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

‚ Study of Ppnq

4

:“: p´∆q2 ` .......

• perator defined on an n-dimensional manifold, which on flat domains in

Rn 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 P4 :“: Pp4q

4 .

‚ In conformal geometry, one study properties which are invariant under ”conformal change” of metrics; i.e. On pMn, gq, one considers metric ˆ g “ ρg for some positive function ρ defined on M.

Sun-Yung Alice Chang Princeton University Modern Aspects of complex analysis and its application In honor of John Garnett and Donald On a fourth order PDE and applications to problems in conformal geometry August 19-23, 2019 2 / 27

Outline of talk

Outline of talk

• 1. Introduction: 2nd order operators, P2 operator, Yamabe problem.
• 2. Ppnq

4

• n n-manifolds when n ě 5, a result of Gursky-Malchiodi.
• 3. P4 on compact 4-manifolds without boundary; Branson’s Q

curvature, some PDE aspect.

• 4. P4 on compact 4-manifolds with boundary, non-local operator P3 on

the boundary.

• 5. Some open question and applications.

Sun-Yung Alice Chang Princeton University Modern Aspects of complex analysis and its application In honor of John Garnett and Donald On a fourth order PDE and applications to problems in conformal geometry August 19-23, 2019 3 / 27

• S1. Second order operator on pMn, gq

‚ On pMn, gq, the Laplace Beltrami operator ∆g is defined as ∆g “ 1 a |g| Bip a |g|gijBjq ‚ On surfaces pM2, gq, the operator ∆g enjoys a ”conformal invariant property; denote gw “ e2wg, then ∆gw “ e´2w∆g ‚ On pM2, gq, ∆g ”prescribe the Gaussian curvature Kg, i.e. ´∆g w ` Kg “ Kgw e2w.

Sun-Yung Alice Chang Princeton University Modern Aspects of complex analysis and its application In honor of John Garnett and Donald On a fourth order PDE and applications to problems in conformal geometry August 19-23, 2019 4 / 27

• S1. Second order operator on pMn, gq

‚ On pMn, gq, n ě 2, the conformal Laplace operator Lg Lg “ ´∆g ` cnRg where cn “

n´2 4pn´1q, and Rg denotes the scalar curvature

• f the metric g.

‚ Under conformal change of metrics ˆ g “ u

4 n´2 g, u ą 0.

Lgu “ cn ˆ R u

n`2 n´2 .

The famous Yamabe problem is to solve above equation for ˆ R a constant c; settled by Yamabe, Trudinger, Aubin and Schoen, ’60-’84. Y pM, gq :“ inf

ˆ gPrgs

ş

M Rˆ gdvˆ g

pvol ˆ gq

n´2 n

.

Sun-Yung Alice Chang Princeton University Modern Aspects of complex analysis and its application In honor of John Garnett and Donald On a fourth order PDE and applications to problems in conformal geometry August 19-23, 2019 5 / 27

• S2. Fourth order operator Ppnq

4

and Qpnq

4

curvature on pMn, gq n ą 4

‚ Recall on pMn, gq, n ą 2, the second order conformal Laplacian

• perator L “ ´∆ `

n´2 4pn´1qR, we have,

Lˆ

gpϕq “ u´ n`2

n´2 Lgpu ϕ q for all ϕ P C 8pMnq, where ˆ

g “ u

4 n´2 g.

‚ Paneitz operator in 1983 on pMn, gq, n ą 4. Ppnq

4

“ p´∆q2 ` δ panR g ` bnRicq d ` n ´ 4 2 Qpnq

4 .

pPpnq

4 qˆ gpϕq “ u´ n`4

n´4 pPpnq

4 qgpu ϕ q for all ϕ P C 8pMnq, where ˆ

g “ u

4 n´4 g.

‚ Notice that Ppnq

4 p1q “ n´4 2 Qpnq 4 , so we can read Qpnq 4

from Ppnq

4

when n ‰ 4.

