# Download Communications In Mathematical Physics - Volume 267 by M. Aizenman (Chief Editor) PDF

By M. Aizenman (Chief Editor)

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Extra info for Communications In Mathematical Physics - Volume 267

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27) in [Ga]. The next proposition asserts that the desired singularities do exist. 4. Suppose that w ∈ M is distinct from all components of z. Let x denote a holomorphic coordinate on a neighborhood of w such that x = 0 is mapped to w and such that the given metric pulls back as |d x|2 + O |x|2 . 6) has the form 2 1 ∂ + x 2 x ∂w t (·) (z,F) − t(·) (z,F) |w (d x)2 + . . 7) where the three dots indicate terms that are bounded as x → 0. 6) has no singularity at x = 0. 6) is jointly holomorphic with respect to both w and w .

13) as z 1 and z 2 converge to z 0 . 14) here the notation uses Eˆ 0 to denote the smallest of the numbers in the set Eˆ v , and φ0 to denote the corresponding version of φv . 14) can vanish. This is the case, for example, when zero is the only constant function in the C ∞ (P) image of φ(W0 ) × V3 × . . × V N via the homomorphism that comes by restricting to the full diagonal in × N −1 P. 1) and the function c are both smooth. 14) need not be on the order of distance between z 1 and z 2 . 14) will shrink slower than this distance when the function a has a reasonably mild divergence on approach to the diagonal.

To set the stage, I treat a surface with n punctures as a compact surface, M, together with a set, ϑ, of n distinct points in M, these are the missing points in the original unpunctured surface. 1) to be a positive multiple of a certain Green’s function for the Laplacian on M. To be precise here, I use a(z, ˆ z) to denote the value of this Green’s function at points z = z in M − ϑ. 1) here is the negative definite Laplacian, δu is the Dirac delta function with mass 1 at the point u ∈ M, and the area of M is computed using the metric’s area measure.