The work of Osgood-Phillips-Sarnak Consider the following scale-invariant version of log det: S [ w ] = − 1 � � � � � � |∇ w | 2 + 2 Kw e 2 w dA �� � � � log dA − K dA . 12 π M M det( − ∆ e 2 w g ) 1 If Area( e 2 w g ) = Area(g), then S [ w ] = log det( − ∆ g ) . 2 S [ w + c ] = S [ w ]. 3 Trudinger inequality gives W 1 , 2 ֒ → e L 2 , hence S : W 1 , 2 → R . M. Gursky (Notre Dame) and A. Malchiodi (SISSA) (Workshop on Geometric Partial Differential Equations ) Determinants and CI-operators June 7, 2012 6 / 30
The work of Osgood-Phillips-Sarnak Consider the following scale-invariant version of log det: S [ w ] = − 1 � � � � � � |∇ w | 2 + 2 Kw e 2 w dA �� � � � log dA − K dA . 12 π M M det( − ∆ e 2 w g ) 1 If Area( e 2 w g ) = Area(g), then S [ w ] = log det( − ∆ g ) . 2 S [ w + c ] = S [ w ]. 3 Trudinger inequality gives W 1 , 2 ֒ → e L 2 , hence S : W 1 , 2 → R . Theorem (O-P-S, ’88) S is maximized by a metric of constant (Gauss) curvature in each conformal class. M. Gursky (Notre Dame) and A. Malchiodi (SISSA) (Workshop on Geometric Partial Differential Equations ) Determinants and CI-operators June 7, 2012 6 / 30
The case of ( S 2 , g 0 ) M. Gursky (Notre Dame) and A. Malchiodi (SISSA) (Workshop on Geometric Partial Differential Equations ) Determinants and CI-operators June 7, 2012 7 / 30
The case of ( S 2 , g 0 ) For S 2 with the round metric we have S [ w ] = − 1 � � � � |∇ w | 2 + 2 w S 2 e 2 w dA �� � � dA − 4 π log . 12 π S 2 M. Gursky (Notre Dame) and A. Malchiodi (SISSA) (Workshop on Geometric Partial Differential Equations ) Determinants and CI-operators June 7, 2012 7 / 30
The case of ( S 2 , g 0 ) For S 2 with the round metric we have S [ w ] = − 1 � � � � |∇ w | 2 + 2 w S 2 e 2 w dA �� � � dA − 4 π log . 12 π S 2 1 The analysis of this case actually precedes the work of OPS, and is due to Onofri. M. Gursky (Notre Dame) and A. Malchiodi (SISSA) (Workshop on Geometric Partial Differential Equations ) Determinants and CI-operators June 7, 2012 7 / 30
The case of ( S 2 , g 0 ) For S 2 with the round metric we have S [ w ] = − 1 � � � � |∇ w | 2 + 2 w S 2 e 2 w dA �� � � dA − 4 π log . 12 π S 2 1 The analysis of this case actually precedes the work of OPS, and is due to Onofri. 2 Critical points of S correspond to constant curvature metrics; hence are given by the orbit of g 0 under the action of the conformal group. M. Gursky (Notre Dame) and A. Malchiodi (SISSA) (Workshop on Geometric Partial Differential Equations ) Determinants and CI-operators June 7, 2012 7 / 30
The case of ( S 2 , g 0 ) For S 2 with the round metric we have S [ w ] = − 1 � � � � |∇ w | 2 + 2 w S 2 e 2 w dA �� � � dA − 4 π log . 12 π S 2 1 The analysis of this case actually precedes the work of OPS, and is due to Onofri. 2 Critical points of S correspond to constant curvature metrics; hence are given by the orbit of g 0 under the action of the conformal group. 3 By Trudinger, Moser inequalities ⇒ S [ w ] ≤ C . M. Gursky (Notre Dame) and A. Malchiodi (SISSA) (Workshop on Geometric Partial Differential Equations ) Determinants and CI-operators June 7, 2012 7 / 30
The case of ( S 2 , g 0 ) For S 2 with the round metric we have S [ w ] = − 1 � � � � |∇ w | 2 + 2 w S 2 e 2 w dA �� � � dA − 4 π log . 12 π S 2 1 The analysis of this case actually precedes the work of OPS, and is due to Onofri. 2 Critical points of S correspond to constant curvature metrics; hence are given by the orbit of g 0 under the action of the conformal group. 3 By Trudinger, Moser inequalities ⇒ S [ w ] ≤ C . Theorem (Onofri, ’82) S [ w ] ≤ 0, with equality if and only if e 2 w g 0 = ϕ ∗ g 0 . M. Gursky (Notre Dame) and A. Malchiodi (SISSA) (Workshop on Geometric Partial Differential Equations ) Determinants and CI-operators June 7, 2012 7 / 30
Some additional remarks about Polakov’s formula M. Gursky (Notre Dame) and A. Malchiodi (SISSA) (Workshop on Geometric Partial Differential Equations ) Determinants and CI-operators June 7, 2012 8 / 30