Sun-Yung Alice Chang Princeton University Modern Aspects of complex analysis and its application In honor of John Garnett and Donald On a fourth order PDE and applications to problems in conformal geometry August 19-23, 2019 6 / 27

• S2. Some more recent development, Qn

4 on Mn, n ě 4

‚ In the last few years, there has been lots of progress made on the study

• f the 4-th order Paneitz operator P4 and its associated curvatures Q4

curvature with surprising results. ‚ Theorem (Gursky-Malchiodi 2014) On pMn, gq, n ě 5 suppose Rg ě 0, pQpnq

4 qg ą 0, then Ppnq 4

is positive, satisfies the strong maximal principle, (i.e. P4u 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 Pp3q

4

is trivial and its Green 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 Modern Aspects of complex analysis and its application In honor of John Garnett and Donald On a fourth order PDE and applications to problems in conformal geometry August 19-23, 2019 7 / 27

• S3. Branson’s Q-curvature on manifold of dimension 4

‚ Branson pointed out that P :“ P4

4 and Q :“ Q4 4 are well defined.

(which we named as Branson’s Q-curvature.) Pgϕ “ p´∆q2ϕ ` δ ˆ2 3Rg ´ 2Ric ˙ dϕ, 2Qg “ ´1 6∆Rg ` 1 6pR2

g ´ 3|Ricg|2q.

‚ Pgw ` 2Qg “ 2Qgw e4w on M4, where gw “ e2wg. Compared to ´∆gw ` Kg “ Kgw e2w on M2 . ‚ For examples: On pR4, |dx|2q, P “ ∆2, On pS4, gcq, P “ ∆2 ´ 2∆, On pM4, gq, g Einstein, P “ p´∆q ˝ pLq.

Sun-Yung Alice Chang Princeton University Modern Aspects of complex analysis and its application In honor of John Garnett and Donald On a fourth order PDE and applications to problems in conformal geometry August 19-23, 2019 8 / 27

Q-curvature

‚ Properties of ∆g on pM2, gq

• 1. ∆gw “ e´2w∆g.
• 2. ´∆gw ` Kg “ Kgw e2w,

Gauss-Bonnet 2πχpMq “ ż

M

Kgdvg

Sun-Yung Alice Chang Princeton University Modern Aspects of complex analysis and its application In honor of John Garnett and Donald On a fourth order PDE and applications to problems in conformal geometry August 19-23, 2019 9 / 27

Branson’s Q-curvature

‚ Properties of Paneitz operator on pM4, gq 1.Pgw “ e´4wPg 2. Pgw ` 2Qg “ 2Qgw e4w Q “ 1 12p´∆R ` R2 ´ 3|Ric|2q. ‚ Gauss-Bonnet-Chern Formula: 4π2χpM4q “ ż pQg ` 1 8|Wg|2q dv, where W denotes the Weyl tensor. ‚ Weyl curvature measures the obstruction to being conformally flat. On pMn, gq, n ě 4. Wg ” 0 in a neighborhood of a point if and only if g “ e2w|dx|2 for some function w. Thus on pSn, gcq, Wgc ” 0. ‚ gw “ e2wg, |Wgw | “ e´2w|Wg| , thus on 4-manifolds |Wgw |2dvgw “ |Wg|2dvg a pointwise conformal invariant; thus g Ñ ş

M |W |2 gdvg is an integral conformal invariant, hence so is

ş

M Qgdvg.

Sun-Yung Alice Chang Princeton University Modern Aspects of complex analysis and its application In honor of John Garnett and Donald On a fourth order PDE and applications to problems in conformal geometry August 19-23, 2019 10 / 27

Q-curvature -some PDE aspect

‚ Pgw ` 2Qg “ 2Qgw e4w ‚ Variational Functional: FQpwq “ă Pgw, w ą `4 ż

M4 Qgw ´

ż

M4 Qgdvglog

ż

M4 Qe4wdvg.

‚ In this case, W 2,2 Ă eL2, and Moser’s constant is 32π2. ‚ On pS4, g0q, ş

S4 Qg0dv0 “ 8π2.

‚ Theorem: Chang-Yang ’95 If ş Qgdvg ă 8π2 AND P is positive with KerP “ constants, then F is bounded from below and minimum is achieved by some gw “ e2wg with Qgw “ constant.

