Uniformization of singular surfaces Andrea Malchiodi (SNS, Pisa) - PowerPoint PPT Presentation

The negative curvature case This case is still rather easy. The sign of K makes the energy coercive I K K < 0 This means that I K → + ∞ as � u � → ∞ . To find a critical point of I K , it is sufficient to pass to the limit along a minimizing sequence Andrea Malchiodi (SNS, Pisa) BMC 2016 6 / 30

The negative curvature case This case is still rather easy. The sign of K makes the energy coercive I K K < 0 This means that I K → + ∞ as � u � → ∞ . To find a critical point of I K , it is sufficient to pass to the limit along a minimizing sequence ( direct methods of the Calculus of Variations ). Andrea Malchiodi (SNS, Pisa) BMC 2016 6 / 30

The positive curvature case Andrea Malchiodi (SNS, Pisa) BMC 2016 7 / 30

The positive curvature case This is the most difficult case: when K > 0 and Σ has the topology of S 2 . Andrea Malchiodi (SNS, Pisa) BMC 2016 7 / 30

The positive curvature case This is the most difficult case: when K > 0 and Σ has the topology of S 2 . Notice that the Gauss-Bonnet formula implies K = 4 π . Andrea Malchiodi (SNS, Pisa) BMC 2016 7 / 30

The positive curvature case This is the most difficult case: when K > 0 and Σ has the topology of S 2 . Notice that the Gauss-Bonnet formula implies K = 4 π . Using the Moser-Trudinger inequality one finds that I K is still bounded from below, but coercivity is lost. Andrea Malchiodi (SNS, Pisa) BMC 2016 7 / 30

The positive curvature case This is the most difficult case: when K > 0 and Σ has the topology of S 2 . Notice that the Gauss-Bonnet formula implies K = 4 π . Using the Moser-Trudinger inequality one finds that I K is still bounded from below, but coercivity is lost. The picture might look like K > 0 I K Andrea Malchiodi (SNS, Pisa) BMC 2016 7 / 30

The positive curvature case This is the most difficult case: when K > 0 and Σ has the topology of S 2 . Notice that the Gauss-Bonnet formula implies K = 4 π . Using the Moser-Trudinger inequality one finds that I K is still bounded from below, but coercivity is lost. The picture might look like K > 0 I K A priori, we cannot minimize any more: minimizing sequences might slide-off to infinity. Andrea Malchiodi (SNS, Pisa) BMC 2016 7 / 30

The positive curvature case This is the most difficult case: when K > 0 and Σ has the topology of S 2 . Notice that the Gauss-Bonnet formula implies K = 4 π . Using the Moser-Trudinger inequality one finds that I K is still bounded from below, but coercivity is lost. The picture might look like K > 0 I K A priori, we cannot minimize any more: minimizing sequences might slide-off to infinity. This is due to a loss of compactness . Andrea Malchiodi (SNS, Pisa) BMC 2016 7 / 30

The Möbius group Andrea Malchiodi (SNS, Pisa) BMC 2016 8 / 30

The Möbius group This is a non-compact family of conformal maps from S 2 to S 2 obtained as follows. Andrea Malchiodi (SNS, Pisa) BMC 2016 8 / 30

The Möbius group This is a non-compact family of conformal maps from S 2 to S 2 obtained as follows. Consider first the stereographic projection π P : S 2 \{ P } → R 2 P 2 (S , g ) 0 y π (y) P Andrea Malchiodi (SNS, Pisa) BMC 2016 8 / 30

The Möbius group This is a non-compact family of conformal maps from S 2 to S 2 obtained as follows. Consider first the stereographic projection π P : S 2 \{ P } → R 2 P 2 (S , g ) 0 y π (y) P Möbius maps are obtained as compositions π − 1 Φ P,λ := ◦ λ Id R 2 ◦ π P . P Andrea Malchiodi (SNS, Pisa) BMC 2016 8 / 30

The Möbius group This is a non-compact family of conformal maps from S 2 to S 2 obtained as follows. Consider first the stereographic projection π P : S 2 \{ P } → R 2 P 2 (S , g ) 0 y π (y) P Möbius maps are obtained as compositions π − 1 Φ P,λ := ◦ λ Id R 2 ◦ π P . P P S 2 Φ P,λ Andrea Malchiodi (SNS, Pisa) BMC 2016 8 / 30

Loss of compactness Andrea Malchiodi (SNS, Pisa) BMC 2016 9 / 30

Loss of compactness The Möbius group acts on functions w ∈ W 1 , 2 ( S 2 ) in this way Andrea Malchiodi (SNS, Pisa) BMC 2016 9 / 30

