scholarly journals On the lower bounds of the $L^2$-norm of the Hermitian scalar curvature

2020 ◽  
Vol 18 (2) ◽  
pp. 537-558
Author(s):  
Julien Keller ◽  
Mehdi Lejmi
Keyword(s):  
Scalar Curvature ◽  
Lower Bounds

scholarly journals Lower bounds on Ricci flow invariant curvatures and geometric applications

2015 ◽  
Vol 2015 (703) ◽  
Author(s):  
Thomas Richard
Keyword(s):  
Scalar Curvature ◽  
Ricci Curvature ◽  
Lower Bounds ◽  
Ricci Flow ◽  
Nonnegative Curvature ◽  
Curvature Operator ◽  
Nonnegative Ricci Curvature ◽  
Invariant Cones

AbstractWe consider Ricci flow invariant cones 𝒞 in the space of curvature operators lying between the cones “nonnegative Ricci curvature” and “nonnegative curvature operator”. Assuming some mild control on the scalar curvature of the Ricci flow, we show that if a solution to the Ricci flow has its curvature operator which satisfies

Lower Bounds on Statistical Submersions with vertical Casorati curvatures

2021 ◽  
Author(s):  
Aliya Naaz Siddiqui ◽  
Mohd Danish Siddiqi ◽  
Ali H. Alkhaldi ◽  
Akram Ali
Keyword(s):  
Scalar Curvature ◽  
Lower Bounds ◽  
Statistical Manifolds ◽  
Normalized Scalar Curvature ◽  
Statistical Submersion

In this paper, we obtain lower bounds for the normalized scalar curvature on statistical submersion with the normalized [Formula: see text]-vertical Casorati curvatures. Also, we discuss the conditions for which the equality cases hold. Beside this, we determine the statistical solitons on statistical submersion from statistical manifolds and illustrate an example of statistical submersions from statistical manifolds.

scholarly journals On Scalar and Ricci Curvatures

2021 ◽  
Author(s):  
Gerard Besson ◽  
◽  
Sylvestre Gallot ◽  
◽  
◽  
...  
Keyword(s):  
Scalar Curvature ◽  
Ricci Curvature ◽  
Lower Bounds ◽  
Riemannian Metric ◽  
Metric Measure Spaces ◽  
Smooth Metric Measure Spaces ◽  
Measure Spaces ◽  
Volume Entropy ◽  
Bounded Below

The purpose of this report is to acknowledge the influence of M. Gromov's vision of geometry on our own works. It is two-fold: in the first part we aim at describing some results, in dimension 3, around the question: which open 3-manifolds carry a complete Riemannian metric of positive or non negative scalar curvature? In the second part we look for weak forms of the notion of ''lower bounds of the Ricci curvature'' on non necessarily smooth metric measure spaces. We describe recent results some of which are already posted in [arXiv:1712.08386] where we proposed to use the volume entropy. We also attempt to give a new synthetic version of Ricci curvature bounded below using Bishop-Gromov's inequality.

scholarly journals SOME COMPACTNESS RESULTS RELATED TO SCALAR CURVATURE DEFORMATION

2007 ◽  
Vol 09 (01) ◽  
pp. 81-120 ◽  
Author(s):  
YU YAN
Keyword(s):  
Scalar Curvature ◽  
Lower Bounds ◽  
Blow Up ◽  
Upper And Lower Bounds ◽  
Conformally Flat ◽  
Conformally Flat Manifolds ◽  
Locally Conformally Flat ◽  
Curvature Deformation ◽  
Flat Manifolds ◽  
Curvature Problem

Motivated by the prescribing scalar curvature problem, we study the equation [Formula: see text] on locally conformally flat manifolds (M,g) with R(g) ≡ 0. We prove that when K satisfies certain conditions and the dimension of M is 3 or 4, any positive solution u of this equation with bounded energy has uniform upper and lower bounds. Similar techniques can also be applied to prove that on four-dimensional locally conformally flat scalar positive manifolds the solutions of [Formula: see text] can only have simple blow-up points.

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