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

Uniformization of singular surfaces

Andrea Malchiodi (SNS, Pisa) BMC 2016

Andrea Malchiodi (SNS, Pisa) BMC 2016 1 / 30

The uniformization problem

Andrea Malchiodi (SNS, Pisa) BMC 2016 2 / 30

The uniformization problem

Let us consider a closed compact (2D) surface (Σ, g), with no boundary.

Andrea Malchiodi (SNS, Pisa) BMC 2016 2 / 30

The uniformization problem

Let us consider a closed compact (2D) surface (Σ, g), with no boundary. A classical geometric problem is to find the best possible metric, namely to deform g so that the Gaussian curvature of Σ becomes constant.

Andrea Malchiodi (SNS, Pisa) BMC 2016 2 / 30

The uniformization problem

Let us consider a closed compact (2D) surface (Σ, g), with no boundary. A classical geometric problem is to find the best possible metric, namely to deform g so that the Gaussian curvature of Σ becomes constant. Due to the classical Uniformization Theorem ((Klein)-Koebe-Poincaré) this is known to be always possible.

Andrea Malchiodi (SNS, Pisa) BMC 2016 2 / 30

The uniformization problem as a PDE

Andrea Malchiodi (SNS, Pisa) BMC 2016 3 / 30

The uniformization problem as a PDE

The problem can be formulated as a partial differential equation on the surface Σ.

Andrea Malchiodi (SNS, Pisa) BMC 2016 3 / 30

The uniformization problem as a PDE

The problem can be formulated as a partial differential equation on the surface Σ. The deformation of the metric g can be taken of conformal type, namely

• f the form ˜

g(x) = λ(x)g(x), with λ smooth and positive function on Σ.

Andrea Malchiodi (SNS, Pisa) BMC 2016 3 / 30

The uniformization problem as a PDE

The problem can be formulated as a partial differential equation on the surface Σ. The deformation of the metric g can be taken of conformal type, namely

• f the form ˜

g(x) = λ(x)g(x), with λ smooth and positive function on Σ. Writing (conveniently) a conformal metric ˜ g on Σ as ˜ g = e2wg, then the Gaussian curvature transforms according to the law −∆gw + Kg = K˜

g e2w

Andrea Malchiodi (SNS, Pisa) BMC 2016 3 / 30

The uniformization problem as a PDE

The problem can be formulated as a partial differential equation on the surface Σ. The deformation of the metric g can be taken of conformal type, namely

• f the form ˜

g(x) = λ(x)g(x), with λ smooth and positive function on Σ. Writing (conveniently) a conformal metric ˜ g on Σ as ˜ g = e2wg, then the Gaussian curvature transforms according to the law −∆gw + Kg = K˜

g e2w,

which reduces the uniformization problem to a PDE

Andrea Malchiodi (SNS, Pisa) BMC 2016 3 / 30

The uniformization problem as a PDE

The problem can be formulated as a partial differential equation on the surface Σ. The deformation of the metric g can be taken of conformal type, namely

• f the form ˜

g(x) = λ(x)g(x), with λ smooth and positive function on Σ. Writing (conveniently) a conformal metric ˜ g on Σ as ˜ g = e2wg, then the Gaussian curvature transforms according to the law −∆gw + Kg = K˜

g e2w,

which reduces the uniformization problem to a PDE: solving the above equation in w with K˜

g = K ∈ R.

Andrea Malchiodi (SNS, Pisa) BMC 2016 3 / 30

A PDE proof of the Uniformization Theorem

Andrea Malchiodi (SNS, Pisa) BMC 2016 4 / 30

A PDE proof of the Uniformization Theorem

The original arguments relied on complex analysis.

Andrea Malchiodi (SNS, Pisa) BMC 2016 4 / 30

A PDE proof of the Uniformization Theorem

The original arguments relied on complex analysis. We will see how to use instead the PDE form of the problem.