Loss of compactness The Möbius group acts on functions w ∈ W 1 , 2 ( S 2 ) in this way: w Φ ( x ) := w (Φ( x )) + 1 w ( x ) �→ 2 log Jac (Φ) . Andrea Malchiodi (SNS, Pisa) BMC 2016 9 / 30

Loss of compactness The Möbius group acts on functions w ∈ W 1 , 2 ( S 2 ) in this way: w Φ ( x ) := w (Φ( x )) + 1 w ( x ) �→ 2 log Jac (Φ) . λ ≫ 1 w Φ λ P S 2 S 2 w All the volume associated to w , e 2 w , concentrates at one point for λ large. Andrea Malchiodi (SNS, Pisa) BMC 2016 9 / 30

Loss of compactness The Möbius group acts on functions w ∈ W 1 , 2 ( S 2 ) in this way: w Φ ( x ) := w (Φ( x )) + 1 w ( x ) �→ 2 log Jac (Φ) . λ ≫ 1 w Φ λ P S 2 S 2 w All the volume associated to w , e 2 w , concentrates at one point for λ large. The energy I K stays bounded despite this loss of compactness. I K K > 0 λ w Φ λ w Andrea Malchiodi (SNS, Pisa) BMC 2016 9 / 30

Compactness recovery Andrea Malchiodi (SNS, Pisa) BMC 2016 10 / 30

Compactness recovery One can recover compactness using some geometric clue ([W.Ding, ’90]). Andrea Malchiodi (SNS, Pisa) BMC 2016 10 / 30

Compactness recovery One can recover compactness using some geometric clue ([W.Ding, ’90]). Fixing P ∈ Σ , consider the equation ( E ) − ∆ w + K g = 4 πδ P on Σ . Andrea Malchiodi (SNS, Pisa) BMC 2016 10 / 30

Compactness recovery One can recover compactness using some geometric clue ([W.Ding, ’90]). Fixing P ∈ Σ , consider the equation ( E ) − ∆ w + K g = 4 πδ P on Σ . w is as singular at P as the 2D Green’s function: w ( x ) ≃ − 2 log d ( x, P ) . Andrea Malchiodi (SNS, Pisa) BMC 2016 10 / 30

Compactness recovery One can recover compactness using some geometric clue ([W.Ding, ’90]). Fixing P ∈ Σ , consider the equation ( E ) − ∆ w + K g = 4 πδ P on Σ . w is as singular at P as the 2D Green’s function: w ( x ) ≃ − 2 log d ( x, P ) . The open manifold (Σ \ { P } , e 2 w g ) is simply connected, complete and with zero Gaussian curvature. Andrea Malchiodi (SNS, Pisa) BMC 2016 10 / 30

Compactness recovery One can recover compactness using some geometric clue ([W.Ding, ’90]). Fixing P ∈ Σ , consider the equation ( E ) − ∆ w + K g = 4 πδ P on Σ . w is as singular at P as the 2D Green’s function: w ( x ) ≃ − 2 log d ( x, P ) . The open manifold (Σ \ { P } , e 2 w g ) is simply connected, complete and with zero Gaussian curvature. A classical theorem in differential geome- try implies that (Σ \ { P } , e 2 w g ) is isometric to the Euclidean plane. Andrea Malchiodi (SNS, Pisa) BMC 2016 10 / 30

Compactness recovery One can recover compactness using some geometric clue ([W.Ding, ’90]). Fixing P ∈ Σ , consider the equation ( E ) − ∆ w + K g = 4 πδ P on Σ . w is as singular at P as the 2D Green’s function: w ( x ) ≃ − 2 log d ( x, P ) . The open manifold (Σ \ { P } , e 2 w g ) is simply connected, complete and with zero Gaussian curvature. A classical theorem in differential geome- try implies that (Σ \ { P } , e 2 w g ) is isometric to the Euclidean plane. Composing then with the inverse stereographic projection, one finds the round sphere, with constant Gaussian curvature 1 . Andrea Malchiodi (SNS, Pisa) BMC 2016 10 / 30

Singular spaces Andrea Malchiodi (SNS, Pisa) BMC 2016 11 / 30

Singular spaces Non-smooth spaces generated a growing interest over the past decades. Andrea Malchiodi (SNS, Pisa) BMC 2016 11 / 30

