Here, in jftRijM-,
w e
discuss mostly analytic topics in Ricci flow includ-
ing weak and strong maximum principles for scalar heat-type equations and
systems on compact and noncompact manifolds, the classification by Bohm
and Wilking of closed manifolds with 2-positive curvature operator, Bando's
result that solutions to the Ricci flow are real analytic in the space variables,
Shi's local derivative of curvature estimates and some variants, and differen-
tial Harnack estimates of Li-Yau-type including Hamilton's matrix estimate
for the Ricci flow and Perelman's estimate for fundamental solutions of the
adjoint heat equation coupled to the Ricci flow. In the appendices we review
aspects of Ricci flow and related geometric analysis and tensor calculus on
the frame bundle. Various topics in this part also include the works of others
as well as the authors.
In Part III of this volume, i.e., in ARijM, we shall discuss aspects of
Perelman's theory of ancient /^-solutions, Perelman's pseudolocality theo-
rem, Hamilton's classification of nonsingular solutions, numerical simula-
tions of Ricci flow, stability of the Ricci flow, the linearized Ricci flow, and
the space-time formulation of the Ricci flow. In the appendices, for the
convenience of the reader, we review and discuss aspects of metric and Rie-
mannian geometry, the reduced distance function and ancient solutions, and
limited aspects of Ricci flat metrics on the K3 surface.
As in previous volumes and as is perhaps typical in geometric analysis,
throughout this book we apply both the techniques of the 'weak maximum
detail principle' and 'exposition by parts'. To wit, we endeavor to supply
the reader with as much detail as possible and we also endeavor, for the
most part, to make the chapters independent of each other.
As such, §iRijk£ (and ARijki as well) may be used either for a topics
course or for self-study, where the lecturer or reader may wish to select
portions from this book -§iRijki, its predecessors gij) T^-, and Ri^i, and its
successor AR^u.
Although the intent of this series of books on the Ricci flow is expository,
there is no substitute for reading the original source material in Ricci flow. In
particular, the papers of Hamilton and Perelman contain a wealth of original
and deep ideas. We encourage interested readers to consult these papers. We
also encourage the reader to consult other sources for Ricci flow including
Cao and Zhu [78], Chen and Zhu [110], Ding [172], Kleiner and Lott [303],
Morgan and Tian [363], Miiller [371], Tao [461], Topping [475], two of
the authors [142], three of the authors [146], The Ricci Flowers [135] and
[136] (Parts I and III of this volume), and sources for geometric evolution
equations (e.g., the mean curvature flow) such as Chou and Zhu [127],
Ecker [179], Zhu [522], and two of the authors [139]. Certain material,
originally intended to appear in a successor volume to [146], has now been
incorporated in Parts I, II, and III of this volume.
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