Some additional remarks about Polakov’s formula 1 As pointed out by B. Chow ( JDG , ’91), the gradient flow for the functional S is (up to a sign) the Ricci flow. M. Gursky (Notre Dame) and A. Malchiodi (SISSA) (Workshop on Geometric Partial Differential Equations ) Determinants and CI-operators June 7, 2012 8 / 30
Some additional remarks about Polakov’s formula 1 As pointed out by B. Chow ( JDG , ’91), the gradient flow for the functional S is (up to a sign) the Ricci flow. 2 The key property exploited by Polyakov to derive his formula: In dimension two, ∆ e 2 w g = e − 2 w ∆ g . This does not hold in dimension n ≥ 3. M. Gursky (Notre Dame) and A. Malchiodi (SISSA) (Workshop on Geometric Partial Differential Equations ) Determinants and CI-operators June 7, 2012 8 / 30
Some additional remarks about Polakov’s formula 1 As pointed out by B. Chow ( JDG , ’91), the gradient flow for the functional S is (up to a sign) the Ricci flow. 2 The key property exploited by Polyakov to derive his formula: In dimension two, ∆ e 2 w g = e − 2 w ∆ g . This does not hold in dimension n ≥ 3. 3 Let L g = ∆ g − ( n − 2) 4( n − 1) R g , where R = R g is the scalar curvature of g . This is called the conformal Laplacian . Note M. Gursky (Notre Dame) and A. Malchiodi (SISSA) (Workshop on Geometric Partial Differential Equations ) Determinants and CI-operators June 7, 2012 8 / 30
Some additional remarks about Polakov’s formula 1 As pointed out by B. Chow ( JDG , ’91), the gradient flow for the functional S is (up to a sign) the Ricci flow. 2 The key property exploited by Polyakov to derive his formula: In dimension two, ∆ e 2 w g = e − 2 w ∆ g . This does not hold in dimension n ≥ 3. 3 Let L g = ∆ g − ( n − 2) 4( n − 1) R g , where R = R g is the scalar curvature of g . This is called the conformal Laplacian . Note g φ = e − n +2 n − 2 g = e 2 w g 2 w L g ( e 2 w φ ) . ˆ ⇒ L ˆ M. Gursky (Notre Dame) and A. Malchiodi (SISSA) (Workshop on Geometric Partial Differential Equations ) Determinants and CI-operators June 7, 2012 8 / 30
Some additional remarks about Polakov’s formula 1 As pointed out by B. Chow ( JDG , ’91), the gradient flow for the functional S is (up to a sign) the Ricci flow. 2 The key property exploited by Polyakov to derive his formula: In dimension two, ∆ e 2 w g = e − 2 w ∆ g . This does not hold in dimension n ≥ 3. 3 Let L g = ∆ g − ( n − 2) 4( n − 1) R g , where R = R g is the scalar curvature of g . This is called the conformal Laplacian . Note g φ = e − n +2 n − 2 g = e 2 w g 2 w L g ( e 2 w φ ) . ˆ ⇒ L ˆ 4 In higher dimensions there are many other examples of conformally invariant operators... M. Gursky (Notre Dame) and A. Malchiodi (SISSA) (Workshop on Geometric Partial Differential Equations ) Determinants and CI-operators June 7, 2012 8 / 30
Four dimensions: the work of Branson-Ørsted ( M , g ) a closed four-manifold. M. Gursky (Notre Dame) and A. Malchiodi (SISSA) (Workshop on Geometric Partial Differential Equations ) Determinants and CI-operators June 7, 2012 9 / 30
Four dimensions: the work of Branson-Ørsted ( M , g ) a closed four-manifold. Suppose A = A g is conformally invariant: M. Gursky (Notre Dame) and A. Malchiodi (SISSA) (Workshop on Geometric Partial Differential Equations ) Determinants and CI-operators June 7, 2012 9 / 30
Four dimensions: the work of Branson-Ørsted ( M , g ) a closed four-manifold. Suppose A = A g is conformally invariant: A e 2 w g φ = e ab A g ( e bw φ ) (and satisfies some additional technical assumptions). M. Gursky (Notre Dame) and A. Malchiodi (SISSA) (Workshop on Geometric Partial Differential Equations ) Determinants and CI-operators June 7, 2012 9 / 30
Four dimensions: the work of Branson-Ørsted ( M , g ) a closed four-manifold. Suppose A = A g is conformally invariant: A e 2 w g φ = e ab A g ( e bw φ ) (and satisfies some additional technical assumptions). For several important examples, Branson-Ørsted gave a Polyakov-type formula for F A [ w ] = log det A ˆ g g = e 2 w g . , ˆ det A g M. Gursky (Notre Dame) and A. Malchiodi (SISSA) (Workshop on Geometric Partial Differential Equations ) Determinants and CI-operators June 7, 2012 9 / 30
