[Paper Review] On Warped Product Gradient Ricci-Harmonic Soliton

Elismar Dias Batista, Levi Adriano+1 more|arXiv (Cornell University)|Jun 27, 2019

Geometric Analysis and Curvature Flows12 references4 citations

TL;DR

This paper investigates gradient Ricci-harmonic solitons (GRHS) on warped product manifolds, deriving necessary and sufficient conditions for their existence by analyzing the harmonic map and potential function structure. It constructs infinitely many geodesically complete, nontrivial examples in the semi-Riemannian setting by assuming the base and fiber are conformal to semi-Euclidean spaces invariant under a co-dimension one translation group, extending beyond Riemannian constraints.

ABSTRACT

In this paper we study gradient Ricci-Harmonic soliton with structure of warped product manifold. We obtain some triviality results for the potential function, warping function and the harmonic map which reaches maximum or minimum. In order to obtain nontrivial examples of warped product gradient Ricci-harmonic soliton, we consider the base and fiber conformal to a semi-Euclidean space which is invariant under the action of a translation group of co-dimension one. This approach provide infinitely many geodesically complete examples in the semi-Riemannian context, which is not contemplated in the Riemannian case by the Theorem 1.2 in [17].

Motivation & Objective

• To characterize the structure of harmonic maps on warped product gradient Ricci-harmonic solitons using the potential function.
• To establish necessary and sufficient conditions for the existence of GRHS on warped product manifolds with harmonic maps of the form $ u = u_B \circ \pi $ or $ u = u_F \circ \sigma $.
• To extend nontrivial GRHS examples beyond the Riemannian case by exploiting semi-Riemannian geometry and group actions.
• To prove triviality results for the warping function, potential function, and harmonic map under maximum/minimum conditions using the maximum principle.
• To construct infinitely many geodesically complete GRHS examples in the semi-Riemannian context by assuming conformal invariance under translation groups.

Proposed method

• Derives a characterization of harmonic maps $ u $ on warped product GRHS by showing $ u = u_B \circ \pi $ or $ u = u_F \circ \sigma $ if and only if the potential function satisfies $ h = h_B \circ \pi $.
• Applies the warped product metric $ g = \pi^*g_B + (f \circ \pi)^2 \sigma^*g_F $, with $ f > 0 $, to define the manifold structure.
• Derives system of PDEs governing GRHS existence: Ricci curvature equations with mixed terms involving $ \theta \nabla u \otimes \nabla u $, harmonic map equations, and curvature conditions on base and fiber.
• Imposes geometric constraints: base $ B $ and fiber $ F $ conformal to semi-Euclidean spaces, invariant under a co-dimension one translation group.
• Uses the maximum principle to prove triviality results: if $ u_B $, $ u_F $, $ h_B $, or $ f $ attain extrema, the soliton reduces to a gradient Ricci soliton or harmonic-Einstein manifold.
• Analyzes geodesic completeness by solving the geodesic equations in the conformal structure, proving all solutions are defined on $ \mathbb{R} $, hence geodesic completeness.

Experimental results

Research questions

• RQ1 Under what conditions does a harmonic map on a warped product GRHS factor through the base or fiber?
• RQ2 What are the necessary and sufficient conditions for the existence of a GRHS on a warped product manifold with $ u = u_B \circ \pi $ or $ u = u_F \circ \sigma $?
• RQ3 Can nontrivial, geodesically complete GRHS be constructed in the semi-Riemannian setting beyond the Riemannian case?
• RQ4 When does the maximum or minimum of the potential function, harmonic map, or warping function force the soliton to degenerate into a gradient Ricci soliton or harmonic-Einstein manifold?
• RQ5 What role does the co-dimension one translation group action play in enabling infinitely many geodesically complete GRHS examples?

Key findings

• A GRHS on a warped product exists if and only if the base satisfies a modified Ricci curvature equation involving $ \theta \nabla u_B \otimes \nabla u_B $, and the fiber is Einstein or Einstein-harmonic.
• If the harmonic map $ u_B $ or $ u_F $ attains a maximum or minimum on the base or fiber, then $ u $ must be constant, reducing the GRHS to a gradient Ricci soliton.
• If $ \lambda \geq 0 $, $ h_B $ attains an extremum, and $ \frac{m \Delta_{g_B} f}{f} \geq \text{scal}_{g_B} $, then $ h_B $ is constant, implying the soliton is harmonic-Einstein.
• If the warping function $ f $ attains a maximum or minimum and $ \lambda \leq \frac{\mu}{f^2} $, then $ f $ is constant, reducing the metric to a semi-Riemannian product.
• By assuming the base and fiber are conformal to semi-Euclidean spaces under a co-dimension one translation group, the paper constructs infinitely many geodesically complete GRHS examples.
• All geodesics on the constructed warped product are defined on $ \mathbb{R} $, confirming geodesic completeness in the semi-Riemannian context.