Chapter 29. Compact 2-Dimensional Ancient Solutions
Picture paragraphs unloaded , wise words being quoted.- From "Hail Mary" by Tupac Shakur
Recall t h at, besides t h e round shrinking metric, there is another ancient solu-
tion on the 2-sphere 52 , called the King-Rosenau solution or sausage model.
We discussed t his solution in Subsection 3.3 of Chapter 2 in Volume One. In this
chapter we shall show that the King- Rosenau solution is the only nonround an-
cient solution on 52. Thus, by the results on noncompact ancient solutions stated
in the previous chapter, we have a complete classification of 2-dimensional complete
a ncient solutions.
One of the main ideas of the proof is the study of a scalar invariant, which we
call Q(x, t). We show that Q is a subsolution to the heat equation and moreovert hat maxx Q(x, t) ---+ 0 as t---+ - oo. This implies t hat Q = 0. A classification result
t hen yields that g(t) is the King- Rosenau solut ion. Geometric methods, such as
the application of isoperimetric monotonicity, are also important to the proof.1. Statement of the classification result and outline of its proof
In t his section we state the main theorem and we give an outline of t he proof
of the t heorem.1.1. Statement of the main theorem.
The main theorem is the following result of P. Daskalopoulos , R. Hamilton, and
N. Sesum. Their proof involves an eclectic combination of monotonicity formulas,
a priori estimates, and geometric arguments, which we shall discuss.
THEOREM 29 .1 (Classification of a ncient solutions on the 2-sphere). Any com-
pact simply-connected ancient solution (M^2 ,g(t)) to the Riccifiow must be either
a round shrinking 2-sphere or the rotationally symmetric King- Rosenau solution.
Note that any compact simply-connected surface is diffeomorphic to the 2-
sphere.
1.2. Outline of the proof of the main theorem.
The proof of the main theorem involves a combination of fine and coarse es-
timates, where the words "fine" and "course" are used in the sense of Chapter 14
in Part II. Roughly, we h ave the following utilities. Coarse estimates are used to
prove convergence to a limit, in our case, backward limits as time tends to minus
infinity. Fine estimates, usually called monotonicity formulas, are used to classify
t hese limits. Fine estimates are also used to obtain stronger control of geometric
quantities for the solution g(t).
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