Search results for "Scalar curvature"

Comparing the relative volume with a revolution manifold as a model

1993

Given a pair (P, M), whereM is ann-dimensional connected compact Riemannian manifold andP is a connected compact hypersurface ofM, the relative volume of (P, M) is the quotient volume(P)/volume(M). In this paper we give a comparison theorem for the relative volume of such a pair, with some bounds on the Ricci curvature ofM and the mean curvature ofP, with respect to that of a model pair\(\left( {\mathcal{P},\mathcal{M}} \right)\) where ℳ is a revolution manifold and\(\mathcal{P}\) a “parallel” of ℳ.

A comparison theorem for the mean exit time from a domain in a K�hler manifold

1992

Let M be a Kahler manifold with Ricci and antiholomorphic Ricci curvature bounded from below. Let ω be a domain in M with some bounds on the mean and JN-mean curvatures of its boundary ∂ω. The main result of this paper is a comparison theorem between the Mean Exit Time function defined on ω and the Mean Exit Time from a geodesic ball of the complex projective space ℂℙ n (λ) which involves a characterization of the geodesic balls among the domain ω. In order to achieve this, we prove a comparison theorem for the mean curvatures of hypersurfaces parallel to the boundary of ω, using the Index Lemma for Submanifolds.

Geometry and analysis of Dirichlet forms (II)

2014

Abstract Given a regular, strongly local Dirichlet form E , under assumption that the lower bound of the Ricci curvature of Bakry–Emery, the local doubling and local Poincare inequalities are satisfied, we obtain that: (i) the intrinsic differential and distance structures of E coincide; (ii) the Cheeger energy functional Ch d E is a quadratic norm. This shows that (ii) is necessary for the Riemannian Ricci curvature defined by Ambrosio–Gigli–Savare to be bounded from below. This together with some recent results of Ambrosio–Gigli–Savare yields that the heat flow gives a gradient flow of Boltzman–Shannon entropy under the above assumptions. We also obtain an improvement on Kuwada's duality …

On nonimmersibility of compact hypersurfaces into a ball of a simply connected space form

1996

We give a nonimmersibility theorem of a compact manifold with nonnegative scalar curvature bounded from above into a geodesic ball of a simply connected space form.

A Quantitative Analysis of Metrics on Rn with Almost Constant Positive Scalar Curvature, with Applications to Fast Diffusion Flows

2017

We prove a quantitative structure theorem for metrics on $\mathbf{R}^n$ that are conformal to the flat metric, have almost constant positive scalar curvature, and cannot concentrate more than one bubble. As an application of our result, we show a quantitative rate of convergence in relative entropy for a fast diffusion equation in $\mathbf{R}^n$ related to the Yamabe flow.

Comparison theorems for the volume of a geodesic ball with a product of space forms as a model

1995

We prove two comparison theorems for the volume of a geodesic ball in a Riemannian manifold taking as a model a geodesic ball in a product of two space forms.

Brane-world and loop cosmology from a gravity–matter coupling perspective

2015

We show that the effective brane-world and the loop quantum cosmology background expansion histories can be reproduced from a modified gravity perspective in terms of an $f(R)$ gravity action plus a $g(R)$ term non-minimally coupled with the matter Lagrangian. The reconstruction algorithm that we provide depends on a free function of the matter density that must be specified in each case and allows to obtain analytical solutions always. In the simplest cases, the function $f(R)$ is quadratic in the Ricci scalar, $R$, whereas $g(R)$ is linear. Our approach is compared with recent results in the literature. We show that working in the Palatini formalism there is no need to impose any constrai…

Investigating the Ultraviolet Properties of Gravity with a Wilsonian Renormalization Group Equation

2008

We review and extend in several directions recent results on the asymptotic safety approach to quantum gravity. The central issue in this approach is the search of a Fixed Point having suitable properties, and the tool that is used is a type of Wilsonian renormalization group equation. We begin by discussing various cutoff schemes, i.e. ways of implementing the Wilsonian cutoff procedure. We compare the beta functions of the gravitational couplings obtained with different schemes, studying first the contribution of matter fields and then the so-called Einstein-Hilbert truncation, where only the cosmological constant and Newton's constant are retained. In this context we make connection with…

Gravity waves from non-minimal quadratic inflation

2015

We discuss non-minimal quadratic inflation in supersymmetric (SUSY) and non-SUSY models which entails a linear coupling of the inflaton to gravity. Imposing a lower bound on the parameter cR, involved in the coupling between the inflaton and the Ricci scalar curvature, inflation can be attained even for subplanckian values of the inflaton while the corresponding effective theory respects the perturbative unitarity up to the Planck scale. Working in the non-SUSY context we also consider radiative corrections to the inflationary potential due to a possible coupling of the inflaton to bosons or fermions. We find ranges of the parameters, depending mildly on the renormalization scale, with adju…

2017

We show that the dynamics of the Higgs field during inflation is not affected by metric fluctuations if the Higgs is an energetically subdominant light spectator. For Standard Model parameters we find that couplings between Higgs and metric fluctuations are suppressed by $\mathcal{O}(10^{-7})$. They are negligible compared to both pure Higgs terms in the effective potential and the unavoidable non-minimal Higgs coupling to background scalar curvature. The question of the electroweak vacuum instability during high energy scale inflation can therefore be studied consistently using the Jordan frame action in a Friedmann--Lemaitre--Robertson--Walker metric, where the Higgs-curvature coupling en…