Singular spaces Non-smooth spaces generated a growing interest over the past decades. They might arise as limits of smooth objects such as limits or limit cones of Einstein metrics in four dimensions [Anderson, ’89], [Biquard, ’11-’13], [Tian-Viaclovsky, ’04], [Colding-Minicozzi, ’14], . . . Andrea Malchiodi (SNS, Pisa) BMC 2016 11 / 30

Singular spaces Non-smooth spaces generated a growing interest over the past decades. They might arise as limits of smooth objects such as limits or limit cones of Einstein metrics in four dimensions [Anderson, ’89], [Biquard, ’11-’13], [Tian-Viaclovsky, ’04], [Colding-Minicozzi, ’14], . . . Sometimes singularities are introduced artificially to simplify a problem, and them smoothed-out via a continuity method. Andrea Malchiodi (SNS, Pisa) BMC 2016 11 / 30

Singular spaces Non-smooth spaces generated a growing interest over the past decades. They might arise as limits of smooth objects such as limits or limit cones of Einstein metrics in four dimensions [Anderson, ’89], [Biquard, ’11-’13], [Tian-Viaclovsky, ’04], [Colding-Minicozzi, ’14], . . . Sometimes singularities are introduced artificially to simplify a problem, and them smoothed-out via a continuity method. For example in the papers [Chen-Donaldson-Sun, ’15] to find Kähler-Einstein metrics. Andrea Malchiodi (SNS, Pisa) BMC 2016 11 / 30

Singular spaces Non-smooth spaces generated a growing interest over the past decades. They might arise as limits of smooth objects such as limits or limit cones of Einstein metrics in four dimensions [Anderson, ’89], [Biquard, ’11-’13], [Tian-Viaclovsky, ’04], [Colding-Minicozzi, ’14], . . . Sometimes singularities are introduced artificially to simplify a problem, and them smoothed-out via a continuity method. For example in the papers [Chen-Donaldson-Sun, ’15] to find Kähler-Einstein metrics. Singularities occur naturally in physical problems: interfaces with triple- quadruple junctions, models of space-times in general relativity, etc. Andrea Malchiodi (SNS, Pisa) BMC 2016 11 / 30

Uniformization prescribing conical singularities Andrea Malchiodi (SNS, Pisa) BMC 2016 12 / 30

Uniformization prescribing conical singularities We consider a uniformization problem on singular surfaces : try to find a constant curvature metric ˜ g on Σ , but having conical singularities of prescribed angles θ 1 , . . . , θ m at given points p 1 , . . . , p m . p i p 1 p 2 Andrea Malchiodi (SNS, Pisa) BMC 2016 12 / 30

Uniformization prescribing conical singularities We consider a uniformization problem on singular surfaces : try to find a constant curvature metric ˜ g on Σ , but having conical singularities of prescribed angles θ 1 , . . . , θ m at given points p 1 , . . . , p m . p i p 1 p 2 If ˜ g is conformal to g , in normal coordinates z near each p i it must be g ( z ) ≃ | z | 2 α i dz ⊗ dz ; ˜ θ i = 2 π (1 + α i ) , α i > − 1 . Andrea Malchiodi (SNS, Pisa) BMC 2016 12 / 30

Uniformization prescribing conical singularities We consider a uniformization problem on singular surfaces : try to find a constant curvature metric ˜ g on Σ , but having conical singularities of prescribed angles θ 1 , . . . , θ m at given points p 1 , . . . , p m . p i p 1 p 2 If ˜ g is conformal to g , in normal coordinates z near each p i it must be g ( z ) ≃ | z | 2 α i dz ⊗ dz ; ˜ θ i = 2 π (1 + α i ) , α i > − 1 . With this notation, the singular structure is encoded in a divisor m � α = α j p j . i =1 Andrea Malchiodi (SNS, Pisa) BMC 2016 12 / 30

Singular Liouville equations Andrea Malchiodi (SNS, Pisa) BMC 2016 13 / 30

Singular Liouville equations For a standard cone of angle θ = 2 π (1 + α ) the curvature is as follows: K = 0 K = − 2 παδ p p Andrea Malchiodi (SNS, Pisa) BMC 2016 13 / 30

Singular Liouville equations For a standard cone of angle θ = 2 π (1 + α ) the curvature is as follows: K = 0 K = − 2 παδ p p Looking for constant K ˜ g ≡ ρ ∈ R , we need to solve for the following singular Liouville equation m − ∆ w + K g = ρ e 2 w − 2 π � ( E ρ,α ) α j δ p j . j =1 Andrea Malchiodi (SNS, Pisa) BMC 2016 13 / 30

