By Artstein-Avidan S.
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Extra resources for A Bernstein-Chernoff deviation inequality, and geometric properties of random families of operators
We need to detect the global geometry of the manifold in order to get estimates, hence existence. 3 An Existence Result: Four Dimensions To ﬁnish our discussion of the σk -Yamabe problem, we want to sketch its solution in four dimensions. This case is special because, in 4-d, the integral σ2 (A) dV is conformally invariant. That is, if g = e−2u g, then σ2 (A) dV = σ2 (A) dV. You can check this by hand using the formulas above along with the fact that dV = e−4u dV. Eventually, you will ﬁnd that σ2 (A) dV = σ2 (A) dV + (divergence terms).
7 it follows that Q dV ≤ 8π 2 , (59) with equality if and only if (M 4 , g) is conformal to the round sphere. 3. If (M 4 , g) has positive scalar curvature, then an extremal for FL exists. 2. It is easy to construct examples of 4-manifold–necessarily with negative scalar curvature–for which Q dV >> 8π 2 . Thus, the existence theory for the functional determinant is quite incomplete. This shows another parallel with the σk -Yamabe problem (and contrast with the classical Yamabe problem): the case of negative curvature is much more diﬃcult than the positive case.
3. The following reproduction property of GΩ holds: GΩ (t, x; τ, ξ) = GΩ (t, x; s, y) GΩ (s, y; τ, ξ)dy, Ω for every t > s > τ and x, ξ ∈ Ω. Proof. We ﬁx τ, ξ, s as above and we set ϕ = GΩ (·, s; τ, ξ). 3-(i). 3-(iii) and obtain that the function x ∈ Ω, t > s, GΩ (t, x; s, y) ϕ(y)dy, u (t, x) = Ω satisﬁes u ∈ C ∞ ((s, ∞) × Ω) ∩ C([s, ∞) × Ω), Hu = 0 in (s, ∞) × Ω, u = 0 in [s, ∞) × ∂Ω, u(s, ·) = ϕ in Ω. It is now suﬃcient to observe that GΩ (·; τ, ξ) has the same properties and to use the Picone maximum principle for H We now specialize the study of Green functions to cylinders based on regular domains which approximate metric balls.
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