Abstract
It is the purpose of this article to establish a technical tool to study regularity of solutions to parabolic equations on manifolds. As applications of this technique, we prove that solutions to the Ricci-DeTurck flow, the surface diffusion flow and the mean curvature flow enjoy joint analyticity in time and space, and solutions to the Ricci flow admit temporal analyticity.
Article PDF
Similar content being viewed by others
References
Amann, H.: Ordinary Differential Equations. An Introduction to Nonlinear Analysis, Translated from the German by Gerhard Metzen. de Gruyter Studies in Mathematics, vol. 13. Walter de Gruyter & Co., Berlin (1990)
Amann, H.: Linear and Quasilinear Parabolic Problems: Volume I, Birkhäuser Boston, Inc., Boston (1995)
Amann H: Elliptic operators with infinite-dimensional state spaces. J. Evol. Equ. 1(2), 143–188 (2001)
Amann H: Function spaces on singular manifolds. Math. Nachr. 286(5–6), 436–475 (2013)
Amann, H.: Anisotropic function spaces on singular manifolds, arXiv.1204.0606.
Amann, H.: Parabolic equations on uniformly regular Riemannian manifolds and degenerate initial boundary value problems, Preprint (2013)
Angenent S.B.: Nonlinear analytic semiflows. Proc. Roy. Soc. Edinburgh Sect. A 115(1–2), 91–107 (1990)
Angenent S.B.: Parabolic equations for curves on surfaces. I. Curves with p-integrable curvature. Ann. of Math. (2) 132(3), 451–483 (1990)
Bando S.: Real analyticity of solutions of Hamilton’s equation. Math. Z. 195(1), 93–97 (1987)
Bergh, J., Löfström, J.: Interpolation Space: An Introduction, Springer, Berlin-New York (1976)
Brakke, K.: The motion of a surface by its mean curvature, Volume 20 of Mathematical Notes. Princeton University Press, Princeton (1978)
Browder, F.E.: Analyticity and partial differential equations I. Am. J. Math. 84:666–710 (1962)
Cai Q., Zhao P.: On stability of Ricci flows based on bounded curvatures. Balkan J. Geom. Appl. 15(2), 34–46 (2010)
Chow, B., Knopf, D.: The Ricci Flow: An Introduction, Mathematical Surveys and Monographs, vol. 110. American Mathematical Society, Providence (2004)
Clément P., Simonett G.: Maximal regularity in continuous interpolation spaces and quasilinear parabolic equations. J. Evol. Equ. 1(1), 39–67 (2001)
DeTurck D.: Deforming metrics in the direction of their Ricci tensors. J. Differential Geom. 18(1), 157–162 (1983)
Escher J., Mayer U., Simonett G.: The surface diffusion flow for immersed hypersurfaces. SIAM J. Math. Anal. 29(6), 1419–1433 (1998)
Escher J., Mucha P.: The surface diffusion flow on rough phase spaces. Discrete Contin. Dyn. Syst. 26(2), 431–453 (2010)
Escher J., Prokert G.: Analyticity of solutions to nonlinear parabolic equations on manifolds and an application to Stokes flow. J. Math. Fluid Mech. 8(1), 1–35 (2006)
Escher, J., Prüss, J., Simonett, G.: A new approach to the regularity of solutions for parabolic equations, evolution equations. In: Lecture Notes in Pure and Appl. Math. vol. 234, pp. 167–190. Dekker, New York (2003)
Escher J., Prüss J., Simonett G.: Analytic solutions for a Stefan problem with Gibbs-Thomson correction. J. Reine Angew. Math. 563, 1–52 (2003)
Escher J., Simonett G.: Analyticity of the interface in a free boundary problem. Math. Ann. 305(3), 439–459 (1996)
Escher J., Simonett G.: The volume preserving mean curvature flow near spheres. Proc. Am. Math. Soc. 126(9), 2789–2796 (1998)
Escher J., Simonett G.: A center manifold analysis for the Mullins-Sekerka Model. J. Differ. Equ. 143(2), 267–292 (1998)