Four dimensions: the work of Branson-Ørsted ( M , g ) a closed four-manifold. Suppose A = A g is conformally invariant: A e 2 w g φ = e ab A g ( e bw φ ) (and satisfies some additional technical assumptions). For several important examples, Branson-Ørsted gave a Polyakov-type formula for F A [ w ] = log det A ˆ g g = e 2 w g . , ˆ det A g Note: In the following, we assume A has a trivial kernel; otherwise the forgoing formulas need to be modified. M. Gursky (Notre Dame) and A. Malchiodi (SISSA) (Workshop on Geometric Partial Differential Equations ) Determinants and CI-operators June 7, 2012 9 / 30
The formula of Branson-Ørsted, cont. M. Gursky (Notre Dame) and A. Malchiodi (SISSA) (Workshop on Geometric Partial Differential Equations ) Determinants and CI-operators June 7, 2012 10 / 30
The formula of Branson-Ørsted, cont. • First observation: F A is always a linear combination of the same three terms: M. Gursky (Notre Dame) and A. Malchiodi (SISSA) (Workshop on Geometric Partial Differential Equations ) Determinants and CI-operators June 7, 2012 10 / 30
The formula of Branson-Ørsted, cont. • First observation: F A is always a linear combination of the same three terms: i.e., F A [ w ] = γ 1 ( A ) · I [ w ] + γ 2 ( A ) · II [ w ] + γ 3 ( A ) · III [ w ] , where I , II , III are functionals on the space of conformal metrics. M. Gursky (Notre Dame) and A. Malchiodi (SISSA) (Workshop on Geometric Partial Differential Equations ) Determinants and CI-operators June 7, 2012 10 / 30
The formula of Branson-Ørsted, cont. • First observation: F A is always a linear combination of the same three terms: i.e., F A [ w ] = γ 1 ( A ) · I [ w ] + γ 2 ( A ) · II [ w ] + γ 3 ( A ) · III [ w ] , where I , II , III are functionals on the space of conformal metrics. • Consequently, there are two parts to Branson-Ørsted’s work: M. Gursky (Notre Dame) and A. Malchiodi (SISSA) (Workshop on Geometric Partial Differential Equations ) Determinants and CI-operators June 7, 2012 10 / 30
The formula of Branson-Ørsted, cont. • First observation: F A is always a linear combination of the same three terms: i.e., F A [ w ] = γ 1 ( A ) · I [ w ] + γ 2 ( A ) · II [ w ] + γ 3 ( A ) · III [ w ] , where I , II , III are functionals on the space of conformal metrics. • Consequently, there are two parts to Branson-Ørsted’s work: 1 Identifying the functionals I , II , III . M. Gursky (Notre Dame) and A. Malchiodi (SISSA) (Workshop on Geometric Partial Differential Equations ) Determinants and CI-operators June 7, 2012 10 / 30
The formula of Branson-Ørsted, cont. • First observation: F A is always a linear combination of the same three terms: i.e., F A [ w ] = γ 1 ( A ) · I [ w ] + γ 2 ( A ) · II [ w ] + γ 3 ( A ) · III [ w ] , where I , II , III are functionals on the space of conformal metrics. • Consequently, there are two parts to Branson-Ørsted’s work: 1 Identifying the functionals I , II , III . 2 For a particular choice of the operator A , computing the coefficients γ 1 , γ 2 , γ 3 . M. Gursky (Notre Dame) and A. Malchiodi (SISSA) (Workshop on Geometric Partial Differential Equations ) Determinants and CI-operators June 7, 2012 10 / 30
The Branson-Ørsted formula for space forms M. Gursky (Notre Dame) and A. Malchiodi (SISSA) (Workshop on Geometric Partial Differential Equations ) Determinants and CI-operators June 7, 2012 11 / 30
The Branson-Ørsted formula for space forms If ( M 4 , g ) is flat, then I [ w ] = 0 , M. Gursky (Notre Dame) and A. Malchiodi (SISSA) (Workshop on Geometric Partial Differential Equations ) Determinants and CI-operators June 7, 2012 11 / 30
The Branson-Ørsted formula for space forms If ( M 4 , g ) is flat, then I [ w ] = 0 , � (∆ w ) 2 , II [ w ] = M. Gursky (Notre Dame) and A. Malchiodi (SISSA) (Workshop on Geometric Partial Differential Equations ) Determinants and CI-operators June 7, 2012 11 / 30
