By Sun-Yung Alice Chang

Non-linear elliptic partial differential equations are a huge device within the research of Riemannian metrics in differential geometry, specifically for difficulties in regards to the conformal swap of metrics in Riemannian geometry. in recent times the position performed via the second one order semi-linear elliptic equations within the examine of Gaussian curvature and scalar curvature has been prolonged to a kin of absolutely non-linear elliptic equations linked to different symmetric capabilities of the Ricci tensor. A case of specific curiosity is the second one symmetric functionality of the Ricci tensor in measurement 4 heavily on the topic of the Pfaffian. In those lectures, ranging from the historical past fabric, the writer stories the matter of prescribing Gaussian curvature on compact surfaces. She then develops the analytic instruments (e.g., greater order conformal invariant operators, Sobolev inequalities, blow-up research) so one can remedy a completely nonlinear equation in prescribing the Chern-Gauss-Bonnet integrand on compact manifolds of measurement 4. the fabric is acceptable for graduate scholars and learn mathematicians attracted to geometry, topology, and differential equations. allotted in the Americas via the yank Mathematical Society.

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**Extra resources for Non-Linear Elliptic Equations in Conformal Geometry**

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7. 10 Suppose there is w ∈ C ∞ (M ), such that (i) Rgw > 0 (ii) σ2 (Agw ) = f > 0 for some f ∈ C ∞ (M ). If (M 4 , g) is not conformally equivalent to (S 4 , gc ), then there exists a constant C1 = C1 ||f ||C 1 , min f (·) −1 M ,g such that ||w||L∞ ≤ C1 . We have to exclude the case of conformal equivalence to (S 4 , gc ), since, for instance, on (S 4 , gc ), if e2w gc = φ∗ (gc ), then one has in Euclidean coordinates, wλ (x) = log λ2 2λ + |x − x0 |2 and σ2 (Agwλ ) ≡ 6 for all λ > 0, but lim ||wλ ||L∞ = ∞.

42 6. 5 If γ2 , γ3 < 0, and if Q0 dv0 < 8π 2 − kg0 = M γ1 γ2 |W0 |20 dv0 , M or equivalently, if |W0 |20 dv0 + γ2 kd : = γ1 M Q0 dv0 > γ2 8π 2 , M then there is wd ∈ C ∞ (M ) such that FA [wd ] = sup Sc1 ,c2 (A) FA [·], and in terms of the metric g = gwd = e2wd g0 , kd . 18) −1 ≤ 0, −1 ≤ 0, and where 1 γ2 + γ3 12 −1 . 6. 7 (Gursky [54]) If Y (M 4 , g0 ) > 0, and if kg0 ≥ 0, then the Paneitz operator (P4 )g0 = P0 is positive, with λ1 (P0 ) = 0 and ker(P0 ) = {R}. Remarks 1. 10). This implies that ker(P ) is a conformally invariant set.

11) with b1 (w) = −6∆20 w − 2∆0 wRg0 − 2 ∇0 w, ∇0 Rg0 b2 (w) = b3 (w) = g0 , −6∆0 (|∇0 w|2g0 ) + 12(∆0 w)2 + 12 ∇0 w, ∇0 ∆0 w g0 , 12∆0 w|∇0 w|2g0 + 12 ∇0 w, ∇0 (|∇0 w|2g0 ) g0 , where each bi (w) is homogeneous of degree i in w. Remarks 1. , 12Qg := Rg2 − 3|Ricg |2g − ∆g Rg . In general, there are only four possible metric invariants of order 4, namely Rg2 , |Ricg |2g , |Wg |2g and ∆g Rg , a linear combination of which furnishes Bk (Ag ). Apart from |Wg |2g these are not pointwise conformal invariants, only the integral of them is.