Itsricci (2024)

1. The Ricci Flow: An Introduction - AMS Bookstore

Intuitively, the idea is to set up a PDE that evolves a metric according to its Ricci curvature. The resulting equation has much in common with the heat ...
The Ricci Flow: An Introduction

Mar 24, 2021 · Research ArticleNurowski's Conformal Class of a Maximally Symmetric (2,3,5)-Distribution and its Ricci-flat RepresentativesMatthew ...
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... its Ricci tensorρhas the form [1]ρ=−12Lvg−λg,whereLdenotes the Lie derivative,vis a vector field andλis a constant. Afterits introduction a detailed study ...

Feb 1, 2022 · For the studied soliton, it is proved that its Ricci tensor is a constant multiple of the vertical component of both metrics. Thus, the ...
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Para-Ricci-like solitons with arbitrary potential on para-Sasaki-like Riemannian $Π$-manifolds are introduced and studied. For the studied soliton, it is proved that its Ricci tensor is a constant multiple of the vertical component of both metrics. Thus, the corresponding scalar curvatures of both considered metrics are equal and constant. An explicit example of the Lie group as the manifold under study is presented.

We also provide an integral representation of the Ricci flow metricitself and of its Ricci tensor in terms of the heat kernel of the conjugate linearized ...

Jun 27, 2023 · On the other hand, we prove that any M-parabolic end is indeed parabolic provided its Ricci curvature is uniformly bounded from below.
In this paper, we define natural capacities using a relative volume of graphs over manifolds, which can be characterized by solutions of bounded variation to Dirichlet problems of minimal hypersurface equation. Using the capacities, we introduce a notion '$M$-parabolicity' for ends of complete manifolds, where a parabolic end must be $M$-parabolic, but not vice versa in general. We study the boundary behavior of solutions associated with capacities in the measure sense, and the existence of minimal graphs over $M$-parabolic or $M$-nonparabolic manifolds outside compact sets. For a $M$-parabolic manifold $P$, we prove a half-space theorem for complete proper minimal hypersurfaces in $P\times\mathbb{R}$. As a corollary, we immediately have a slice theorem for smooth mean concave domains in $P\times\mathbb{R}^+$, where the $M$-parabolic condition is sharp by our example. On the other hand, we prove that any $M$-parabolic end is indeed parabolic provided its Ricci curvature is uniformly bounded from below. Compared to harmonic functions, we get the asymptotic estimates with sharp orders for minimal graphic functions on nonparabolic manifolds of nonnegative Ricci curvature outside compact sets.

Dec 10, 2020 · Nurowski's Conformal Class of a Maximally Symmetric (2,3,5)-Distribution and its Ricci-flat Representatives. Authors. Matthew Randall.
We show that the solutions to the second-order differential equation associated to the generalised Chazy equation with parameters k = 2 and k = 3 naturally show up in the conformal rescaling that takes a representative metric in Nurowski’s conformal class associated to a maximally symmetric (2,3,5)-distribution (described locally by a certain function...

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May 21, 2018 · ... properties of the underlying quantum geometry, its Ricci curvature can be matched well to that of a five-dimensional round sphere.
Quantum Ricci curvature has been introduced recently as a new, geometric observable characterizing the curvature properties of metric spaces, without the need for a smooth structure. Besides coordinate invariance, its key features are scalability, computability, and robustness. We demonstrate that these properties continue to hold in the context of nonperturbative quantum gravity, by evaluating the quantum Ricci curvature numerically in two-dimensional Euclidean quantum gravity, defined in terms of dynamical triangulations. Despite the well-known, highly nonclassical properties of the underlying quantum geometry, its Ricci curvature can be matched well to that of a five-dimensional round sphere.

Abstract. Given a three-dimensional Riemannian manifold containing a ball with an explicit lower bound on its Ricci curvature and positive lower bound on ...
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Oct 3, 2017 · Abstract. It is shown that a shrinking gradient Ricci soliton must be smoothly asymptotic to a cone if its Ricci curvature goes to zero at ...
Ovidiu Munteanu, Jiaping Wang

It is shown that a shrinking gradient Ricci soliton must be smoothly asymptotic to a cone if its Ricci curvature goes to zero at infinity. Original language ...
It is shown that a shrinking gradient Ricci soliton must be smoothly asymptotic to a cone if its Ricci curvature goes to zero at infinity.

Aug 10, 2021 · In Riemannian geometry, the Ricci flow is the analogue of heat diffusion; a deformation of the metric tensor driven by its Ricci curvature.
SciPost Submission Detail General Relativity and the Ricci Flow

... its Ricci curvature. Original language, English (US). Pages (from-to), 435-437. Number of pages, 3. Journal, Journal of Soviet Mathematics. Volume, 10. Issue ...
We give a simple proof of a somewhat stronger theorem of Yu. A. Aminov concerning a lower estimate of the diameter of a surface immersed in R, this estimate being a function of the mean curvature of the surface and of its Ricci curvature.