We show how to view the equations for a cohomogeneity one Ricci soliton as a Hamiltonian system with a constraint. We investigate conserved quantities and superpotentials and use this to find some explicit formulae for Ricci solitons not of Kähler type in five dimensions.

1.
Alexakis
,
S.
,
Chen
,
D. Z.
, and
Fournodvalos
,
G.
, “
Singular Ricci solitons and their stability under the Ricci flow
,”
Comm. Partial Diff. Eqs.
40
(
12
),
2123
2172
(
2015
).
2.
Bernstein
,
J.
and
Mettler
,
T.
, “
Two-dimensional gradient Ricci solitons revisited
,”
Int. Math. Res. Not. IMRN
2015
(
1
),
78
98
.
3.
Betancourt de la Parra
,
A.
, “
Painlevé analysis of the Bryant soliton
,” e-print arXiv:1310.7254 (math.DG).
4.
Buzano
,
M.
,
Dancer
,
A. S.
,
Gallaugher
,
M.
, and
Wang
,
M.
, “
A family of steady Ricci solitons and Ricci-flat metrics
,”
Comm. Anal. Geom.
23
(
3
),
611
638
(
2015
).
5.
Chow
,
B.
,
Chu
,
S. C.
,
Glickenstein
,
D.
,
Guenther
,
C.
,
Isenberg
,
J.
,
Ivey
,
T.
,
Knopf
,
D.
,
Lu
,
P.
,
Luo
,
F.
, and
Ni
,
L.
, “
The Ricci flow: Techniques and applications, I: Geometric aspects
,” in
Mathematical Surveys and Monographs
(
American Mathematical Society
,
2007
), Vol.
135
.
6.
Dancer
,
A.
,
Hall
,
S.
, and
Wang
,
M.
, “
Cohomogeneity one shrinking Ricci solitons: An analytic and numerical study
,”
Asian J. Math.
17
(
1
),
33
61
(
2013
).
7.
Dancer
,
A.
and
Wang
,
M.
, “
The cohomogeneity one Einstein equations from the Hamiltonian viewpoint
,”
J. Reine Angew. Math.
524
,
97
128
(
2000
).
8.
Dancer
,
A.
and
Wang
,
M.
, “
Superpotentials for the cohomogeneity one Einstein equations
,”
Commun. Math. Phys.
260
,
75
115
(
2005
).
9.
Dancer
,
A.
and
Wang
,
M.
, “
Classification of superpotentials
,”
Comm. Math. Phys.
284
,
583
647
(
2008
).
10.
Dancer
,
A.
and
Wang
,
M.
, “
New examples of non-Kähler Ricci solitons
,”
Math. Res. Lett.
16
,
349
363
(
2009
).
11.
Dancer
,
A.
and
Wang
,
M.
, “
Non-Kähler expanding Ricci solitons
,”
Int. Math. Res. Not.
2009
(
6
),
1107
1133
.
12.
Dancer
,
A.
and
Wang
,
M.
, “
On Ricci solitons of cohomogeneity one
,”
Ann. Global Anal. Geom.
39
,
259
292
(
2011
).
13.
Dancer
,
A.
and
Wang
,
M.
, “
Classifying superpotentials: Three summands case
,”
J. Geom. Phys.
61
,
675
692
(
2011
).
14.
Goriely
,
A.
, “
Integrability and nonintegrability of dynamical systems
,” in
Advanced Series in Nonlinear Dynamics
(
World Scientific
,
2001
), Vol.
19
.
15.
Hamilton
,
R. S.
, “
The formation of singularities in the Ricci flow
,”
Surv. Differ. Geom.
2
,
7
136
(
1995
).
16.
Ivey
,
T.
, “
New examples of complete Ricci solitons
,”
Proc. AMS
122
,
241
245
(
1994
).
17.
Perelman
,
G.
, “
The entropy formula for the Ricci flow and its geometric applications
,” e-print arXiv:math.DG/0211159.
18.
Prelle
,
M. J.
and
Singer
,
M. F.
, “
Elementary first integrals of differential equations
,”
Trans. Am. Math. Soc.
279
,
215
229
(
1983
).
You do not currently have access to this content.