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Kazuo Akutagawa

2012-03-19
11:10:00 - 12:00:00

308 , Mathematics Research Center Building (ori. New Math. Bldg.)

This is a joint work with Julie Rowlett (Max Planck Institute, Germany). In this talk, we consider the Yamabe problem on a given compact $n$-manifold $(n \geq 3)$ with conic singularities and equipped with a ``conic'' metric $g$. For the Yamabe constant $Y(M, [g])$, a refined Aubin's inequality holds, that is, there exists a universal positive constant ${\bf Y}$ (corresponding to the asymptotic structure of $g$ near conic singularities) such that $$ Y(M, [g]) \leq {\bf Y}. $$ If $Y(M, [g]) < {\bf Y}$, we show the existence of a Yamabe metric in $[g]$ (see Akutagawa-Botvinnik [GAFA 13 (2003), 259--333] for the case of a ``rigid conic'' metric). However, the important problem here is whether a ``conic'' Yamabe metric exists or not. There is a necessary condition for $g$ to the existence of a conic Yamabe metric in $[g]$. Under this condition, if $Y(M, [g]) \leq 0$, we prove the existence of a conic Yamabe metric in $[g]$. This is an extension of the result due to Jeffres-Rowlett [Math. Res. Lett. 17 (2010), 441--465].

For material related to this talk, click here.