Singular Liouville equations For a standard cone of angle θ = 2 π (1 + α ) the curvature is as follows: K = 0 K = − 2 παδ p p Looking for constant K ˜ g ≡ ρ ∈ R , we need to solve for the following singular Liouville equation m − ∆ w + K g = ρ e 2 w − 2 π � ( E ρ,α ) α j δ p j . j =1 By the Gauss-Bonnet formula ρ must satisfy the constraint m � ρ = 2 π α j + 2 πχ (Σ) ( w.l.o.g. assume V ol g (Σ) = 1) . j =1 Andrea Malchiodi (SNS, Pisa) BMC 2016 13 / 30

Variational structure of ( E ρ,α ) Andrea Malchiodi (SNS, Pisa) BMC 2016 14 / 30

Variational structure of ( E ρ,α ) ( E ρ,α ) can be desingularized by a substitution. Andrea Malchiodi (SNS, Pisa) BMC 2016 14 / 30

Variational structure of ( E ρ,α ) ( E ρ,α ) can be desingularized by a substitution. For p ∈ Σ , let w p solve − ∆ w p = δ p − 1 on Σ . Andrea Malchiodi (SNS, Pisa) BMC 2016 14 / 30

Variational structure of ( E ρ,α ) ( E ρ,α ) can be desingularized by a substitution. For p ∈ Σ , let w p solve − ∆ w p = δ p − 1 on Σ . Again, w p ( x ) ∼ − 1 2 π log | x − p | for x close to p . Andrea Malchiodi (SNS, Pisa) BMC 2016 14 / 30

Variational structure of ( E ρ,α ) ( E ρ,α ) can be desingularized by a substitution. For p ∈ Σ , let w p solve − ∆ w p = δ p − 1 on Σ . Again, w p ( x ) ∼ − 1 2 π log | x − p | for x close to p . Changing variables as u �→ u + 2 π � j αw p j , we obtain the equivalent problem: h ( x ) e 2 u − 1 ( ˜ h ( x ) ∼ dist ( x, p j ) 2 α j . � � − ∆ u = ρ ; E ρ,α ) Andrea Malchiodi (SNS, Pisa) BMC 2016 14 / 30

Variational structure of ( E ρ,α ) ( E ρ,α ) can be desingularized by a substitution. For p ∈ Σ , let w p solve − ∆ w p = δ p − 1 on Σ . Again, w p ( x ) ∼ − 1 2 π log | x − p | for x close to p . Changing variables as u �→ u + 2 π � j αw p j , we obtain the equivalent problem: h ( x ) e 2 u − 1 ( ˜ h ( x ) ∼ dist ( x, p j ) 2 α j . � � − ∆ u = ρ ; E ρ,α ) ( ˜ E ρ,α ) is the Euler-Lagrange eq. for the functional I ρ,α : H 1 (Σ) → R � � � |∇ u | 2 + 2 ρ h ( x ) e 2 u , I ρ,α ( u ) = u − ρ log Σ Σ Σ Andrea Malchiodi (SNS, Pisa) BMC 2016 14 / 30

Variational structure of ( E ρ,α ) ( E ρ,α ) can be desingularized by a substitution. For p ∈ Σ , let w p solve − ∆ w p = δ p − 1 on Σ . Again, w p ( x ) ∼ − 1 2 π log | x − p | for x close to p . Changing variables as u �→ u + 2 π � j αw p j , we obtain the equivalent problem: h ( x ) e 2 u − 1 ( ˜ h ( x ) ∼ dist ( x, p j ) 2 α j . � � − ∆ u = ρ ; E ρ,α ) ( ˜ E ρ,α ) is the Euler-Lagrange eq. for the functional I ρ,α : H 1 (Σ) → R � � � |∇ u | 2 + 2 ρ h ( x ) e 2 u , I ρ,α ( u ) = u − ρ log Σ Σ Σ similarly to ( U ) . Notice the weight h ( x ) in the last term. Andrea Malchiodi (SNS, Pisa) BMC 2016 14 / 30

Moser-Trudinger inequality Andrea Malchiodi (SNS, Pisa) BMC 2016 15 / 30

Moser-Trudinger inequality Recall the classical Moser-Trudinger inequality � e 2( u − u ) ≤ 1 � |∇ u | 2 + C ; u ∈ H 1 (Σ) . (MT) log 4 π Σ Σ Andrea Malchiodi (SNS, Pisa) BMC 2016 15 / 30

