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Continuous maximal regularity on uniformly regular Riemannian manifolds. (English) Zbl 1295.35161
Summary: We establish continuous maximal regularity results for parabolic differential operators acting on sections of tensor bundles on uniformly regular Riemannian manifolds $$\mathsf M$$. As an application, we show that solutions to the Yamabe flow on $$\mathsf M$$ instantaneously regularize and become real analytic in space and time. The regularity result is obtained by introducing a family of parameter-dependent diffeomorphisms acting on functions on $$\mathsf M$$ in conjunction with maximal regularity and the implicit function theorem.

##### MSC:
 35B65 Smoothness and regularity of solutions to PDEs 35K55 Nonlinear parabolic equations 53C44 Geometric evolution equations (mean curvature flow, Ricci flow, etc.) (MSC2010) 53A30 Conformal differential geometry (MSC2010) 35K90 Abstract parabolic equations 35R01 PDEs on manifolds 58J35 Heat and other parabolic equation methods for PDEs on manifolds
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