Sun-Yung Alice Chang Princeton University Modern Aspects of complex analysis and its application In honor of John Garnett and Donald On a fourth order PDE and applications to problems in conformal geometry August 19-23, 2019 11 / 27

Q-curvature -some PDE aspect

‚ Recall Yamabe constant Y pMn, gq ” inf

w

ş Rgw dvgw volpgwq

n´2 n

, it is a (second order) conformally invariant constant. ‚ Theorem: Gursky, late ’90 (a) When Y pM, gq ą 0, then ş Qgdvg ď 8π2, with equality only on pS4, g0q. (b) If Y pM, gq ą 0 and if ş Qgdvg ą 0 ù ñ Pg ě 0 with KerP “ tconstantsu. 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 Modern Aspects of complex analysis and its application In honor of John Garnett and Donald On a fourth order PDE and applications to problems in conformal geometry August 19-23, 2019 12 / 27

GJMS operators

‚ Conformal Laplace operator, Paneitz operator are special cases of GJMS (Graham-Jenney-Manning-Sparling, ’85) opertors P2k, these are conformal covariant operators of order 2k defined on pMn, gq, where 2k ď n. ‚ P2k “ p´∆qk ` lower order terms with associated Q2k curvature. ‚ Given ˆ g “ u

4 n´2k g,

pP2kqgpuq “ n ´ 2k 2 pQ2kqˆ

gu

n`2k n´2k

‚ P2 is the conformal Laplace operator, P4 is the Paneitz operator.

Sun-Yung Alice Chang Princeton University Modern Aspects of complex analysis and its application In honor of John Garnett and Donald On a fourth order PDE and applications to problems in conformal geometry August 19-23, 2019 13 / 27

Fractional GJMS operators, boundary operators

‚ Graham-Zworski,’02 re-derived the GJMS operator P2k, as the residue of the scattering opertor on its simple pole k on conformally compact Poincare Einstein manifolds. In this setting, they defined a class of ”non-local” pseudo-differential operator P2γ operators as the scattering

• perator at γ on the conformal infinity of the space where 2γ P r0, nq ´ N.

In general, leading symbol of P2γ is p´∆q2γ. ‚ Examples of conformally (semi) compact Einstein manifolds. ‚ Example 1 pRn`1

`

, Rn, gHq gH “ dy2 ` dx2 y2 g “ y2gH “ dy2 ` dx2 is the flat metric on Rn`1. In this case, P2γf “ p´∆xqγf .

Sun-Yung Alice Chang Princeton University Modern Aspects of complex analysis and its application In honor of John Garnett and Donald On a fourth order PDE and applications to problems in conformal geometry August 19-23, 2019 14 / 27

• S4. Fractional GJMS operators, boundary operators

‚ Example 2. On pBn`1, Sn, gHq, where pBn`1, gH “ p

2 1´|y|2 q2|dy|2q. Choose

r “ 2 1 ´ |y| 1 ` |y|, gH “ g` “ r´2 ˜ dr2 ` p1 ´ r2 4 q

2

gc ¸ . with pSn, rgcsq as conformal infinity. ‚ Example 3. AdS-Schwarzchild space

Sun-Yung Alice Chang Princeton University Modern Aspects of complex analysis and its application In honor of John Garnett and Donald On a fourth order PDE and applications to problems in conformal geometry August 19-23, 2019 15 / 27

Fractional GJMS operator P2γ when γ “ 1

2 when n ě 1

‚ when γ “ 1

2, on general manifolds X n`1 with boundary,

B1 “ B Bµ ` n ´ 1 2n H the Robin boundary operator, where H is the mean curvature and Q1 “ 1

nH, B1 has conformal covariant property.

‚ For a given function f defined on BX, denote Uf the solution of the P2pUf q “ 0 with Dirichlet data f , P1pf q :“ B1pUf q , i.e. P1 is the Dirichlet to Neumann operator. ‚ On pX n`1, gq with boundary, when n ą 1, Escobar has applied B1 to study boundary Yamabe problem.

Sun-Yung Alice Chang Princeton University Modern Aspects of complex analysis and its application In honor of John Garnett and Donald On a fourth order PDE and applications to problems in conformal geometry August 19-23, 2019 16 / 27

When n “ 1, curvature associated with fractional GJMS

• perator P1.

On compact surface pX 2, gq with boundary, the Gauss Bonnet formula becomes: 2πχpX, BXq “ ż

X

Kgdvg ` ż

BX

kgdsigmag where kg is the geodesic curvature. Under conformal change, ´ B Bnw ` kg “ kgw ew onBX. The PDE (special case of P1 operator) is applied to study boundary value problem on complete, non-compact surfaces.

