Uniqueness of compact ancient solutions to the higher-dimensional Ricci flow

A The Bryant soliton

In [16] Bryant showed that up to constant multiples, there is only one complete, steady, rotationally symmetric soliton in dimension three that is not flat. It has positive sectional curvature. The maximum scalar curvature is equal to 1, and is attained at the center of rotation. The complete soliton can be written in the form g = d ⁢ z ⊗ d ⁢ z + B ⁢ ( z ) 2 ⁢ g S n - 1 , where z is the distance from the center of rotation. For large z, the metric has the following asymptotics: the aperature B ⁢ ( z ) has leading order term 2 ⁢ ( n - 2 ) ⁢ z , the orbital sectional curvature K orb has leading order term 1 2 ⁢ ( n - 2 ) ⁢ z , and the radial sectional curvature K rad has leading order term 1 4 ⁢ z 2 .

Sometimes it is more convenient to write the metric in the form Φ ⁢ ( r ) - 1 ⁢ d ⁢ r 2 + r 2 ⁢ g S n - 1 , where the function Φ ⁢ ( r ) is defined by

The function Φ ⁢ ( r ) is known to satisfy the equation

The orbital and radial sectional curvatures are given by

It is known that Φ ⁢ ( r ) has the following asymptotics. Near r = 0 , Φ is smooth and has the asymptotic expansion

where b 0 is a negative constant (since the curvature is positive). As r → ∞ , Φ is smooth and has the asymptotic expansion

where c 0 is a positive constant.

We will next find (for the convenience of the reader) the exact values of the constants b 0 and c 0 in the above asymptotics for the Bryant soliton of maximal scalar curvature one.

Recall that the scalar curvature is given by R = ( n - 1 ) ⁢ ( n - 2 ) ⁢ K orb + 2 ⁢ ( n - 1 ) ⁢ K rad . The maximal scalar curvature is attained at z = 0 , at which point K orb = K rad . The maximal scalar curvature being equal to 1 is equivalent to K orb = K rad = 1 n ⁢ ( n - 1 ) at z = 0 . On the other hand, the asymptotic expansion of Φ ⁢ ( r ) gives

Consequently, b 0 = - 1 n ⁢ ( n - 1 ) .

Bryant’s asymptotics imply that for z sufficiently large, the aperture satisfies

implying that 2 ⁢ ( n - 2 ) ⁢ z = ( 1 + o ⁢ ( 1 ) ) ⁢ r 2 . The radial sectional curvature satisfies

for r large. On the other hand, the asymptotic expansion of Φ ⁢ ( r ) implies

for r large. Comparing the two formulae, we conclude that c 0 = ( n - 2 ) 2 .

Summarizing the above discussion we conclude the following asymptotics for the Bryant soliton with maximal scalar curvature equal to one: