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Chern-Gauss-Bonnet formula for singular Yamabe metrics in dimension four. (arXiv:1902.01562v1 [math.DG])
来源于:arXiv
We derive a formula of Chern-Gauss-Bonnet type for the Euler characteristic
of a four dimensional manifold-with-boundary in terms of the geometry of the
Loewner-Nirenberg singular Yamabe metric in a prescribed conformal class. The
formula involves the renormalized volume and a boundary integral. It is shown
that if the boundary is umbilic, then the sum of the renormalized volume and
the boundary integral is a conformal invariant. Analogous results are proved
for asymptotically hyperbolic metrics in dimension four for which the second
elementary symmetric function of the eigenvalues of the Schouten tensor is
constant. Extensions and generalizations of these results are discussed.
Finally, a general result is proved identifying the infinitesimal anomaly of
the renormalized volume of an asymptotically hyperbolic metric in terms of its
renormalized volume coefficients, and used to outline alternate proofs of the
conformal invariance of the renormalized volume plus boundary integral. 查看全文>>