Monotonicity formulas for the first eigenvalue of the weighted p-Laplacian under the Ricci-harmonic flow
Tóm tắt
Let
$\Delta _{p,\phi }$
be the weighted p-Laplacian defined on a smooth metric measure space. We study the evolution and monotonicity formulas for the first eigenvalue,
$\lambda _{1}=\lambda (\Delta _{p,\phi })$
, of
$\Delta _{p,\phi }$
under the Ricci-harmonic flow. We derive some monotonic quantities involving the first eigenvalue, and as a consequence, this shows that
$\lambda _{1}$
is monotonically nondecreasing and almost everywhere differentiable along the flow existence.
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