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A Structural Analysis of Warped Product Yamabe Gradient Solitons

Jahnabi Chakraborti, Anandateertha Mangasuli

TL;DR

The paper investigates warped-product Yamabe gradient solitons, proving that in nontrivial cases the soliton function $h$ depends only on the base and the fiber must have constant scalar curvature, which reduces the problem to a base almost Yamabe gradient soliton with an explicit $\lambda$-characterization. It then derives structural results linking critical points of the soliton function to global geometry, showing that complete-base solitons have no critical points and that, with at most two critical points, the base and total space admit rigid isometric decompositions. Scalar-curvature estimates are developed, including a lower-bound criterion for $R$ and quadratic growth bounds for the potential function under completeness and Ricci curvature conditions, complemented by explicit complete and incomplete examples. Together, these results advance the classification and curvature control of noncompact warped-product Yamabe gradient solitons and illustrate the geometric consequences of the soliton structure.

Abstract

We investigate Yamabe gradient solitons, which are warped product manifolds. We show that the fiber of a nontrivial warped product Yamabe gradient soliton has constant scalar curvature. Based on this result, we obtain a specific class of warped products that cannot be nontrivial solitons. Furthermore, we derive estimates of the scalar curvature and the soliton function of warped product solitons.

A Structural Analysis of Warped Product Yamabe Gradient Solitons

TL;DR

The paper investigates warped-product Yamabe gradient solitons, proving that in nontrivial cases the soliton function depends only on the base and the fiber must have constant scalar curvature, which reduces the problem to a base almost Yamabe gradient soliton with an explicit -characterization. It then derives structural results linking critical points of the soliton function to global geometry, showing that complete-base solitons have no critical points and that, with at most two critical points, the base and total space admit rigid isometric decompositions. Scalar-curvature estimates are developed, including a lower-bound criterion for and quadratic growth bounds for the potential function under completeness and Ricci curvature conditions, complemented by explicit complete and incomplete examples. Together, these results advance the classification and curvature control of noncompact warped-product Yamabe gradient solitons and illustrate the geometric consequences of the soliton structure.

Abstract

We investigate Yamabe gradient solitons, which are warped product manifolds. We show that the fiber of a nontrivial warped product Yamabe gradient soliton has constant scalar curvature. Based on this result, we obtain a specific class of warped products that cannot be nontrivial solitons. Furthermore, we derive estimates of the scalar curvature and the soliton function of warped product solitons.

Paper Structure

This paper contains 4 sections, 14 theorems, 33 equations.

Table of Contents

  1. Introduction
  2. Triviality of Warperd Product Yamabe gradient Solitons
  3. Critical points of the soliton function
  4. Scalar Curvature and Potential Function Estimation

Key Result

Theorem 1.1

TBAP1 Let $(M^{n+m}=B^n\times_f F^m, g= g_{B}+f^2 g_{F}, h, \rho)$ be a complete and nontrivial warped product Yamabe gradient soliton. Then the soliton function $h$ does not depend on $F^m$ if and only if the fiber manifold $F^m$ has constant scalar curvature.

Theorems & Definitions (30)

  • Theorem 1.1
  • Theorem 2.1
  • proof
  • Example 2.2
  • Corollary 2.3
  • Corollary 2.4
  • Remark 2.5
  • Example 2.6
  • Example 2.7
  • Theorem 2.8
  • ...and 20 more