Sun-Yung Alice Chang Princeton University Modern Aspects of complex analysis and its application In honor of John Garnett and Donald On a fourth order PDE and applications to problems in conformal geometry August 19-23, 2019 17 / 27

When n “ 1, curvature associated with fractional GJMS

• perator P1.

On compact surface pX 2, gq with boundary, the Gauss Bonnet formula becomes: 2πχpX, BXq “ ż

X

Kgdvg ` ż

BX

kgdsigmag where kg is the geodesic curvature. Under conformal change, ´ B Bnw ` kg “ kgw ew onBX. The PDE (special case of P1 operator) is applied to study boundary value problem on complete, non-compact surfaces. We now give some geometric motivation to study higher order P2γ

• perator, in particular on the 3rd order P3 when γ “ 3

2.

Sun-Yung Alice Chang Princeton University Modern Aspects of complex analysis and its application In honor of John Garnett and Donald On a fourth order PDE and applications to problems in conformal geometry August 19-23, 2019 17 / 27

• S5. 3rd order boundary operator on 4-manifolds

‚ (Chang-Qing, Branson-Gilkey 2000) On pX 4, BX, gq, the Gauss-Bonnet-Chern formula can be written as 4π2χpX 4q “ ż

X

p1 8|Wg|2dvg ` Qgdvgq ` ż

BX

pLg ` Tgqdσg (1) (2) where Lgdσg is a pointwise conformal invariant term, T a 3rd order curvature term. ‚ Furthermore T is the curvature of a 3rd order conformal covariant

• perator, which we call B3,

pB3qgw ` Tg “ Tgw e3w onBX ‚ B3w “ ´1 2 B Bn∆gw ´ ˜ ∆ B Bnw ` l.o.t T “ ´ 1 12 B BnR ` 1 6RH ` boundary curvature terms .

Sun-Yung Alice Chang Princeton University Modern Aspects of complex analysis and its application In honor of John Garnett and Donald On a fourth order PDE and applications to problems in conformal geometry August 19-23, 2019 18 / 27

Positivity of P2γ operator

‚ Relation of B3 to P3 is that, given a function f , consider the solution Uf , where pP4qgpUf q “ 0 onX 4, with Uf |BX “ f and BUf

Bµ “ 0 then

P3pf q “ B3pUf q. ‚ Theorem (Case-Chang, ’14) On a pX n`1, Mn, g`q a conformal compact Einstein manifold. (a) When 0 ă γ ă 1, and Λ1p´∆g`q ą n2

4 ´ γ2, Q2γ ą 0 implies P2γ ą 0.

(b) When 1 ă γ ă 2, n ě 4, RpM,g0q ą 0 and Q2γ ą 0 implies P2γ ą 0. (c) When n “ 3, the same result holds when 1 ă γ ď 3

2.

‚ We have also strong maximal principle, positive Green’s function for P2γ

• perators under the same curvature assumptions.

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

Proof of Theorem

‚ Two ideas in the proof of Theorem:

1 Extension Theorem for the flat case (result of Caffarelli and Silvestre

• n Rn`1

`

for 0 ă γ ă 1 cases ), extend it to the case of γ ą 1, and generalize the result to CCE manifolds.

2 Some ”right” choice of the conformal compactified Einstein metric

g˚ “ pρ˚q2g` in X n`1. ‚ Key property of g˚ metric on X 4 for the case γ “ 3

2 is Q4pg˚q ” 0.

Sun-Yung Alice Chang Princeton University Modern Aspects of complex analysis and its application In honor of John Garnett and Donald On a fourth order PDE and applications to problems in conformal geometry August 19-23, 2019 20 / 27

An open question

‚ It turns out on 4-manifolds with boundary, besides the operator B3 of

• rder 3, by the work of J. Case, there are other natural ”conformal

covariant” operators B2, B1 of order 1 and 2. ‚ It remains open what are the ”well -posed” set of boundary value problems of P4pUq “ 0 on the 4-manifold M4 with U satisfies 2-boundary conditions (e.g. with given U|BM “ f and B3pUq. ‚ Regularity of such PDE also remain open –one expect the earlier works

• f Kenig-Pipher and others be helpful here.

