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Well-posedness of nonlinear flows on manifolds of bounded geometry

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Abstract

We present straightforward conditions which ensure that a strongly elliptic linear operator L generates an analytic semigroup on Hölder spaces on an arbitrary complete manifold of bounded geometry. This is done by establishing the equivalent property that L is ‘sectorial,’ a condition that specifies the decay of the resolvent \((\lambda I - L)^{-1}\) as \(\lambda \) diverges from the Hölder spectrum of L. A key step is that we prove existence of this resolvent if \(\lambda \) is sufficiently large using a geometric microlocal version of the semiclassical pseudodifferential calculus. The properties of L and \(e^{-tL}\) we obtain can then be used to prove well-posedness of a wide class of nonlinear flows. We illustrate this by proving well-posedness on Hölder spaces of the flow associated with the ambient obstruction tensor on complete manifolds of bounded geometry. This new result for a higher-order flow on a noncompact manifold exhibits the broader applicability of our technique.

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Acknowledgements

The authors thank Jack Lee, Yoshihiko Matsumoto, András Vasy, and Guofang Wei for useful conversations during this work. This work was supported by collaboration grants from the Simons Foundation (#426628, E. Bahuaud and #283083, C. Guenther). J. Isenberg was supported by NSF grant PHY-1707427.

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Authors and Affiliations

  1. Department of Mathematics, Seattle University, Seattle, WA, 98122, USA

    Eric Bahuaud

  2. Department of Mathematics, Pacific University, Forest Grove, OR, 97116, USA

    Christine Guenther

  3. Department of Mathematics, University of Oregon, Eugene, OR, 97403-1222, USA

    James Isenberg

  4. Department of Mathematics, Stanford University, Stanford, CA, 94305, USA

    Rafe Mazzeo

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  1. Eric Bahuaud
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  2. Christine Guenther
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  4. Rafe Mazzeo
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Correspondence to Christine Guenther.

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Bahuaud, E., Guenther, C., Isenberg, J. et al. Well-posedness of nonlinear flows on manifolds of bounded geometry. Ann Glob Anal Geom 65, 25 (2024). https://doi.org/10.1007/s10455-023-09940-x

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