Abstract
Let \((M^n, g(t))\) be a compact Riemannian manifold. In this paper, we derive the evolution formula for the geometric constant \(\lambda _{a}^{b} (g)\) as an infimum of a certain energy function when the following partial differential equation:
with \(\int _M u^2 d\mu = 1\), has positive solutions, where a and b are real constants along the extended Ricci flow and the normalized extended Ricci flow. In addition, we derive some monotonicity formulas by imposing some conditions along both the extended Ricci flow and the normalized extended Ricci flow.
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Acknowledgements
The authors are thankful to the respected reviewer for her/his valuable suggestions to improve the paper. The first author (A. Saha) gratefully acknowledges to the CSIR (File No.: 09/025(0273)/2019-EMR-I), Government of India for the award of Junior Research Fellowship. This research work is also partially supported by DST FIST programme (No.: SR/FST/MSII/2017/10(C)).
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Saha, A., Azami, S. & Hui, S.K. Evolution and Monotonicity of Geometric Constants Along the Extended Ricci Flow. Mediterr. J. Math. 18, 199 (2021). https://doi.org/10.1007/s00009-021-01848-9
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DOI: https://doi.org/10.1007/s00009-021-01848-9