On a fourth order PDE and applications to problems in conformal - PowerPoint PPT Presentation

On a fourth order PDE and applications to problems in conformal geometry Sun-Yung Alice Chang Princeton University Modern Aspects of complex analysis and its application In honor of John Garnett and Donald Marshall August 19-23, 2019 Sun-Yung Alice Chang Princeton University On a fourth order PDE and applications to problems in conformal geometry Modern Aspects of complex analysis and its application In honor of John Garnett and Donald August 19-23, 2019 1 / 27

‚ Study of : “ : p´ ∆ q 2 ` ....... P p n q 4 operator defined on an n-dimensional manifold, which on flat domains in R n is the bi-Laplace operator. It turns out this operator has played special roles in the study of curvatures problems on manifolds. ‚ When n=4, we denote P 4 : “ : P p 4 q 4 . ‚ In conformal geometry, one study properties which are invariant under ”conformal change” of metrics; i.e. On p M n , g q , one considers metric g “ ρ g for some positive function ρ defined on M. ˆ Sun-Yung Alice Chang Princeton University On a fourth order PDE and applications to problems in conformal geometry Modern Aspects of complex analysis and its application In honor of John Garnett and Donald August 19-23, 2019 2 / 27

Outline of talk Outline of talk 1. Introduction: 2nd order operators, P 2 operator, Yamabe problem. 2. P p n q on n-manifolds when n ě 5, a result of Gursky-Malchiodi. 4 3. P 4 on compact 4-manifolds without boundary; Branson’s Q curvature, some PDE aspect. 4. P 4 on compact 4-manifolds with boundary, non-local operator P 3 on the boundary. 5. Some open question and applications. Sun-Yung Alice Chang Princeton University On a fourth order PDE and applications to problems in conformal geometry Modern Aspects of complex analysis and its application In honor of John Garnett and Donald August 19-23, 2019 3 / 27

S1. Second order operator on p M n , g q ‚ On p M n , g q , the Laplace Beltrami operator ∆ g is defined as 1 a | g | g ij B j q ∆ g “ B i p a | g | ‚ On surfaces p M 2 , g q , the operator ∆ g enjoys a ”conformal invariant property; denote g w “ e 2 w g , then ∆ g w “ e ´ 2 w ∆ g ‚ On p M 2 , g q , ∆ g ”prescribe the Gaussian curvature K g , i.e. ´ ∆ g w ` K g “ K g w e 2 w . Sun-Yung Alice Chang Princeton University On a fourth order PDE and applications to problems in conformal geometry Modern Aspects of complex analysis and its application In honor of John Garnett and Donald August 19-23, 2019 4 / 27

S1. Second order operator on p M n , g q On p M n , g q , n ě 2, the conformal Laplace operator L g ‚ n ´ 2 L g “ ´ ∆ g ` c n R g where c n “ 4 p n ´ 1 q , and R g denotes the scalar curvature of the metric g . 4 n ´ 2 g , u ą 0. ‚ Under conformal change of metrics ˆ g “ u n ` 2 L g u “ c n ˆ n ´ 2 . R u The famous Yamabe problem is to solve above equation for ˆ R a constant c ; settled by Yamabe, Trudinger, Aubin and Schoen, ’60-’84. ş M R ˆ g dv ˆ g Y p M , g q : “ inf . n ´ 2 g Pr g s ˆ p vol ˆ g q n Sun-Yung Alice Chang Princeton University On a fourth order PDE and applications to problems in conformal geometry Modern Aspects of complex analysis and its application In honor of John Garnett and Donald August 19-23, 2019 5 / 27

S2. Fourth order operator P p n q and Q p n q curvature on 4 4 p M n , g q n ą 4 Recall on p M n , g q , n ą 2, the second order conformal Laplacian ‚ n ´ 2 operator L “ ´ ∆ ` 4 p n ´ 1 q R , we have, g p ϕ q “ u ´ n ` 2 4 n ´ 2 L g p u ϕ q for all ϕ P C 8 p M n q , w here ˆ n ´ 2 g . L ˆ g “ u ‚ Paneitz operator in 1983 on p M n , g q , n ą 4. “ p´ ∆ q 2 ` δ p a n R g ` b n Ric q d ` n ´ 4 P p n q Q p n q 4 . 4 2 g p ϕ q “ u ´ n ` 4 4 p P p n q n ´ 4 p P p n q 4 q g p u ϕ q for all ϕ P C 8 p M n q , w here ˆ n ´ 4 g . 4 q ˆ g “ u Notice that P p n q 2 Q p n q 4 , so we can read Q p n q from P p n q 4 p 1 q “ n ´ 4 ‚ when 4 4 n ‰ 4. Sun-Yung Alice Chang Princeton University On a fourth order PDE and applications to problems in conformal geometry Modern Aspects of complex analysis and its application In honor of John Garnett and Donald August 19-23, 2019 6 / 27

