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Second Variation of F-Einstein-Hilbert Functional

Ahmed Mohammed Cherif

TL;DR

This work develops the second variation theory for the generalized Einstein-Hilbert functional $\mathcal{E}_{F}$ on closed manifolds, extending stable Einstein metrics to the $F$-dependent setting. It computes the first variation to obtain the divergence-free generalized Einstein tensor $E_F(g)$ and then derives the second variation under a fixed-volume constraint, yielding explicit operators $T_0(h)$ and $T_1(h)$ and the induced $F$-Einstein operator $\Delta_E^F$. The results provide a criterion for (strict) stability of $F$-Einstein metrics via transverse-traceless tensors and showcase reductions to the classical case $F(s)=s$, along with curvature-based stability conditions. These findings broaden the toolkit for rigidity, extremality, and potential dynamics of Riemannian metrics in $f(R)$-type gravity settings and related geometric analysis.

Abstract

This article describes a formula for second variation of generalized Einstein-Hilbert functional on Riemannian manifolds. This work extends the definition of stable Einstein manifolds, and we present some properties.

Second Variation of F-Einstein-Hilbert Functional

TL;DR

This work develops the second variation theory for the generalized Einstein-Hilbert functional on closed manifolds, extending stable Einstein metrics to the -dependent setting. It computes the first variation to obtain the divergence-free generalized Einstein tensor and then derives the second variation under a fixed-volume constraint, yielding explicit operators and and the induced -Einstein operator . The results provide a criterion for (strict) stability of -Einstein metrics via transverse-traceless tensors and showcase reductions to the classical case , along with curvature-based stability conditions. These findings broaden the toolkit for rigidity, extremality, and potential dynamics of Riemannian metrics in -type gravity settings and related geometric analysis.

Abstract

This article describes a formula for second variation of generalized Einstein-Hilbert functional on Riemannian manifolds. This work extends the definition of stable Einstein manifolds, and we present some properties.

Paper Structure

This paper contains 3 sections, 10 theorems, 88 equations.

Table of Contents

  1. Introduction
  2. $F$-Einstein-Hilbert functional
  3. The second variation of $\mathcal{E}_{F}$

Key Result

Theorem 2

The first variation of the $F$-Einstein-Hilbert functional in the direction of $h$ is given by the formula where $\langle ,\rangle$ is the induced Riemannian metric on $\otimes^2T^*M$, and $F'$ is the derivative of the function $F$.

Theorems & Definitions (29)

  • Definition 1: Felice, Sotiriou
  • Theorem 2: Felice, Sotiriou
  • Definition 3
  • Lemma 4: Reto, Peter
  • proof : Proof of Theorem \ref{['th1']}
  • Remark 5
  • Remark 6
  • Theorem 7: Felice, Sotiriou
  • Theorem 8
  • proof
  • ...and 19 more