The Branson-Ørsted formula for space forms If ( M 4 , g ) is flat, then I [ w ] = 0 , � (∆ w ) 2 , II [ w ] = � (∆ w + |∇ w | 2 ) 2 . III [ w ] = 12 Note that all three are non-negative. M. Gursky (Notre Dame) and A. Malchiodi (SISSA) (Workshop on Geometric Partial Differential Equations ) Determinants and CI-operators June 7, 2012 11 / 30
The Branson-Ørsted formula for space forms If ( M 4 , g ) is flat, then I [ w ] = 0 , � (∆ w ) 2 , II [ w ] = � (∆ w + |∇ w | 2 ) 2 . III [ w ] = 12 Note that all three are non-negative. • Hence log det formula is a linear combination of three basic functionals: � � � (∆ w ) 2 , H [ w ] = (∆ w ) |∇ w | 2 , P [ w ] = |∇ w | 4 , B [ w ] = with gradients M. Gursky (Notre Dame) and A. Malchiodi (SISSA) (Workshop on Geometric Partial Differential Equations ) Determinants and CI-operators June 7, 2012 11 / 30
The Branson-Ørsted formula for space forms If ( M 4 , g ) is flat, then I [ w ] = 0 , � (∆ w ) 2 , II [ w ] = � (∆ w + |∇ w | 2 ) 2 . III [ w ] = 12 Note that all three are non-negative. • Hence log det formula is a linear combination of three basic functionals: � � � (∆ w ) 2 , H [ w ] = (∆ w ) |∇ w | 2 , P [ w ] = |∇ w | 4 , B [ w ] = with gradients ∇B = ∆ 2 w , ∇H = σ 2 ( ∇ 2 w ) , ∇P = div( |∇ w | 2 ∇ w ) . M. Gursky (Notre Dame) and A. Malchiodi (SISSA) (Workshop on Geometric Partial Differential Equations ) Determinants and CI-operators June 7, 2012 11 / 30
The Branson-Ørsted formula for space forms: the sphere M. Gursky (Notre Dame) and A. Malchiodi (SISSA) (Workshop on Geometric Partial Differential Equations ) Determinants and CI-operators June 7, 2012 12 / 30
The Branson-Ørsted formula for space forms: the sphere On S 4 with the round metric, I [ w ] = 0 , M. Gursky (Notre Dame) and A. Malchiodi (SISSA) (Workshop on Geometric Partial Differential Equations ) Determinants and CI-operators June 7, 2012 12 / 30
The Branson-Ørsted formula for space forms: the sphere On S 4 with the round metric, I [ w ] = 0 , � � � [(∆ w ) 2 + 2 |∇ w | 2 ] dv − 8 π 2 log e 4( w − w ) dv � II [ w ] = (compare with the formula for S on the 2-sphere); M. Gursky (Notre Dame) and A. Malchiodi (SISSA) (Workshop on Geometric Partial Differential Equations ) Determinants and CI-operators June 7, 2012 12 / 30
The Branson-Ørsted formula for space forms: the sphere On S 4 with the round metric, I [ w ] = 0 , � � � [(∆ w ) 2 + 2 |∇ w | 2 ] dv − 8 π 2 log e 4( w − w ) dv � II [ w ] = (compare with the formula for S on the 2-sphere); � � (∆ w + |∇ w | 2 ) 2 dv − 48 |∇ w | 2 dv . III [ w ] = 12 M. Gursky (Notre Dame) and A. Malchiodi (SISSA) (Workshop on Geometric Partial Differential Equations ) Determinants and CI-operators June 7, 2012 12 / 30
The Branson-Ørsted formula for space forms: the sphere On S 4 with the round metric, I [ w ] = 0 , � � � [(∆ w ) 2 + 2 |∇ w | 2 ] dv − 8 π 2 log e 4( w − w ) dv � II [ w ] = (compare with the formula for S on the 2-sphere); � � (∆ w + |∇ w | 2 ) 2 dv − 48 |∇ w | 2 dv . III [ w ] = 12 Theorem ( i ) (Beckner, ’93) II [ w ] ≥ 0 with equality iff e 2 w g = ϕ ∗ g . M. Gursky (Notre Dame) and A. Malchiodi (SISSA) (Workshop on Geometric Partial Differential Equations ) Determinants and CI-operators June 7, 2012 12 / 30
The Branson-Ørsted formula for space forms: the sphere On S 4 with the round metric, I [ w ] = 0 , � � � [(∆ w ) 2 + 2 |∇ w | 2 ] dv − 8 π 2 log e 4( w − w ) dv � II [ w ] = (compare with the formula for S on the 2-sphere); � � (∆ w + |∇ w | 2 ) 2 dv − 48 |∇ w | 2 dv . III [ w ] = 12 Theorem ( i ) (Beckner, ’93) II [ w ] ≥ 0 with equality iff e 2 w g = ϕ ∗ g . ( ii ) (Beckner, Chang-Yang) III [ w ] ≥ 0 with equality iff e 2 w g = ϕ ∗ g . M. Gursky (Notre Dame) and A. Malchiodi (SISSA) (Workshop on Geometric Partial Differential Equations ) Determinants and CI-operators June 7, 2012 12 / 30
The coefficients γ i . M. Gursky (Notre Dame) and A. Malchiodi (SISSA) (Workshop on Geometric Partial Differential Equations ) Determinants and CI-operators June 7, 2012 13 / 30
The coefficients γ i . 1 Let A = ∆ − 1 6 R , the conformal laplacian. In this case, ( γ 1 , γ 2 , γ 3 ) = (1 , − 4 , − 2 / 3). M. Gursky (Notre Dame) and A. Malchiodi (SISSA) (Workshop on Geometric Partial Differential Equations ) Determinants and CI-operators June 7, 2012 13 / 30
