Well-Posedness of the Variable Exponent Ricci Flow
About this article
Keywords:
Variable exponent Ricci flow; degenerate parabolic equations; well-posedness; geometric analysis; parabolic Hölder spacesAbstract
We establish local well-posedness for the variable exponent Ricci flow
∂t/∂g=−2∣Ricg(t)∣p(x)−2Ricg(t), g(0)=g0
on closed Riemannian manifolds, where p:M→(1,∞) is a Hölder continuous function. We identify precise conditions on the exponent function p(x) and initial metric g0 that guarantee existence, uniqueness, and continuous dependence on initial data. The main challenges arise from the degenerate/singular parabolic nature of the equation when p(x) ≠2 and the spatially varying nonlinearity. Our results reveal a trichotomy: for p(x)≥2 we obtain strong solutions under mild conditions; for 1<p(x)<2 well-posedness requires a curvature gap condition; and at the critical interface where p(x) crosses 2, we demonstrate potential non-uniqueness. Applications to adaptive geometric regularization and connections to variable exponent Sobolev spaces are discussed.
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