Andrea Malchiodi (SNS, Pisa) BMC 2016 4 / 30

A PDE proof of the Uniformization Theorem

The original arguments relied on complex analysis. We will see how to use instead the PDE form of the problem. By the above transformation law for Kg we have to solve (U) − ∆gu + Kg = Ke2u

Andrea Malchiodi (SNS, Pisa) BMC 2016 4 / 30

A PDE proof of the Uniformization Theorem

The original arguments relied on complex analysis. We will see how to use instead the PDE form of the problem. By the above transformation law for Kg we have to solve (U) − ∆gu + Kg = Ke2u, K =

• Σ

KgdVg.

Andrea Malchiodi (SNS, Pisa) BMC 2016 4 / 30

A PDE proof of the Uniformization Theorem

The original arguments relied on complex analysis. We will see how to use instead the PDE form of the problem. By the above transformation law for Kg we have to solve (U) − ∆gu + Kg = Ke2u, K =

• Σ

KgdVg. Easy case: K = 0.

Andrea Malchiodi (SNS, Pisa) BMC 2016 4 / 30

A PDE proof of the Uniformization Theorem

The original arguments relied on complex analysis. We will see how to use instead the PDE form of the problem. By the above transformation law for Kg we have to solve (U) − ∆gu + Kg = Ke2u, K =

• Σ

KgdVg. Easy case: K = 0. The equation simply becomes ∆gu = Kg.

Andrea Malchiodi (SNS, Pisa) BMC 2016 4 / 30

A PDE proof of the Uniformization Theorem

The original arguments relied on complex analysis. We will see how to use instead the PDE form of the problem. By the above transformation law for Kg we have to solve (U) − ∆gu + Kg = Ke2u, K =

• Σ

KgdVg. Easy case: K = 0. The equation simply becomes ∆gu = Kg. By Fredholm’s alternative the equation ∆gu = f is solvable if and only if f integrates to zero

Andrea Malchiodi (SNS, Pisa) BMC 2016 4 / 30

A PDE proof of the Uniformization Theorem

The original arguments relied on complex analysis. We will see how to use instead the PDE form of the problem. By the above transformation law for Kg we have to solve (U) − ∆gu + Kg = Ke2u, K =

• Σ

KgdVg. Easy case: K = 0. The equation simply becomes ∆gu = Kg. By Fredholm’s alternative the equation ∆gu = f is solvable if and only if f integrates to zero, which is the case here.

Andrea Malchiodi (SNS, Pisa) BMC 2016 4 / 30

Variational structure of (U)

Andrea Malchiodi (SNS, Pisa) BMC 2016 5 / 30

Variational structure of (U)

It turns out that solutions of (U) are stationary points of an energy, defined on the Sobolev space W 1,2(Σ), given by IK(u) =

• Σ

|∇gu|2dVg + 2

• Σ

Kgu dVg − K log

• Σ

e2udVg.

Andrea Malchiodi (SNS, Pisa) BMC 2016 5 / 30

Variational structure of (U)

It turns out that solutions of (U) are stationary points of an energy, defined on the Sobolev space W 1,2(Σ), given by IK(u) =

• Σ

|∇gu|2dVg + 2

• Σ

Kgu dVg − K log

• Σ

e2udVg. This energy has also interest in string theory ([Polyakov, ’81]) as well as in spectral theory ([Osgood-Phillips-Sarnak, ’88]).

Andrea Malchiodi (SNS, Pisa) BMC 2016 5 / 30

Variational structure of (U)

It turns out that solutions of (U) are stationary points of an energy, defined on the Sobolev space W 1,2(Σ), given by IK(u) =

• Σ

|∇gu|2dVg + 2

• Σ

Kgu dVg − K log

• Σ

e2udVg. This energy has also interest in string theory ([Polyakov, ’81]) as well as in spectral theory ([Osgood-Phillips-Sarnak, ’88]). In 2D we have the Sobolev embedding of W 1,2(Σ) into every Lp(Σ)

Andrea Malchiodi (SNS, Pisa) BMC 2016 5 / 30

Variational structure of (U)

It turns out that solutions of (U) are stationary points of an energy, defined on the Sobolev space W 1,2(Σ), given by IK(u) =

• Σ

|∇gu|2dVg + 2

• Σ

Kgu dVg − K log

• Σ

e2udVg. This energy has also interest in string theory ([Polyakov, ’81]) as well as in spectral theory ([Osgood-Phillips-Sarnak, ’88]). In 2D we have the Sobolev embedding of W 1,2(Σ) into every Lp(Σ), which can be pushed to exponential class by the Moser-Trudinger inequality (MT) log

• Σ

e2(u−u)dVg ≤ 1 4π

• Σ

|∇gu|2dVg + CΣ.