The coefficients γ i . 1 Let A = ∆ − 1 6 R , the conformal laplacian. In this case, ( γ 1 , γ 2 , γ 3 ) = (1 , − 4 , − 2 / 3). 2 A = P , the Paneitz operator : (2 R ij − 2 P g φ = ( − ∆) 2 φ + ∇ j � 3 Rg ij ) ∇ i φ � , where R ij are the components of the Ricci tensor. log det P was considered by Connes ’94 in connection with quantum gravity. M. Gursky (Notre Dame) and A. Malchiodi (SISSA) (Workshop on Geometric Partial Differential Equations ) Determinants and CI-operators June 7, 2012 13 / 30
The coefficients γ i . 1 Let A = ∆ − 1 6 R , the conformal laplacian. In this case, ( γ 1 , γ 2 , γ 3 ) = (1 , − 4 , − 2 / 3). 2 A = P , the Paneitz operator : (2 R ij − 2 P g φ = ( − ∆) 2 φ + ∇ j � 3 Rg ij ) ∇ i φ � , where R ij are the components of the Ricci tensor. log det P was considered by Connes ’94 in connection with quantum gravity. Branson ’96 computed the coefficients in a separate paper, and found ( γ 1 , γ 2 , γ 3 ) = ( − 1 / 4 , − 14 , 8 / 3). M. Gursky (Notre Dame) and A. Malchiodi (SISSA) (Workshop on Geometric Partial Differential Equations ) Determinants and CI-operators June 7, 2012 13 / 30
Another example: Cheeger’s half-torsion M. Gursky (Notre Dame) and A. Malchiodi (SISSA) (Workshop on Geometric Partial Differential Equations ) Determinants and CI-operators June 7, 2012 14 / 30
Another example: Cheeger’s half-torsion For ( M 2 k , g ) an even-dimensional manifold, define (det( − ∆ 0 )) n (det( − ∆ 2 )) n − 4 . . . τ h = (det( − ∆ 1 )) n − 2 (det( − ∆ 3 )) n − 6 · · · , where ∆ p denotes the Hodge laplacian on p -forms. M. Gursky (Notre Dame) and A. Malchiodi (SISSA) (Workshop on Geometric Partial Differential Equations ) Determinants and CI-operators June 7, 2012 14 / 30
Another example: Cheeger’s half-torsion For ( M 2 k , g ) an even-dimensional manifold, define (det( − ∆ 0 )) n (det( − ∆ 2 )) n − 4 . . . τ h = (det( − ∆ 1 )) n − 2 (det( − ∆ 3 )) n − 6 · · · , where ∆ p denotes the Hodge laplacian on p -forms. • Notice this only involves p for p < n / 2. In four dimensions, log τ h = 4 log det( − ∆ 0 ) − 2 log det( − ∆ 1 ) . M. Gursky (Notre Dame) and A. Malchiodi (SISSA) (Workshop on Geometric Partial Differential Equations ) Determinants and CI-operators June 7, 2012 14 / 30
Another example: Cheeger’s half-torsion For ( M 2 k , g ) an even-dimensional manifold, define (det( − ∆ 0 )) n (det( − ∆ 2 )) n − 4 . . . τ h = (det( − ∆ 1 )) n − 2 (det( − ∆ 3 )) n − 6 · · · , where ∆ p denotes the Hodge laplacian on p -forms. • Notice this only involves p for p < n / 2. In four dimensions, log τ h = 4 log det( − ∆ 0 ) − 2 log det( − ∆ 1 ) . • The half-torsion plays a role in self-dual field theory; cf. Witten ’97. M. Gursky (Notre Dame) and A. Malchiodi (SISSA) (Workshop on Geometric Partial Differential Equations ) Determinants and CI-operators June 7, 2012 14 / 30
Another example: Cheeger’s half-torsion For ( M 2 k , g ) an even-dimensional manifold, define (det( − ∆ 0 )) n (det( − ∆ 2 )) n − 4 . . . τ h = (det( − ∆ 1 )) n − 2 (det( − ∆ 3 )) n − 6 · · · , where ∆ p denotes the Hodge laplacian on p -forms. • Notice this only involves p for p < n / 2. In four dimensions, log τ h = 4 log det( − ∆ 0 ) − 2 log det( − ∆ 1 ) . • The half-torsion plays a role in self-dual field theory; cf. Witten ’97. • In general, the Hodge laplacian is not conformally covariant. However, the ratio above has the requisite conformal properties for deriving a Polyakov-type formula (cf. Juhl ’09). The coefficients for the corresponding functional are ( γ 1 , γ 2 , γ 3 ) = ( − 13 , − 248 , 116 / 3). M. Gursky (Notre Dame) and A. Malchiodi (SISSA) (Workshop on Geometric Partial Differential Equations ) Determinants and CI-operators June 7, 2012 14 / 30
Key point: log det L is a convex combination of II and III ; for log det P (and log τ h ), γ 2 and γ 3 have different signs. M. Gursky (Notre Dame) and A. Malchiodi (SISSA) (Workshop on Geometric Partial Differential Equations ) Determinants and CI-operators June 7, 2012 15 / 30
