(v,v)-2Hess(h)_x(v,v) \right)$. Here $Ric_x$ denotes the Ricci curvature at $x$ and $Hess(h)$ is the Hessian of $h$. Then $M$ has finite fundamental group if $Δ^h-ρ^h<0$. Here $Δ^h=: Δ+2L_{\nabla h}$ is the Bis…

New 5-manifold found with zero Ricci curvature.

Ricci flow can change metrics with intermediate curvatures.

In this paper we employ numerical methods to study the Einstein equation \[ Ric(g)=λ\, g, \] where $Ric$ is the Ricci tensor and $λ$ is the Einstein constant, restricted to a class of full flag manifolds. These metrics describe the gravitational field of a vacuum with cosmological constant (vacuum is the case $λ=0$). I…

We study a parabolic complex Monge-Ampère type equation of the form \eqref{MA} on a complete noncompact \K manifold. We prove a short time existence result and obtain basic estimates. Applying these results, we prove that under certain assumptions on a given real and closed (1,1) form $Ω$ and initial \K metric $g_0$ on…

This paper concerns the $L^2$ essential spectrum of the Laplacian $Δ$ and the drift Laplacian $Δ_f$ on complete Riemannian manifolds endowed with a weighted measure $e^{-f}d\;vol_g$. We prove that the essential spectrum of the drift Laplacian $Δ_f$ is $[0,+\infty)$ provided the Bakry-Émery curvature tensor $Ric_f$ is …

We prove Li-Yau type gradient bounds for the heat equation either on manifolds with fixed metric or under the Ricci flow. In the former case the curvature condition is $|Ric^-| \in L^p$ for some $p>n/2$, or $\sup_\M \int_\M |Ric^-|^2(y)d^{2-n}(x,y)dy<\infty$, where $n$ is the dimension of the manifold. In the later cas…

Unified approach to various energy conditions in spacetime geometry.

Stock return predictability is an important research theme as it reflects our economic and social organization, and significant efforts are made to explain the dynamism therein. Statistics of strong explanative power, called "factor" have been proposed to summarize the essence of predictive stock returns. Although mach…

In this paper, we show that along $\mathbb Q$-Fano fibration, when general fibres, base and central fiber (with at worst Kawamata log terminal singularities)are K-poly stable then there exists a relative Kähler-Einstein metric. We introduce the fiberwise Kähler-Einstein foliation and we mention that the main difficulty…

Study nonexistence and gradient estimates for solutions on manifolds with bounded Ricci curvature.

New proof shows 3-manifolds with Ricci curvature bound are either flat or grow non-Euclidean.

RaSE ensemble framework improves sparse classification accuracy.