A geometrical mass and its extremal properties for metrics on \(S^2\). (English) Zbl 1144.53055
The paper provides some \(2\)-dimensional analogon to the notion of “positive mass” as it enters conformal geometry in dimensions \(3,4,5\) via R. Schoen’s solution of the Yamabe problem (using the positive mass theorem) [J. Differ. Geom. 20, 479–495 (1984; Zbl 0576.53028)]. The geometric quantity defined on closed \(2\)-manifolds is the “geometric mass” given by \(GM(x):=m(x)-{1\over2\pi}\Delta^{-1}K(x)\), where \(K\) is the scalar curvature of the manifold and \(m(x)=\lim_{y\to x} [G(x,y)+{1\over2\pi}\log d(x,y)]\), called “Robin’s mass” by the author, is the manifold’s Green’s function at \(x\) reduced by its obvious singular asymptotics.
It turns out that \(GM(x)\) is actually independent of \(x\) if the manifold is a sphere, while on manifolds of higher genus it usually does depend on \(x\). In the spherical case, \(GM\) is minimized (volumes being fixed) at the standard round metric.
It turns out that \(GM(x)\) is actually independent of \(x\) if the manifold is a sphere, while on manifolds of higher genus it usually does depend on \(x\). In the spherical case, \(GM\) is minimized (volumes being fixed) at the standard round metric.
Reviewer: Andreas Gastel (Erlangen)
MSC:
| 53C21 | Methods of global Riemannian geometry, including PDE methods; curvature restrictions |
| 35J60 | Nonlinear elliptic equations |
| 53A30 | Conformal differential geometry (MSC2010) |
Citations:
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