Key point: log det L is a convex combination of II and III ; for log det P (and log τ h ), γ 2 and γ 3 have different signs. • In particular, on S 4 we have log det L ≤ 0, with equality only for the round metric (modulo the conformal group). However.... M. Gursky (Notre Dame) and A. Malchiodi (SISSA) (Workshop on Geometric Partial Differential Equations ) Determinants and CI-operators June 7, 2012 15 / 30
Key point: log det L is a convex combination of II and III ; for log det P (and log τ h ), γ 2 and γ 3 have different signs. • In particular, on S 4 we have log det L ≤ 0, with equality only for the round metric (modulo the conformal group). However.... Proposition (G-Malchiodi) For any conformal four-manifold, inf F P = −∞ , sup F P = + ∞ . Thus log det L and log det P exhibit quite different variational properties. M. Gursky (Notre Dame) and A. Malchiodi (SISSA) (Workshop on Geometric Partial Differential Equations ) Determinants and CI-operators June 7, 2012 15 / 30
A geometric detour: the geometry of the Euler equations M. Gursky (Notre Dame) and A. Malchiodi (SISSA) (Workshop on Geometric Partial Differential Equations ) Determinants and CI-operators June 7, 2012 16 / 30
A geometric detour: the geometry of the Euler equations • If we view the functionals I , II , III separately, what (geometric) property does a critial metric satisfy? M. Gursky (Notre Dame) and A. Malchiodi (SISSA) (Workshop on Geometric Partial Differential Equations ) Determinants and CI-operators June 7, 2012 16 / 30
A geometric detour: the geometry of the Euler equations • If we view the functionals I , II , III separately, what (geometric) property does a critial metric satisfy? I: g = e 2 w g is critical for I ⇔ ˆ ˆ g satisfies M. Gursky (Notre Dame) and A. Malchiodi (SISSA) (Workshop on Geometric Partial Differential Equations ) Determinants and CI-operators June 7, 2012 16 / 30
A geometric detour: the geometry of the Euler equations • If we view the functionals I , II , III separately, what (geometric) property does a critial metric satisfy? I: g = e 2 w g is critical for I ⇔ ˆ ˆ g satisfies g ) | 2 ≡ const , | W (ˆ where W is the Weyl tensor. M. Gursky (Notre Dame) and A. Malchiodi (SISSA) (Workshop on Geometric Partial Differential Equations ) Determinants and CI-operators June 7, 2012 16 / 30
A geometric detour: the geometry of the Euler equations • If we view the functionals I , II , III separately, what (geometric) property does a critial metric satisfy? I: g = e 2 w g is critical for I ⇔ ˆ ˆ g satisfies g ) | 2 ≡ const , | W (ˆ where W is the Weyl tensor. III: g = e 2 w g is critical for I ⇔ ˆ ˆ g satisfies M. Gursky (Notre Dame) and A. Malchiodi (SISSA) (Workshop on Geometric Partial Differential Equations ) Determinants and CI-operators June 7, 2012 16 / 30
A geometric detour: the geometry of the Euler equations • If we view the functionals I , II , III separately, what (geometric) property does a critial metric satisfy? I: g = e 2 w g is critical for I ⇔ ˆ ˆ g satisfies g ) | 2 ≡ const , | W (ˆ where W is the Weyl tensor. III: g = e 2 w g is critical for I ⇔ ˆ ˆ g satisfies ( − ∆ ˆ g ) R ˆ g = 0 , where R is the scalar curvature. (Consequently, R ˆ g ≡ const.) M. Gursky (Notre Dame) and A. Malchiodi (SISSA) (Workshop on Geometric Partial Differential Equations ) Determinants and CI-operators June 7, 2012 16 / 30
The geometry of the Euler equations, cont. II: g = e 2 w g is critical for II ⇔ ˆ ˆ g satisfies M. Gursky (Notre Dame) and A. Malchiodi (SISSA) (Workshop on Geometric Partial Differential Equations ) Determinants and CI-operators June 7, 2012 17 / 30
The geometry of the Euler equations, cont. II: g = e 2 w g is critical for II ⇔ ˆ ˆ g satisfies Q ˆ g ≡ const . where Q is the Q-curvature : M. Gursky (Notre Dame) and A. Malchiodi (SISSA) (Workshop on Geometric Partial Differential Equations ) Determinants and CI-operators June 7, 2012 17 / 30