Moser-Trudinger inequality Recall the classical Moser-Trudinger inequality � e 2( u − u ) ≤ 1 � |∇ u | 2 + C ; u ∈ H 1 (Σ) . (MT) log 4 π Σ Σ With power-type weights, it turns out that ([Chen,’90], [Troyanov,’91]) � 1 � h ( x ) e 2( u − u ) ≤ |∇ u | 2 + C. (CT) log 4 π min { 1 , 1 + min j α j } Σ Σ Andrea Malchiodi (SNS, Pisa) BMC 2016 15 / 30

Moser-Trudinger inequality Recall the classical Moser-Trudinger inequality � e 2( u − u ) ≤ 1 � |∇ u | 2 + C ; u ∈ H 1 (Σ) . (MT) log 4 π Σ Σ With power-type weights, it turns out that ([Chen,’90], [Troyanov,’91]) � 1 � h ( x ) e 2( u − u ) ≤ |∇ u | 2 + C. (CT) log 4 π min { 1 , 1 + min j α j } Σ Σ • The best constant picks-up the most singular behaviour of h ( x ) , and coincides with the classical Andrea Malchiodi (SNS, Pisa) BMC 2016 15 / 30

Three geometric cases: ρ = 2 π � m j =1 α j + 2 πχ (Σ) Andrea Malchiodi (SNS, Pisa) BMC 2016 16 / 30

Three geometric cases: ρ = 2 π � m j =1 α j + 2 πχ (Σ) Sub-critical case ρ < 4 π min { 1 , 1 + min j α j } . Andrea Malchiodi (SNS, Pisa) BMC 2016 16 / 30

Three geometric cases: ρ = 2 π � m j =1 α j + 2 πχ (Σ) Sub-critical case ρ < 4 π min { 1 , 1 + min j α j } . I ρ,α coercive, and one can solve by minimization ([Troyanov, ’91], [Thurston, ’98]). Andrea Malchiodi (SNS, Pisa) BMC 2016 16 / 30

Three geometric cases: ρ = 2 π � m j =1 α j + 2 πχ (Σ) Sub-critical case ρ < 4 π min { 1 , 1 + min j α j } . I ρ,α coercive, and one can solve by minimization ([Troyanov, ’91], [Thurston, ’98]). Critical case ρ = 4 π min { 1 , 1+min j α j } . Andrea Malchiodi (SNS, Pisa) BMC 2016 16 / 30

Three geometric cases: ρ = 2 π � m j =1 α j + 2 πχ (Σ) Sub-critical case ρ < 4 π min { 1 , 1 + min j α j } . I ρ,α coercive, and one can solve by minimization ([Troyanov, ’91], [Thurston, ’98]). Critical case ρ = 4 π min { 1 , 1+min j α j } . I ρ,α bd. below but not coercive. Andrea Malchiodi (SNS, Pisa) BMC 2016 16 / 30

Three geometric cases: ρ = 2 π � m j =1 α j + 2 πχ (Σ) Sub-critical case ρ < 4 π min { 1 , 1 + min j α j } . I ρ,α coercive, and one can solve by minimization ([Troyanov, ’91], [Thurston, ’98]). Critical case ρ = 4 π min { 1 , 1+min j α j } . I ρ,α bd. below but not coercive. Supercritical case ρ > 4 π min { 1 , 1 + min j α j } . Andrea Malchiodi (SNS, Pisa) BMC 2016 16 / 30

Three geometric cases: ρ = 2 π � m j =1 α j + 2 πχ (Σ) Sub-critical case ρ < 4 π min { 1 , 1 + min j α j } . I ρ,α coercive, and one can solve by minimization ([Troyanov, ’91], [Thurston, ’98]). Critical case ρ = 4 π min { 1 , 1+min j α j } . I ρ,α bd. below but not coercive. Supercritical case ρ > 4 π min { 1 , 1 + min j α j } . I ρ,α unbounded below. Andrea Malchiodi (SNS, Pisa) BMC 2016 16 / 30

Three geometric cases: ρ = 2 π � m j =1 α j + 2 πχ (Σ) Sub-critical case ρ < 4 π min { 1 , 1 + min j α j } . I ρ,α coercive, and one can solve by minimization ([Troyanov, ’91], [Thurston, ’98]). Critical case ρ = 4 π min { 1 , 1+min j α j } . I ρ,α bd. below but not coercive. Supercritical case ρ > 4 π min { 1 , 1 + min j α j } . I ρ,α unbounded below. In some supercritical cases the problem is not solvable, as for example the tear-drop : Andrea Malchiodi (SNS, Pisa) BMC 2016 16 / 30

