On the Stability of Certain Riemannian Functionals
Abstract
Given a compact smooth manifold Mn without boundary and n ≥ 3, the Lpnorm of the curvature tensor,
defines a Riemannian functional on the space of Riemannian metrics with unit volume M1. Consider C2,αtopology on M1 Rp remains invariant under the action of the group of diffeomorphisms D of M. So, Rp is defined on M1/ D. Our first result is that Rp restricted to the space M1/D has strict local minima at Riemannian metrics with constant sectional curvature for certain values of p. The product of spherical space forms and the product of compact hyperbolic manifolds are also critical point for Rp if they are product of same dimensional manifolds. We prove that these spaces are strict local minima for Rp restricted to M1/D. Compact locally symmetric isotropy irreducible metrics are critical points for Rp. We give a criteria for the local minima of Rp restricted to the conformal class of metrics of a given irreducible symmetric metric. We also prove that the metrics with constant bisectional curvature are strict local minima for Rp restricted to the space of Kahlar metrics with unite volume quotient by D.
Next we consider the Riemannian functional given by
In [GV], M. J. Gursky and J. A. Viaclovsky studied the local properties of the moduli space of critical metrics for the functional Ric2.We generalize their results for any p > 0.
Collections
 Mathematics (MA) [118]
Related items
Showing items related by title, author, creator and subject.

On an ODE Associated to the Ricci Flow
Bhattacharya, Atreyee (20180418)We discuss two topics in this talk. First we study compact Ricciflat four dimensional manifolds without boundary and obtain point wise restrictions on curvature( not involving global quantities such as volume and diameter) ... 
An Introduction to Minimal Surfaces
Ram Mohan, Devang S (20171210)In the first chapter of this report, our aim is to introduce harmonic maps between Riemann surfaces using the Energy integral of a map. Once we have the desired prerequisites, we move on to show how to continuously deform ... 
Compactness Theorems for The Spaces of Distance Measure Spaces and Riemann Surface Laminations
Divakaran, D (20180218)Gromov’s compactness theorem for metric spaces, a compactness theorem for the space of compact metric spaces equipped with the GromovHausdorﬀ distance, is a theorem with many applications. In this thesis, we give a ...