The geometry of the Euler equations, cont. II: g = e 2 w g is critical for II ⇔ ˆ ˆ g satisfies Q ˆ g ≡ const . where Q is the Q-curvature : Q = 1 − ∆ R + R 2 − 3 | Ric | 2 � � . 12 M. Gursky (Notre Dame) and A. Malchiodi (SISSA) (Workshop on Geometric Partial Differential Equations ) Determinants and CI-operators June 7, 2012 17 / 30
The geometry of the Euler equations, cont. II: g = e 2 w g is critical for II ⇔ ˆ ˆ g satisfies Q ˆ g ≡ const . where Q is the Q-curvature : Q = 1 − ∆ R + R 2 − 3 | Ric | 2 � � . 12 Euler equation of F A g = e 2 w g is critical for F A ⇔ ˆ ˆ g satisfies g ) | 2 + γ 2 Q ˆ γ 1 | W (ˆ g + γ 3 ( − ∆ ˆ g ) R ˆ g ≡ const . (E-L) M. Gursky (Notre Dame) and A. Malchiodi (SISSA) (Workshop on Geometric Partial Differential Equations ) Determinants and CI-operators June 7, 2012 17 / 30
The geometry of the Euler equations, cont. Some applications of the Euler equation: M. Gursky (Notre Dame) and A. Malchiodi (SISSA) (Workshop on Geometric Partial Differential Equations ) Determinants and CI-operators June 7, 2012 18 / 30
The geometry of the Euler equations, cont. Some applications of the Euler equation: Vanishing theorems for Betti numbers of four-manifolds under integral curvature conditions; M. Gursky (Notre Dame) and A. Malchiodi (SISSA) (Workshop on Geometric Partial Differential Equations ) Determinants and CI-operators June 7, 2012 18 / 30
The geometry of the Euler equations, cont. Some applications of the Euler equation: Vanishing theorems for Betti numbers of four-manifolds under integral curvature conditions; A conformally invariant sphere theorem. More precisely: sharp characterizations of S 4 , CP 2 , and S 3 × S 1 via L 2 -curvature estimates. (This uses Ricci flow and properties of the Euler equation); M. Gursky (Notre Dame) and A. Malchiodi (SISSA) (Workshop on Geometric Partial Differential Equations ) Determinants and CI-operators June 7, 2012 18 / 30
The geometry of the Euler equations, cont. Some applications of the Euler equation: Vanishing theorems for Betti numbers of four-manifolds under integral curvature conditions; A conformally invariant sphere theorem. More precisely: sharp characterizations of S 4 , CP 2 , and S 3 × S 1 via L 2 -curvature estimates. (This uses Ricci flow and properties of the Euler equation); M. Gursky (Notre Dame) and A. Malchiodi (SISSA) (Workshop on Geometric Partial Differential Equations ) Determinants and CI-operators June 7, 2012 18 / 30
The geometry of the Euler equations, cont. Some applications of the Euler equation: Vanishing theorems for Betti numbers of four-manifolds under integral curvature conditions; A conformally invariant sphere theorem. More precisely: sharp characterizations of S 4 , CP 2 , and S 3 × S 1 via L 2 -curvature estimates. (This uses Ricci flow and properties of the Euler equation); Note 4 π 2 χ ( M ) = 1 � � � W � 2 dV + Q dV . 2 Hence, the LHS of (E-L) integrates to a conformal invariant. (Compare with prescribing Gauss curvature in 2-d). M. Gursky (Notre Dame) and A. Malchiodi (SISSA) (Workshop on Geometric Partial Differential Equations ) Determinants and CI-operators June 7, 2012 18 / 30
Existence results for critical metrics for general 4-manifolds M. Gursky (Notre Dame) and A. Malchiodi (SISSA) (Workshop on Geometric Partial Differential Equations ) Determinants and CI-operators June 7, 2012 19 / 30
Existence results for critical metrics for general 4-manifolds Theorem (Chang-Yang, ’95) Assume M. Gursky (Notre Dame) and A. Malchiodi (SISSA) (Workshop on Geometric Partial Differential Equations ) Determinants and CI-operators June 7, 2012 19 / 30
Existence results for critical metrics for general 4-manifolds Theorem (Chang-Yang, ’95) Assume γ 2 and γ 3 have the same sign; M. Gursky (Notre Dame) and A. Malchiodi (SISSA) (Workshop on Geometric Partial Differential Equations ) Determinants and CI-operators June 7, 2012 19 / 30
Existence results for critical metrics for general 4-manifolds Theorem (Chang-Yang, ’95) Assume γ 2 and γ 3 have the same sign; A smallness condition on the total Q-curvature (alternatively, | W | 2 dv ≤ c ( χ ( M 4 )) ); � M. Gursky (Notre Dame) and A. Malchiodi (SISSA) (Workshop on Geometric Partial Differential Equations ) Determinants and CI-operators June 7, 2012 19 / 30
