## Tuesday Seminar on Topology

Seminar information archive ～09/12｜Next seminar｜Future seminars 09/13～

Date, time & place | Tuesday 17:00 - 18:30 056Room #056 (Graduate School of Math. Sci. Bldg.) |
---|---|

Organizer(s) | KAWAZUMI Nariya, KITAYAMA Takahiro, SAKASAI Takuya |

### 2006/11/28

17:00-18:00 Room #056 (Graduate School of Math. Sci. Bldg.)

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 $Y(M, C) \\leq 0$, 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 $(M_{\\infty}, C_{\\infty})$ of its {\\it infinite} conformal coverings,

under a certain topological condition on the relation between $\\pi_1(M)$ and $\\pi_1(M_{\\infty})$.

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 $Y(M, C) \\leq 0$, 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 $(M_{\\infty}, C_{\\infty})$ of its {\\it infinite} conformal coverings,

under a certain topological condition on the relation between $\\pi_1(M)$ and $\\pi_1(M_{\\infty})$.

For the proof of this, we aslo establish a version of positive mass theorem

for a specific class of asymptotically flat manifolds with singularities.