Highlights and interdependencies of Part II
Half my life is in books' written pages.
- From "Dream On" by Steven Tyler of Aerosmith
0.1. Highlights. In this book we consider the following mostly analytic
topics, described in more detail in the section "Contents of Part II of Volume
Two" below.
(1) Proofs are given of the weak maximum principles for scalars and
systems on both compact and complete noncompact manifolds. In
the noncompact case we have strived to present complete proofs
of general results which are readily applicable. A proof is given
of the strong maximum principle for systems with an emphasis
on the evolution of the curvature operator under the Ricci flow.
The application of the maximum principle to the Ricci flow was
pioneered by Hamilton.
(2) We present the solution of Bohm and Wilking of the conjecture of
Rauch and Hamilton that closed manifolds, in any dimension, with
positive curvature operator are diffeomorphic to spherical space
forms. Bohm and Wilking prove that the normalized Ricci flow
evolves Riemannian metrics on closed manifolds with 2-positive
curvature operator to constant positive sectional curvature met-
(3) We discuss the following two topics: (i) nonnegative curvature
conditions which are not preserved under the Ricci flow and (ii)
Bando's result that solutions of the Ricci flow on closed manifolds
are real analytic.
(4) We present Shi's local derivative of curvature estimates (including
all higher derivatives) based on the Bernstein technique. We also
present a refinement, due to one of the authors, where bounds
on some higher derivatives of the initial metric are assumed and
consequently improved bounds of all higher derivatives are obtained
in space and time.
(5) We discuss the differential Harnack estimates of Li-Yau-Hamilton-
type—giving a detailed proof of Hamilton's matrix estimate for
complete solutions of the Ricci flow with bounded nonnegative cur-
vature operator (this includes the noncompact case). An applica-
tion is Hamilton's result that eternal solutions are steady gradient
Ricci solitons. We also discuss a variant on Hamilton's proof of the
matrix Harnack estimate.
Partly based on Bohm and Wilking's work is the recent result of Brendle and Schoen
[48] proving that positively |-pinched closed manifolds are diffeomorphic to spherical
space forms. Related to this, Brendle and Schoen [48] and Nguyen [376] independently
proved that the condition of positive isotropic curvature is preserved in all dimensions (a
result previously known only in dimension 4 by the work of Hamilton).
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