Three geometric cases: ρ = 2 π � m j =1 α j + 2 πχ (Σ) Sub-critical case ρ < 4 π min { 1 , 1 + min j α j } . I ρ,α coercive, and one can solve by minimization ([Troyanov, ’91], [Thurston, ’98]). Critical case ρ = 4 π min { 1 , 1+min j α j } . I ρ,α bd. below but not coercive. Supercritical case ρ > 4 π min { 1 , 1 + min j α j } . I ρ,α unbounded below. In some supercritical cases the problem is not solvable, as for example the tear-drop : S 2 with one singular point. p S 2 Andrea Malchiodi (SNS, Pisa) BMC 2016 16 / 30

Three geometric cases: ρ = 2 π � m j =1 α j + 2 πχ (Σ) Sub-critical case ρ < 4 π min { 1 , 1 + min j α j } . I ρ,α coercive, and one can solve by minimization ([Troyanov, ’91], [Thurston, ’98]). Critical case ρ = 4 π min { 1 , 1+min j α j } . I ρ,α bd. below but not coercive. Supercritical case ρ > 4 π min { 1 , 1 + min j α j } . I ρ,α unbounded below. In some supercritical cases the problem is not solvable, as for example the tear-drop : S 2 with one singular point. p S 2 From the PDE point of view, the difficulty involves blow-up phenomena (indefinite concentration of energy/volume). Andrea Malchiodi (SNS, Pisa) BMC 2016 16 / 30

Blowing-up solutions Andrea Malchiodi (SNS, Pisa) BMC 2016 17 / 30

Blowing-up solutions A sequence of solutions is said to blow-up if it becomes unbounded. Andrea Malchiodi (SNS, Pisa) BMC 2016 17 / 30

Blowing-up solutions A sequence of solutions is said to blow-up if it becomes unbounded. This is bad in principle, but one can use the Möbius invariance of the equation to bring it back to finite values and obtain a blow-up profile . Andrea Malchiodi (SNS, Pisa) BMC 2016 17 / 30

Blowing-up solutions A sequence of solutions is said to blow-up if it becomes unbounded. This is bad in principle, but one can use the Möbius invariance of the equation to bring it back to finite values and obtain a blow-up profile . For the singular Liouville problem there are two different profiles 2) (north) American Football 1) Sphere � K = 4 π � K = 4 π (1 + α ) Andrea Malchiodi (SNS, Pisa) BMC 2016 17 / 30

Blow-up implies quantization Andrea Malchiodi (SNS, Pisa) BMC 2016 18 / 30

Blow-up implies quantization Theorem ([Bartolucci-Tarantello, ’02]) Suppose a sequence of solutions u n to ( ˜ E ρ,α ) blows-up. Andrea Malchiodi (SNS, Pisa) BMC 2016 18 / 30

Blow-up implies quantization Theorem ([Bartolucci-Tarantello, ’02]) Suppose a sequence of solutions u n to ( ˜ E ρ,α ) blows-up. Then u n concentrates at finitely-many points, developing k spheres, k ≥ 0 , plus possibly American Footballs at some of the singular points p i . Andrea Malchiodi (SNS, Pisa) BMC 2016 18 / 30

Blow-up implies quantization Theorem ([Bartolucci-Tarantello, ’02]) Suppose a sequence of solutions u n to ( ˜ E ρ,α ) blows-up. Then u n concentrates at finitely-many points, developing k spheres, k ≥ 0 , plus possibly American Footballs at some of the singular points p i . Therefore ρ ∈ Λ α , where � � 4 π (1 + α j ) : k ∈ N ∗ , J ⊆ { 1 , . . . , m } � Λ α := 4 kπ + . j ∈ J Andrea Malchiodi (SNS, Pisa) BMC 2016 18 / 30

Blow-up implies quantization Theorem ([Bartolucci-Tarantello, ’02]) Suppose a sequence of solutions u n to ( ˜ E ρ,α ) blows-up. Then u n concentrates at finitely-many points, developing k spheres, k ≥ 0 , plus possibly American Footballs at some of the singular points p i . Therefore ρ ∈ Λ α , where � � 4 π (1 + α j ) : k ∈ N ∗ , J ⊆ { 1 , . . . , m } � Λ α := 4 kπ + . j ∈ J • Notice that Λ α is discrete, for example Λ α = 4 π N when α = 0 . Andrea Malchiodi (SNS, Pisa) BMC 2016 18 / 30