Editing Ricci flow (section)

=== Ricci flow on manifolds with boundary ===
The study of the Ricci flow on manifolds with boundary was started by Ying Shen.<ref>{{Cite journal |last=Shen |first=Ying |date=1996-03-01 |title=On Ricci deformation of a Riemannian metric on manifold with boundary |url=https://msp.org/pjm/1996/173-1/p12.xhtml |journal=Pacific Journal of Mathematics |volume=173 |issue=1 |pages=203–221 |doi=10.2140/pjm.1996.173.203 |issn=0030-8730}}</ref> Shen introduced a boundary value problem for  manifolds with weakly umbilic boundaries, that is, the Second Fundamental Form of the boundary is a constant multiple of the metric, and then he proved that when the initial metric has positive Ricci curvature and the boundary is totally geodesic, the solution to the flow converges to a metric of constant positive curvature and totally geodesic boundary. [[Simon Brendle]]<ref>{{Cite journal |last=Brendle |first=S. |date=2002-11-01 |title=Curvature flows on surfaces with boundary |url=https://link.springer.com/article/10.1007/s00208-002-0350-4 |journal=Mathematische Annalen |language=en |volume=324 |issue=3 |pages=491–519 |doi=10.1007/s00208-002-0350-4 |issn=1432-1807}}</ref> showed that Shen's theorem is also valid for surfaces with totally geodesic boundary, and also introduced dynamic boundary conditions coupled to the Ricci flow.<ref>{{Cite journal |last=Brendle |first=S. |date=2002-12-01 |title=A family of curvature flows on surfaces with boundary |url=https://link.springer.com/article/10.1007/s00209-002-0439-1 |journal=Mathematische Zeitschrift |language=en |volume=241 |issue=4 |pages=829–869 |doi=10.1007/s00209-002-0439-1 |issn=1432-1823}}</ref> The first results for boundaries that are not totally geodesic, which include convergence results, were given by Jean Cortissoz<ref>{{Cite journal |last=Cortissoz |first=Jean C. |date=2009-02-01 |title=Three-manifolds of positive curvature and convex weakly umbilic boundary |url=https://link.springer.com/article/10.1007/s10711-008-9300-y |journal=Geometriae Dedicata |language=en |volume=138 |issue=1 |pages=83–98 |doi=10.1007/s10711-008-9300-y |issn=1572-9168}}</ref> in the case of 3-manifolds with convex weakly umbilic boundary, with subsequent developments, together with Alexander Murcia<ref>{{Cite journal |last1=Cortissoz |first1=Jean C. |last2=Murcia |first2=Alexander |date=2019-08-23 |title=The Ricci flow on surfaces with boundary |url=https://intlpress.com/JDetail/1805783967360049154 |journal=Communications in Analysis and Geometry |language=EN |volume=27 |issue=2 |pages=377–420 |doi=10.4310/CAG.2019.v27.n2.a5 |arxiv=1209.2386 |issn=1944-9992}}</ref> and César Reyes,<ref>{{Cite journal |last1=Cortissoz |first1=Jean C. |last2=Reyes |first2=César |date=2023 |title=Classical solutions to the one-dimensional logarithmic diffusion equation with nonlinear Robin boundary conditions |url=https://onlinelibrary.wiley.com/doi/10.1002/mana.202100415 |journal=Mathematische Nachrichten |language=en |volume=296 |issue=9 |pages=4086–4107 |doi=10.1002/mana.202100415 |issn=1522-2616}}</ref> to metrics on a disk and a cylinder. Artem Pulemotov<ref>{{Cite journal |last=Pulemotov |first=Artem |date=2013-10-01 |title=Quasilinear parabolic equations and the Ricci flow on manifolds with boundary |url=https://www.degruyterbrill.com/document/doi/10.1515/crelle-2012-0004/html?lang=en |journal=Journal für die reine und angewandte Mathematik (Crelles Journal) |language=en |volume=2013 |issue=683 |pages=97–118 |doi=10.1515/crelle-2012-0004 |arxiv=1012.2941 |issn=1435-5345}}</ref> and then Panagiotis Gianniotis<ref>{{cite journal
 | last = Gianniotis | first = Panagiotis
 | arxiv = 1210.0813
 | doi = 10.4310/jdg/1476367059
 | issue = 2
 | journal = Journal of Differential Geometry
 | mr = 3557306
 | pages = 291–324
 | title = The Ricci flow on manifolds with boundary
 | volume = 104
 | year = 2016}}</ref> introduced an interesting boundary value problem for the Ricci flow, and proved short time existence and uniqueness. Recently, Gang Li<ref>{{Citation |last=Li |first=Gang |title=A normalized Ricci flow on surfaces with boundary towards the complete hyperbolic metric |date=2025-03-14 |url=https://arxiv.org/abs/2502.00660 |access-date=2025-04-10 |arxiv=2502.00660 }}</ref> showed that in the case of surfaces with boundary, there can be existence of a solution for all time without convergence. Tsz-Kiu Aaron Chow<ref>{{Cite journal |last=Chow |first=Tsz-Kiu Aaron |date=2022-02-01 |title=Ricci flow on manifolds with boundary with arbitrary initial metric |url=https://www.degruyterbrill.com/document/doi/10.1515/crelle-2021-0060/html |journal=Journal für die reine und angewandte Mathematik (Crelles Journal) |language=en |volume=2022 |issue=783 |pages=159–216 |doi=10.1515/crelle-2021-0060 |arxiv=2012.04430 |issn=1435-5345}}</ref> has introduced a family of solutions to the Ricci flow which preserves some geometric properties of the initial data, and has used them to provide geometric applications to manifolds with boundary.