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Evolution of Functionals Under Extended Ricci Flow

Shouvik Datta Choudhury

Abstract

In this paper, we investigate the evolution of certain functionals involving higher powers of a scalar quantity $F$ under Bernard List's extended Ricci flow on a compact Riemannian manifold. By deriving explicit expressions for the time derivative of integrals of the form $\int_M F^n \cdot \frac{\partial F}{\partial t} \, dμ$ for various powers $n$, we explore the intricate interplay between geometric quantities and scalar functions without making any assumptions about the manifold, the scalar field $Φ$, or the function $u$.

Evolution of Functionals Under Extended Ricci Flow

Abstract

In this paper, we investigate the evolution of certain functionals involving higher powers of a scalar quantity under Bernard List's extended Ricci flow on a compact Riemannian manifold. By deriving explicit expressions for the time derivative of integrals of the form for various powers , we explore the intricate interplay between geometric quantities and scalar functions without making any assumptions about the manifold, the scalar field , or the function .

Paper Structure

This paper contains 5 sections, 3 theorems, 100 equations.

Table of Contents

  1. MSC Classification
  2. Introduction
  3. Literature Review
  4. Mathematical Preliminaries
  5. Theorems and Proofs

Key Result

Theorem 1

Under Bernard List's extended Ricci flow, applying DeTurck's trick and Uhlenbeck's gauge fixing, the curvature tensors evolve according to the following equations:

Theorems & Definitions (6)

  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • Theorem 3
  • proof