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.

Show description

Read Online or Download Non-Linear Elliptic Equations in Conformal Geometry PDF

Similar geometry books

Conceptual Spaces: The Geometry of Thought

Inside cognitive technological know-how, techniques at the moment dominate the matter of modeling representations. The symbolic procedure perspectives cognition as computation regarding symbolic manipulation. Connectionism, a unique case of associationism, types institutions utilizing synthetic neuron networks. Peter Gardenfors bargains his conception of conceptual representations as a bridge among the symbolic and connectionist ways.

The Art of the Intelligible (survey of mathematics in its conceptual development)

A compact survey, on the straight forward point, of a few of the main very important thoughts of arithmetic. realization is paid to their technical beneficial properties, ancient improvement and broader philosophical value. all of the a variety of branches of arithmetic is mentioned individually, yet their interdependence is emphasized all through.

Der Goldene Schnitt

Der Goldene Schnitt tritt seit der Antike in vielen Bereichen der Geometrie, Architektur, Musik, Kunst sowie der Philosophie auf, aber er erscheint auch in neueren Gebieten der Technik und der Fraktale. Dabei ist der Goldene Schnitt kein isoliertes Phänomen, sondern in vielen Fällen das erste und somit einfachste nichttriviale Beispiel im Rahmen weiterführender Verallgemeinerungen.

Complex Manifolds and Hyperbolic Geometry: II Iberoamerican Congress on Geometry, January 4-9, 2001, Cimat, Guanajuato, Mexico

This quantity derives from the second one Iberoamerican Congress on Geometry, held in 2001 in Mexico on the Centro de Investigacion en Matematicas A. C. , an across the world famous software of analysis in natural arithmetic. The convention themes have been selected with an eye fixed towards the presentation of latest equipment, fresh effects, and the production of extra interconnections among different study teams operating in advanced manifolds and hyperbolic geometry.

Extra resources for Non-Linear Elliptic Equations in Conformal Geometry

Sample text

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.

Download PDF sample

Rated 4.61 of 5 – based on 25 votes