Gage, M.: On an area-preserving evolution equation for plane curves. In: Nonlinear problems in geometry (Mobile, Ala., 1985), vol. 51, pp. 51–62. Contemp. Math. Am. Math. Soc., Providence (1986)
Greene, R.E.: Complete metrics of bounded curvature on noncompact manifolds. Arch. Math. (Basel) 31(1), 89–95 (1978/79)
Hamilton R.: Three-manifolds with positive Ricci curvature. J. Differ. Geom. 17(2), 255–306 (1982)
Huisken G.: The volume preserving mean curvature flow. J. Reine Angew. Math. 382, 35–48 (1987)
Huisken G., Sinestrari C.: Mean curvature flow singularities for mean convex surfaces. Calc. Var. Partial Differential Equations 8(1), 1–14 (1999)
Huisken G., Sinestrari C.: Convexity estimates for mean curvature flow and singularities of mean convex surfaces. Acta Math. 183(1), 45–70 (1999)
Huisken G., Sinestrari C.: Mean curvature flow with surgeries of two-convex hypersurfaces. Invent. Math. 175(1), 137–221 (2009)
Koch H., Lamm T.: Geometric flows with rough initial data. Asian J. Math. 16(2), 209–235 (2012)
Kotschwar B.: A local version of Bando’s theorem on the real-analyticity of solutions to the Ricci flow. Bull. Lond. Math. Soc. 45(1), 153–158 (2013)
Kotschwar, B.: Time-analyticity of solutions to the Ricci flow, arXiv:1210.3083v2.
LeCrone, J., Simonett, G.: On well-posedness, stability, and bifurcation for the axisymmetric surface diffusion flow, SIAM J. Math. Anal. (to appear) arXiv:1209.3998v2
Lunardi, A.: Analytic Semigroups and Optimal Regularity in Parabolic Problems, Birkhäuser, Basel (1995)
Morgan, J., Tian, G.: Ricci flow and the Poincaré conjecture, Clay Mathematics Monographs, vol. 3. American Mathematical Society, Providence; Clay Mathematics Institute, Cambridge (2007)
Morrey, C.B.Jr: The analytic embedding of abstract real-analytic manifolds. Ann. of Math. 68(2), 159–201 (1958)
Müller, O., Nardmann, M.: Every conformal class contains a metric of bounded geometry, arXiv:1303.5957
Perelman, G.: The entropy formula for the Ricci flow and its geometric applications, arXiv:math/0211159
Perelman, G.: Ricci flow with surgery on three-manifolds, arXiv:math/0303109
Prüss, J., Simonett, G.: On the manifold of closed hypersurfaces in \({\mathbb{R}^n}\) . Discrete Contin. Dyn. Syst. 33, 5407–5428. arXiv:1212.6445 (2013)
Shao, Y.: Real analytic solutions to the Willmore flow. Electron. J. Differ. Equ. (to appear). arXiv:1302.3994
Shao, Y., Simonett, G.: Continuous maximal regularity on uniformly regular Riemannian manifolds, arXiv:1309.2041
Triebel, H.: Interpolation Theory, Function Spaces, Differential Operator, North-Holland Publishing Co., Amsterdam-New York (1978)
Wheeler G.: Lifespan theorem for simple constrained surface diffusion flows. J. Math. Anal. Appl. 375(2), 685–698 (2011)
Wheeler G.: Surface diffusion flow near spheres. Calc. Var. Partial Differential Equations 44(1–2), 131–151 (2012)
White B.: The size of the singular set in mean curvature flow of mean convex sets. J. Am. Math. Soc. 13(3), 665–695 (2000)
White B.: The nature of singularities in mean curvature flow of mean-convex sets. J. Am. Math. Soc. 16(1), 123–138 (2003)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Shao, Y. A family of parameter-dependent diffeomorphisms acting on function spaces over a Riemannian manifold and applications to geometric flows. Nonlinear Differ. Equ. Appl. 22, 45–85 (2015). https://doi.org/10.1007/s00030-014-0275-0
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00030-014-0275-0