52 1. RICCI SOLITONS
PROBLEM 1.88. Are there any 3-dimensional steady gradient solitons
besides a flat solution, the Bryant soliton, and a quotient of the product
of the cigar and~? This is equivalent to asking if a steady gradient Ricci
soliton with n = 3 and sect (g ( t)) > 0 is isometric to a Bryant soliton.
10.3. Gradient Ricci solitons in higher dimensions. Here we as-
sume n 2: 3. Note that any expanding or steady Ricci soliton on a closed
n-dimensional manifold is Einstein.
PROBLEM 1.89. Is a shrinking gradient Ricci soliton with n 2: 4 and
Rm (g (t)) > 0 compact?
PROBLEM 1.90. Is a compact shrinking gradient Ricci soliton, n 2: 4, andRm (g (t)) > 0 isometric to a shrinking constant positive sectional curvature
solution?^16
PROBLEM 1.91. Are there n-dimensional steadies with positive curvature
operator besides the Bryant soliton?
PROBLEM 1.92. Are there any n-dimensional expanders with positive
curvature operator besides the rotationally symmetric one?
' PROBLEM 1.93. Does there exist an expander with n 2: 3 and positivelypinched Ricci curvature, i.e., Rij 2: sRgij for some E > 0, where R > O? By
Theorem 1.85, such an expander has R decaying exponentially.
11. Notes and commentary
Section 1. The first occurrence of the notion of a Ricci soliton in the
literature is in Friedan [145], where non-Einstein Ricci solitons are called
quasi-Einstein metrics. See Besse [27] for a comprehensive treatment of
Einstein manifolds.
The Gaussian soliton first appeared in §2.1 of Perelman [297].
In the case of surfaces we have encountered Ricci solitons in Chapter
5 of Volume One. There, solitons motivated several aspects of Hamilton's
original proof of convergence of the Ricci fl.ow for surfaces with R (go) > 0,
including the scalar curvature, Harnack, and entropy estimates, as well as
the estimate which shows the metric approaches a soliton. See in Volume
One, Corollary 5.17 on p. 115, Proposition 5.57 on p. 145, Proposition 5.39
on p. 134, and Corollary 5.35 on p. 130.
Expanding solitons are of interest because they constitute borderline
cases for Hamilton's Harnack inequality. (Expanding solitons also model
the formation of certain Type III singularities.) In fact, as we mentioned in
Section 2 of this chapter and will discuss in more detail in Part II, the con-
sideration of quantities which vanish on solitons led to the discovery of the
Harnack quadratic. Steady solitons, where the isometry class of the metric
is independerrt oft E ( -oo, oo), occur as limits of Type II singularities.
·^16 The answer to this problem is 'yes' and follows from the recent work of Bohm and
Wilking [30] (see Part II of this volume).