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Tuesday Seminar on Topology
Seminar information archive ～06/25｜Next seminar｜Future seminars 06/26～
| Date, time & place | Tuesday 17:00 - 18:30 056Room #056 (Graduate School of Math. Sci. Bldg.) |
|---|---|
| Organizer(s) | HABIRO Kazuo, KAWAZUMI Nariya, KITAYAMA Takahiro, SAKASAI Takuya |
2006/11/28
17:00-18:00 Room #056 (Graduate School of Math. Sci. Bldg.)
芥川 和雄 (東京理科大学理工学部)
The Yamabe constants of infinite coverings and a positive mass theorem
芥川 和雄 (東京理科大学理工学部)
The Yamabe constants of infinite coverings and a positive mass theorem
[ Abstract ]
The {\\it Yamabe constant} Y(M,C) of a given closed conformal manifold
(M,C) is defined by the infimum of
the normalized total-scalar-curavarure functional E
among all metrics in C.
The study of the second variation of this functional E led O.Kobayashi and Schoen
to independently introduce a natural differential-topological invariant Y(M),
which is obtained by taking the supremum of Y(M,C) over the space of all conformal classes.
This invariant Y(M) is called the {\\it Yamabe invariant} of M.
For the study of the Yamabe invariant,
the relationship between Y(M,C) and those of its conformal coverings
is important, the case when Y(M,C)>0 particularly.
When Y(M,C)leq0, by the uniqueness of unit-volume constant scalar curvature metrics in C,
the desired relation is clear.
When Y(M,C)>0, such a uniqueness does not hold.
However, Aubin proved that Y(M,C) is strictly less than
the Yamabe constant of any of its non-trivial {\\it finite} conformal coverings,
called {\\it Aubin's Lemma}.
In this talk, we generalize this lemma to the one for the Yamabe constant of
any (Minfty,Cinfty) of its {\\it infinite} conformal coverings,
under a certain topological condition on the relation between pi1(M) and pi1(Minfty).
For the proof of this, we aslo establish a version of positive mass theorem
for a specific class of asymptotically flat manifolds with singularities.
The {\\it Yamabe constant} Y(M,C) of a given closed conformal manifold
(M,C) is defined by the infimum of
the normalized total-scalar-curavarure functional E
among all metrics in C.
The study of the second variation of this functional E led O.Kobayashi and Schoen
to independently introduce a natural differential-topological invariant Y(M),
which is obtained by taking the supremum of Y(M,C) over the space of all conformal classes.
This invariant Y(M) is called the {\\it Yamabe invariant} of M.
For the study of the Yamabe invariant,
the relationship between Y(M,C) and those of its conformal coverings
is important, the case when Y(M,C)>0 particularly.
When Y(M,C)leq0, by the uniqueness of unit-volume constant scalar curvature metrics in C,
the desired relation is clear.
When Y(M,C)>0, such a uniqueness does not hold.
However, Aubin proved that Y(M,C) is strictly less than
the Yamabe constant of any of its non-trivial {\\it finite} conformal coverings,
called {\\it Aubin's Lemma}.
In this talk, we generalize this lemma to the one for the Yamabe constant of
any (Minfty,Cinfty) of its {\\it infinite} conformal coverings,
under a certain topological condition on the relation between pi1(M) and pi1(Minfty).
For the proof of this, we aslo establish a version of positive mass theorem
for a specific class of asymptotically flat manifolds with singularities.
