On the spectrum of the weighted p-Laplacian under the Ricci-harmonic flow

This paper studies the behaviour of the spectrum of the weighted p -Laplacian on a complete Riemannian manifold evolving by the Ricci-harmonic flow. Precisely, the first eigenvalue diverges in a finite time along this flow. It is further shown that the same divergence result holds on gradient shrink...

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Bibliographic Details
Published in:Journal of inequalities and applications 2020-03, Vol.2020 (1), p.1-14, Article 58
Main Authors: Abolarinwa, Abimbola, Edeki, Sunday O., Ehigie, Julius O.
Format: Article
Language:English
Subjects:
Analysis
Applications of Mathematics
Eigenvalue
Eigenvalues
Laplace–Beltrami operator
Mathematics
Mathematics and Statistics
Monotonicity
Ricci harmonic flow
Ricci solitons
Riemann manifold
Solitary waves
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Summary:This paper studies the behaviour of the spectrum of the weighted p -Laplacian on a complete Riemannian manifold evolving by the Ricci-harmonic flow. Precisely, the first eigenvalue diverges in a finite time along this flow. It is further shown that the same divergence result holds on gradient shrinking and steady almost Ricci-harmonic solitons under the condition that the soliton function is nonnegative and superharmonic. We also continue the program in (Abolarinwa, Adebimpe and Bakare in J. Ineq. Appl. 2019:10, 2019 ) to the case of volume-preserving Ricci-harmonic flow.
ISSN:1029-242X
1025-5834
1029-242X
DOI:10.1186/s13660-020-02322-y