A Dual Yamabe Flow and Related Integral Flows
Copyright policy )A Dual Yamabe Flow and Related Integral Flows
We study a family of nonlinear integral flows that involve Riesz potentials on Riemannian manifolds. In the Hardy-Littlewood-Sobolev (HLS) subcritical regime, we present a precise blow-up profile exhibited by the flows. In the HLS critical regime, by introducing a \textit{dual $Q$ curvature} we demonstrate the concentration-compactness phenomenon. If, in addition, the integral kernel matches with the Green's function of a conformally invariant elliptic operator, this critical flow can be considered as a dual Yamabe flow. Convergence is then established on the unit spheres, which is also valid on certain locally conformally flat manifolds.
35 pages. Submitted
- Beijing Normal University China (People's Republic of)
Mathematics - Differential Geometry, Diffusion processes and stochastic analysis on manifolds, Methods of global Riemannian geometry, including PDE methods; curvature restrictions, Hardy-Littlewood-Sobolev functional, Flows related to symplectic and contact structures, integral flow, Local Riemannian geometry, Mathematics - Analysis of PDEs, Differential Geometry (math.DG), Mathematics - Classical Analysis and ODEs, Classical Analysis and ODEs (math.CA), FOS: Mathematics, dual \(Q\) curvature, Analysis of PDEs (math.AP)
Mathematics - Differential Geometry, Diffusion processes and stochastic analysis on manifolds, Methods of global Riemannian geometry, including PDE methods; curvature restrictions, Hardy-Littlewood-Sobolev functional, Flows related to symplectic and contact structures, integral flow, Local Riemannian geometry, Mathematics - Analysis of PDEs, Differential Geometry (math.DG), Mathematics - Classical Analysis and ODEs, Classical Analysis and ODEs (math.CA), FOS: Mathematics, dual \(Q\) curvature, Analysis of PDEs (math.AP)
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