Sun-Yung Alice Chang Princeton University Modern Aspects of complex analysis and its application In honor of John Garnett and Donald On a fourth order PDE and applications to problems in conformal geometry August 19-23, 2019 21 / 27

An application in analysis: Sharp Sobolev Trace inequality

• f order 4 on pBd, Sd´1, |dx|2q

‚ In Ache-Chang ’15, we applied the strategy in the above work of Case-Chang to derive sharp Sobolev Trace inequality of order 4. ‚ As an application we can use the inequality when d “ 4 can be used to identify the extremal metric of a main term in the log-determinant functional for the pair operators pP2, P1q on 4-manifolds with boundary.

Sun-Yung Alice Chang Princeton University Modern Aspects of complex analysis and its application In honor of John Garnett and Donald On a fourth order PDE and applications to problems in conformal geometry August 19-23, 2019 22 / 27

Classical Milin-Lebedev inequality on pD, S1, |dx|2q

‚ The classical Milin-Lebedev inequality log ¨ ˝ 1 π ¿

S1

ef dσ ˛ ‚ď 1 4π ż

D

|∇v|2dx ` 1 π ¿

S1

fdσ, (3) where ∇v is the gradient of v with respect to the Euclidean metric on the disk D “ tx P R2 : |x| ď 1u. ‚ Milin-Lebedev inequality has been applied to a wide variety of problems in classical analysis, including the Bieberbach conjecture and by Osgood-Phillips-Sarnak in the study of the compactness of isospectral planar domains.

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

Sharp Sobolev Trace inequality on pB4, S3, |dx|2q

When d “ 4, the inequality is of Milin-Lebedev type. Theorem: Let f P C 8pS3q and let v be a C 8 extension of f to the ball B4 satisfying

B Bnv|BB4 “ 0. Then we have

log ¨ ˝ 1 2π2 ¿

S3

e3pf ´¯

f qdσ

˛ ‚ď 3 16π2 ż

B4p∆g0vq2dx

(4) ` 3 8π2 ¿

S3

| ˜ ∇f |2dσ. Equality holds for any f pξq “ log |1 ´ xz0, ξy| ` c with ξ P Sd´1, |z0| ă 1, c a constant and v is a bi-harmonic extension of f satisfying the Neumann boundary condition.

Sun-Yung Alice Chang Princeton University Modern Aspects of complex analysis and its application In honor of John Garnett and Donald On a fourth order PDE and applications to problems in conformal geometry August 19-23, 2019 24 / 27

Sharp Sobolev Trace inequality of order 4 on pBd, Sd´1, |dx|2q

‚ Main idea in the proof of the inequality: Establish the inequalities in the g˚ metric using the Extension theorem, then apply the conformal covariant property of the P4 operator to transform the inequalities back from g˚ back to g0 “ |dx|2. ‚ Lemma: On the model case pB4, S3, gHq, where gH “ ρ´2|dx|2 and ρ “ 1´|x|2

2

, we have when d “ 4, g˚ “ e2wgH “ e2ρ|dx|2 “ ep1´|x|2q|dx|2 with ´∆gHw “ 3 on B4. ‚ We remark when we apply the same scheme to deal with Sobolev trace inequalities of order 2 on pBd, Sd´1q, it turns out g˚ “ |dx|2

Sun-Yung Alice Chang Princeton University Modern Aspects of complex analysis and its application In honor of John Garnett and Donald On a fourth order PDE and applications to problems in conformal geometry August 19-23, 2019 25 / 27

A geometric application

Study of Q4 leads to the study of σ2 “ 1

6pR2 ´ 3|Ric|2q, which is the

(Weyl free part of the ) Gauss-Bonnet integrand. Under conformal change

• f metric, study of σ2pˆ

gq “ 1 is a fully non-linear PDE problem. 2Q4 “ ´1 6∆R ` σ2.

Sun-Yung Alice Chang Princeton University Modern Aspects of complex analysis and its application In honor of John Garnett and Donald On a fourth order PDE and applications to problems in conformal geometry August 19-23, 2019 26 / 27

Congratulations to Don and John, may your life become even more enjoyable; and you remain continuously productive in the days to come!

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