Geometric constant on Riemannian manifold evolves by geometric flow

Sah Apurba, Azami, Shahroud, Pişcoran, Laurian-Ioan and Hui, Shyamal Kumar

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Issue no: Vol 39/2023 no. 3 Tags: Geometric flow, Laplace operator, Riemannian manifold

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In this article, we consider $(M^n, g(t))$ an $n$ -dimensional closed Riemannian manifold whose metric $g(t)$ evolves by the abstract geometric flow and the geometric constant $\lambda_a^b$ as the lowest constant such that the equation

$- \Delta u + a u \log u + b S u = \lambda_a^b u$

with $\int_M u^2 d\mu = 1$ has a positive solution, where $a\ (>0)$ and $b$ are two real constants. Here we find the evolution formula for $\lambda_a^b$ on $(M^n, g(t))$ evolving along the abstract geometric flow.