Sharp parabolic regularity and geometric flows on singular spaces
Journées équations aux dérivées partielles (2015), article no. 6, 11 p.

This is a brief survey about regularity expansions for solutions of elliptic and parabolic problems on spaces with conic singularities. The results themselves are closely related to classical results about elliptic boundary problems, and analogues of these are expected to hold on quite general stratified spaces with incomplete iterated edge metrics. The emphasis here is on the interpretation and application of these expansions to geometric problems.

DOI : 10.5802/jedp.635
Mazzeo, Rafe 1

1 Department of Mathematics Stanford University Stanford CA 94305-2125 USA
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Mazzeo, Rafe. Sharp parabolic regularity  and geometric flows on singular spaces. Journées équations aux dérivées partielles (2015), article  no. 6, 11 p. doi : 10.5802/jedp.635. http://www.numdam.org/articles/10.5802/jedp.635/

[1] P. Albin, E. Leichtnam, R. Mazzeo and P. Piazza, The signature package on Witt spaces Ann. Sci. l’ENS 45 (2012), no. 2, 241-310. | Numdam | MR | Zbl

[2] P. Albin, E. Leichtnam, R. Mazzeo and P. Piazza, Hodge theory on Cheeger spaces, To appear J. Reine Ang. Math.

[3] E. Bahuaud, B. Vertman, Yamabe flow on manifolds with edges, Math. Nach. 287 (2014), no. 2-3, 127-159. | MR | Zbl

[4] E. Bahuaud, B. Vertman, Mapping properties of the heat operator on edge manifolds, Math. Nachr. 288 (2015), no. 2-3, 126–157. | MR

[5] D. Bartolucci, F. De Marchis and A. Malchiodi, Supercritical conformal metrics on surfaces with conical singularities, IMRN (2011), 5625–5643. | MR | Zbl

[6] K. Brakke, The motion of a surface by its mean curvature Mathematical Notes # 20, Princeton University Press, Princeton (1978). | MR | Zbl

[7] L. Bronsard and F. Reitich, On three-phase boundary motion and the singular limit of a vector-valued Ginzburg-Landau equation Arch. Rational Mech. Anal. 124 (1993), no. 4, 355-379. | MR | Zbl

[8] A. Carlotto, A. Malchiodi Weighted barycentric sets and singular Liouville equations on compact surfaces, J. Funct. Anal., 262 (2012), no. 2, 409–450. | MR | Zbl

[9] X.-X. Chen, Y.-Q. Wang, Bessel functions, heat kernels and the conical Kähler–Ricci flow, J. Func. Anal. 269 (2015) no. 2, 551-632. | MR

[10] S. Donaldson, Kähler metrics with cone singularities along a divisor, in Essays in Mathematics and its Applications, 47-49, Springer-Verlag, Heidelberg (2012). | MR

[11] A. Freire, Mean curvature motion of triple junctions of graphs in two dimensions, Comm. PDE, 35 (2010) no. 2, 302-327. | MR | Zbl

[12] G. Giesen, P. Topping, Ricci flow of negatively curved incomplete surfaces, Calc. Var. and PDE, 38 No. 3 (2010), 357–367. | MR | Zbl

[13] G. Giesen, P. Topping, Existence of Ricci flows of incomplete surfaces, Comm. PDE, 36 (2011), 1860–1880. | MR | Zbl

[14] R. S. Hamilton, Ricci flow on surfaces, in: Mathematics and General Relativity, Contemporary Math. 71 (1988), 237–261. | MR | Zbl

[15] T. Ilmanen, A. Neves and F. Schulze On short time existence for the planar network flow . | arXiv

[16] T. Jeffres, P. Loya, Regularity of solutions of the heat equation on a cone, Int. Math. Res. Not. (2003) No. 3, 161–178. | MR | Zbl

[17] T. Jeffres, R. Mazzeo, Y.A. Rubinstein, Kähler–Einstein metrics with edge singularities, with an appendix by C. Li and Y.A. Rubinstein, Annals of Math, 183 (2016) no. 1, 95-176.

[18] H. Koch and T. Lamm, Geometric flows with rough initial data, Asian J. Math. 16 (2012), no. 2, 209–235. | MR | Zbl

[19] J. Lira, R. Mazzeo and M. Saez, In preparation.

[20] A. Malchiodi, D. Ruiz New improved Moser-Trudinger inequalities and singular Liouville equations on compact surfaces, GAFA, 21-5 (2011), 1196–1217. | MR | Zbl

[21] C. Mantegazza, M. Novaga, and V. M. Tortorelli, Motion by curvature of planar networks, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 3 No. 2 (2004), 235–324. | Numdam | MR | Zbl

[22] R. Mazzeo, Elliptic theory of differential edge operators, I, Comm. PDE 16 (1991), 1616-1664. | MR | Zbl

[23] R. Mazzeo, Y. Rubinstein and N. Sesum, Ricci flow on surfaces with conic singularities, Anal. and PDE 8 (2015) no. 4, 839-882. | MR | Zbl

[24] R. Mazzeo and M. Saez, Self similar expanding solutions for the planar network flow in Analytic aspects of problems in Riemannian geometry: Elliptic PDEs, solitons and computer imaging Eds. Paul Baird, Ahmad El Soufi, Ali Fardoun, Rachid Regbaoui, Séminaires et Congrès, Société Mathématique de France, 19 (2009), 159–173. | MR | Zbl

[25] R. Mazzeo and B. Vertman, Elliptic theory of differential edge operators, II: boundary value problems, Indiana Univ. Math. J. 63 No. 6 (2014), 1911–1955. | MR

[26] R. Mazzeo and H. Weiss, The Teichmüller theory of conic surfaces, arXiv:1305.0255.

[27] R. C. McOwen, Point singularities and conformal metrics on Riemann surfaces, Proc. Amer. Math. Soc. 103 (1988), 222–224. | MR | Zbl

[28] G. Mondello and D. Panov, Spherical metrics with conical singularities on a 2-sphere: angle constraints, . | arXiv

[29] E. Mooers, Heat kernel asymptotics on manifolds with conic singularities, J. Anal. Math. 78 (1999), 1–36. | MR | Zbl

[30] D. H. Phong, J. Song, J. Sturm, X.-W. Wang, The Ricci flow on the sphere with marked points, . | arXiv

[31] D. Ramos, Smoothening cone points with Ricci flow, . | arXiv

[32] J. Ross, R. Thomas, Weighted projective embeddings, stability of orbifolds and constant scalar curvature Kähler metrics, J. Diff. Geom. 88 (2011) No. 1, 109–159. | MR | Zbl

[33] O. Schnürer and F. Schulze, Self-similar expanding networks to curve shortening flow, Ann. Sc. Norm. Super. Pisa Cl. Sci. 5 (2007), no. 4, 511–528. | Numdam | MR | Zbl

[34] M. Simon, Deformation of 𝒞 0 Riemannian metrics in the direction of their Ricci curvature, Comm. Anal. Geom. 16 (2008) No. 1, 1033–1074. | MR | Zbl

[35] A. Tromba, Teichmüller theory in Riemannian geometry, Lecture Notes in Mathematics, ETH Zürich, Birkhäuser, Basel, 1992. | MR | Zbl

[36] M. Troyanov, Prescribing curvature on compact surfaces with conical singularities, Trans. Amer. Math. Soc. 324 (1991), 793–821. | MR | Zbl

[37] H. Yin, Ricci flow on surfaces with conical singularities, J. Geom. Anal. 20 (2010), no. 4, 970–995. | MR | Zbl

[38] H. Yin, Ricci flow on surfaces with conical singularities, II, . | arXiv | MR

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