Existence results for critical metrics for general 4-manifolds Theorem (Chang-Yang, ’95) Assume γ 2 and γ 3 have the same sign; A smallness condition on the total Q-curvature (alternatively, | W | 2 dv ≤ c ( χ ( M 4 )) ); � Then an extremal for F = γ 1 I + γ 2 II + γ 3 III exists. M. Gursky (Notre Dame) and A. Malchiodi (SISSA) (Workshop on Geometric Partial Differential Equations ) Determinants and CI-operators June 7, 2012 19 / 30
Existence results for critical metrics for general 4-manifolds Theorem (Chang-Yang, ’95) Assume γ 2 and γ 3 have the same sign; A smallness condition on the total Q-curvature (alternatively, | W | 2 dv ≤ c ( χ ( M 4 )) ); � Then an extremal for F = γ 1 I + γ 2 II + γ 3 III exists. Moreover, it is smooth (G-Chang-Yang, Viaclovsky-Uhlenbeck). M. Gursky (Notre Dame) and A. Malchiodi (SISSA) (Workshop on Geometric Partial Differential Equations ) Determinants and CI-operators June 7, 2012 19 / 30
Existence results for critical metrics for general 4-manifolds Theorem (Chang-Yang, ’95) Assume γ 2 and γ 3 have the same sign; A smallness condition on the total Q-curvature (alternatively, | W | 2 dv ≤ c ( χ ( M 4 )) ); � Then an extremal for F = γ 1 I + γ 2 II + γ 3 III exists. Moreover, it is smooth (G-Chang-Yang, Viaclovsky-Uhlenbeck). • The “smallness” condition is related to the best constant in the Moser-Trudinger inequality of Adams. M. Gursky (Notre Dame) and A. Malchiodi (SISSA) (Workshop on Geometric Partial Differential Equations ) Determinants and CI-operators June 7, 2012 19 / 30
Existence results for critical metrics for general 4-manifolds Theorem (Chang-Yang, ’95) Assume γ 2 and γ 3 have the same sign; A smallness condition on the total Q-curvature (alternatively, | W | 2 dv ≤ c ( χ ( M 4 )) ); � Then an extremal for F = γ 1 I + γ 2 II + γ 3 III exists. Moreover, it is smooth (G-Chang-Yang, Viaclovsky-Uhlenbeck). • The “smallness” condition is related to the best constant in the Moser-Trudinger inequality of Adams. • This result does not apply to log det P (or log τ h ). M. Gursky (Notre Dame) and A. Malchiodi (SISSA) (Workshop on Geometric Partial Differential Equations ) Determinants and CI-operators June 7, 2012 19 / 30
Back to the sphere For ( M , g ) the standard 4-sphere, M. Gursky (Notre Dame) and A. Malchiodi (SISSA) (Workshop on Geometric Partial Differential Equations ) Determinants and CI-operators June 7, 2012 20 / 30
Back to the sphere For ( M , g ) the standard 4-sphere, � � − 12(∆ w ) 2 − 16 |∇ w | 2 ∆ w − 8 |∇ w | 4 + 24 |∇ w | 2 � F L [ w ] = dv S 4 � � + 32 π 2 log S 4 e 4( w − w ) dv � , M. Gursky (Notre Dame) and A. Malchiodi (SISSA) (Workshop on Geometric Partial Differential Equations ) Determinants and CI-operators June 7, 2012 20 / 30
Back to the sphere For ( M , g ) the standard 4-sphere, � � − 12(∆ w ) 2 − 16 |∇ w | 2 ∆ w − 8 |∇ w | 4 + 24 |∇ w | 2 � F L [ w ] = dv S 4 � � + 32 π 2 log S 4 e 4( w − w ) dv � , while � � 18(∆ w ) 2 + 64 |∇ w | 2 ∆ w + 32 |∇ w | 4 − 60 |∇ w | 2 � F P [ w ] = dv S 4 � � + 112 π 2 log S 4 e 4( w − w ) dv � . M. Gursky (Notre Dame) and A. Malchiodi (SISSA) (Workshop on Geometric Partial Differential Equations ) Determinants and CI-operators June 7, 2012 20 / 30
Back to the sphere For ( M , g ) the standard 4-sphere, � � − 12(∆ w ) 2 − 16 |∇ w | 2 ∆ w − 8 |∇ w | 4 + 24 |∇ w | 2 � F L [ w ] = dv S 4 � � + 32 π 2 log S 4 e 4( w − w ) dv � , while � � 18(∆ w ) 2 + 64 |∇ w | 2 ∆ w + 32 |∇ w | 4 − 60 |∇ w | 2 � F P [ w ] = dv S 4 � � + 112 π 2 log S 4 e 4( w − w ) dv � . • Note: The cross-term (∆ w ) |∇ w | 2 compared to positive terms, and the interaction of logarithmic term with derivative terms. This accounts for the lack of boundedness. M. Gursky (Notre Dame) and A. Malchiodi (SISSA) (Workshop on Geometric Partial Differential Equations ) Determinants and CI-operators June 7, 2012 20 / 30
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