By Sun-Yung Alice Chang
Non-linear elliptic partial differential equations are a big device within the examine of Riemannian metrics in differential geometry, particularly for difficulties in regards to the conformal swap of metrics in Riemannian geometry. lately the position performed via the second one order semi-linear elliptic equations within the learn of Gaussian curvature and scalar curvature has been prolonged to a relatives of absolutely non-linear elliptic equations linked to different symmetric features of the Ricci tensor. A case of specific curiosity is the second one symmetric functionality of the Ricci tensor in measurement 4 heavily regarding the Pfaffian. In those lectures, ranging from the history fabric, the writer experiences the matter of prescribing Gaussian curvature on compact surfaces. She then develops the analytic instruments (e.g., larger order conformal invariant operators, Sobolev inequalities, blow-up research) to be able to resolve a completely nonlinear equation in prescribing the Chern-Gauss-Bonnet integrand on compact manifolds of measurement 4. the cloth is appropriate for graduate scholars and learn mathematicians attracted to geometry, topology, and differential equations. allotted in the Americas by way of the yankee Mathematical Society.
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4. , in terms of the corresponding metric g : = gwd = e2wd g0 , − 1 1 kd γ2 + γ3 ∆R = −γ1 |W |2 − γ2 σ2 + , 12 2 vol(M, g) where kd : = γ1 M |W0 |20 dv0 + γ2 M Q0 dv0 . 1 If one chooses γ2 = 1, γ3 = 24 (3δ − 2), δ > 0, and finally γ1 , such that kd = 0, then the Euler–Lagrange equations for the functional F δ [w] : = γ1 I[w] + II[w] + 1 (3δ − 2)III[w] 24 read as (in terms of g) δ∆R = 8γ1 |W |2 + 4σ2 , (∗)δ or equivalently, (for σ2 = σ2 (Ag ) as in Chapter 7) σ2 (Ag ) = δ ∆R − 2γ1 |W |2 . 8 to the solution of the equation (∗)δ .
21), the integrand vanishes at zero, if Re(s) is sufficiently large. 1 + · · · , hence The last integral is holomorphic in s. In addition, Γ(s) = 1s − s+1 d s2 s = s2 − + · · · , in particular Γ(s) s+1 ds |s=0 s Γ(s) = 0. 26). ϕdvgu , M ϕdvgu M ∞ k=4 ak (ϕ, ∆u ) k + 2s − 2 ϕdvgu M 24 3. 9 (Osgood–Phillips–Sarnak [73], [74]) Isospectral metrics on a closed compact surface (M 2 , g) are C ∞ -compact modulo the isometry class. The basic idea in the proof is that on a compact closed surface (M 2 , g0 ), each heat coefficient a2i for each i ≥ 2 controls the Sobolev W i,2 -norm modulo some lower order W l,2 -norm for l < i of the conformal factor w for the metric gw = e2w g0 .
10 in Chapter 10. 9. 1 [23] On (M 4 , g0 ) closed, compact, assume (i) Y (M 4 , g0 ) > 0, (ii) M σ2 (A0 ) dv0 > 0; then there is f ∈ C ∞ (M ), f > 0, and w ∈ C ∞ (M ), such that σ2 (Agw ) = f. Remark. Conditions (i) and (ii) are invariant under conformal change of the metric, so sometimes we will simply write Y (M ) or M σ2 (A) dv without specifying the metric. Outline of the proof. We will use a continuity method on the “regularized equation” (in terms of g = e2w g0 ) δ∆R = 8γ1 |W |2 + 4σ2 (A).