S2. Some more recent development, Q n 4 on M n , n ě 4 ‚ In the last few years, there has been lots of progress made on the study of the 4-th order Paneitz operator P 4 and its associated curvatures Q 4 curvature with surprising results. ‚ Theorem (Gursky-Malchiodi 2014) On p M n , g q , n ě 5 suppose R g ě 0, p Q p n q 4 q g ą 0, then P p n q is positive, satisfies the strong maximal principle, 4 (i.e. P 4 u ě 0, then either u ą 0 or u ” 0 on M), and its Green’s function is positive. ‚ (Hang-Yang 2014) In 3d, assuming a metric g is of positive Yamabe class, then there exists a metric in the same conformal class of g with positive Q curvature if and only if the kernel of P p 3 q is trivial and its Green 4 function is negative away from the pole. ‚ Most recent works of Gursky-Hang-Lin (2015), and series of papers by Hang-Yang (2015). Sun-Yung Alice Chang Princeton University On a fourth order PDE and applications to problems in conformal geometry Modern Aspects of complex analysis and its application In honor of John Garnett and Donald August 19-23, 2019 7 / 27

S3. Branson’s Q-curvature on manifold of dimension 4 Branson pointed out that P : “ P 4 4 and Q : “ Q 4 ‚ 4 are well defined. (which we named as Branson’s Q-curvature.) ˆ 2 ˙ P g ϕ “ p´ ∆ q 2 ϕ ` δ 3 Rg ´ 2 Ric d ϕ, 2 Q g “ ´ 1 6∆ R g ` 1 6 p R 2 g ´ 3 | Ric g | 2 q . ‚ P g w ` 2 Q g “ 2 Q g w e 4 w on M 4 , where g w “ e 2 w g . Compared to ´ ∆ g w ` K g “ K g w e 2 w on M 2 . ‚ For examples: On p R 4 , | dx | 2 q , P “ ∆ 2 , On p S 4 , g c q , P “ ∆ 2 ´ 2∆, On p M 4 , g q , g Einstein, P “ p´ ∆ q ˝ p L q . Sun-Yung Alice Chang Princeton University On a fourth order PDE and applications to problems in conformal geometry Modern Aspects of complex analysis and its application In honor of John Garnett and Donald August 19-23, 2019 8 / 27

Q-curvature ‚ Properties of ∆ g on p M 2 , g q 1. ∆ g w “ e ´ 2 w ∆ g . 2. ´ ∆ g w ` K g “ K g w e 2 w , Gauss-Bonnet ż 2 πχ p M q “ K g dv g M Sun-Yung Alice Chang Princeton University On a fourth order PDE and applications to problems in conformal geometry Modern Aspects of complex analysis and its application In honor of John Garnett and Donald August 19-23, 2019 9 / 27

Branson’s Q-curvature ‚ Properties of Paneitz operator on p M 4 , g q 1. P g w “ e ´ 4 w P g 2. P g w ` 2 Q g “ 2 Q g w e 4 w 1 12 p´ ∆ R ` R 2 ´ 3 | Ric | 2 q . Q “ ‚ Gauss-Bonnet-Chern Formula: ż p Q g ` 1 4 π 2 χ p M 4 q “ 8 | W g | 2 q dv , where W denotes the Weyl tensor. ‚ Weyl curvature measures the obstruction to being conformally flat. On p M n , g q , n ě 4. W g ” 0 in a neighborhood of a point if and only if g “ e 2 w | dx | 2 for some function w . Thus on p S n , g c q , W g c ” 0. g w “ e 2 w g , | W g w | “ e ´ 2 w | W g | , thus on 4-manifolds ‚ | W g w | 2 dv g w “ | W g | 2 dv g a pointwise conformal invariant; thus M | W | 2 ş ş g Ñ g dv g is an integral conformal invariant, hence so is M Q g dv g . Sun-Yung Alice Chang Princeton University On a fourth order PDE and applications to problems in conformal geometry Modern Aspects of complex analysis and its application In honor of John Garnett and Donald August 19-23, 2019 10 / 27

Q-curvature -some PDE aspect ‚ P g w ` 2 Q g “ 2 Q g w e 4 w ‚ Variational Functional: ż ż ż M 4 Qe 4 w dv g . F Q p w q “ă P g w , w ą ` 4 M 4 Q g w ´ M 4 Q g dv g log ‚ In this case, W 2 , 2 Ă e L 2 , and Moser’s constant is 32 π 2 . ‚ On p S 4 , g 0 q , ş S 4 Q g 0 dv 0 “ 8 π 2 . ‚ Theorem: Chang-Yang ’95 Q g dv g ă 8 π 2 AND P is positive with KerP “ constants , then F is ş If bounded from below and minimum is achieved by some g w “ e 2 w g with Q g w “ constant . Sun-Yung Alice Chang Princeton University On a fourth order PDE and applications to problems in conformal geometry Modern Aspects of complex analysis and its application In honor of John Garnett and Donald August 19-23, 2019 11 / 27

Q-curvature -some PDE aspect ‚ Recall Yamabe constant ş R g w dv g w Y p M n , g q ” inf , n ´ 2 w vol p g w q n it is a (second order) conformally invariant constant. ‚ Theorem: Gursky, late ’90 Q g dv g ď 8 π 2 , with equality only on ş (a) When Y p M , g q ą 0, then p S 4 , g 0 q . ş (b) If Y p M , g q ą 0 and if Q g dv g ą 0 ù ñ P g ě 0 with KerP “ t constants u . The proof of the result uses the extremal metrics of a generalized Polyakov type formula of log determinant of the conformal Laplace operators. Sun-Yung Alice Chang Princeton University On a fourth order PDE and applications to problems in conformal geometry Modern Aspects of complex analysis and its application In honor of John Garnett and Donald August 19-23, 2019 12 / 27