Andrea Malchiodi (SNS, Pisa) BMC 2016 5 / 30

Variational structure of (U)

It turns out that solutions of (U) are stationary points of an energy, defined on the Sobolev space W 1,2(Σ), given by IK(u) =

• Σ

|∇gu|2dVg + 2

• Σ

Kgu dVg − K log

• Σ

e2udVg. This energy has also interest in string theory ([Polyakov, ’81]) as well as in spectral theory ([Osgood-Phillips-Sarnak, ’88]). In 2D we have the Sobolev embedding of W 1,2(Σ) into every Lp(Σ), which can be pushed to exponential class by the Moser-Trudinger inequality (MT) log

• Σ

e2(u−u)dVg ≤ 1 4π

• Σ

|∇gu|2dVg + CΣ. This shows that IK is well defined on W 1,2(Σ).

Andrea Malchiodi (SNS, Pisa) BMC 2016 5 / 30

The negative curvature case

Andrea Malchiodi (SNS, Pisa) BMC 2016 6 / 30

The negative curvature case

This case is still rather easy.

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

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

IK K < 0

This means that IK → +∞ as u → ∞.

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

IK K < 0

This means that IK → +∞ as u → ∞. To find a critical point of IK, 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

IK K < 0

This means that IK → +∞ as u → ∞. To find a critical point of IK, 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: whenK > 0and Σ has the topology of S2.

Andrea Malchiodi (SNS, Pisa) BMC 2016 7 / 30

The positive curvature case

This is the most difficult case: whenK > 0and Σ has the topology of S2. Notice that the Gauss-Bonnet formula implies K = 4π.

Andrea Malchiodi (SNS, Pisa) BMC 2016 7 / 30

The positive curvature case

This is the most difficult case: whenK > 0and Σ has the topology of S2. Notice that the Gauss-Bonnet formula implies K = 4π. Using the Moser-Trudinger inequality one finds that IK 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: whenK > 0and Σ has the topology of S2. Notice that the Gauss-Bonnet formula implies K = 4π. Using the Moser-Trudinger inequality one finds that IK is still bounded from below, but coercivity is lost. The picture might look like IK K > 0

Andrea Malchiodi (SNS, Pisa) BMC 2016 7 / 30

The positive curvature case

This is the most difficult case: whenK > 0and Σ has the topology of S2. Notice that the Gauss-Bonnet formula implies K = 4π. Using the Moser-Trudinger inequality one finds that IK is still bounded from below, but coercivity is lost. The picture might look like IK K > 0 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: whenK > 0and Σ has the topology of S2. Notice that the Gauss-Bonnet formula implies K = 4π. Using the Moser-Trudinger inequality one finds that IK is still bounded from below, but coercivity is lost. The picture might look like IK K > 0 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 S2 to S2 obtained as follows.

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

The Möbius group

This is a non-compact family of conformal maps from S2 to S2 obtained as follows. Consider first the stereographic projection πP : S2\{P} → R2

P

P

π y (y) (S , g )

2

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

The Möbius group

This is a non-compact family of conformal maps from S2 to S2 obtained as follows. Consider first the stereographic projection πP : S2\{P} → R2

P

P

π y (y) (S , g )

2

Möbius maps are obtained as compositions ΦP,λ := π−1

P

• λ IdR2
• πP .

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

The Möbius group

This is a non-compact family of conformal maps from S2 to S2 obtained as follows. Consider first the stereographic projection πP : S2\{P} → R2

P

P

π y (y) (S , g )

2

Möbius maps are obtained as compositions ΦP,λ := π−1

P

• λ IdR2
• πP .

S2 P ΦP,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(S2) in this way

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

Loss of compactness

The Möbius group acts on functions w ∈ W 1,2(S2) in this way: w(x) → wΦ(x) := w(Φ(x)) + 1

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(S2) in this way: w(x) → wΦ(x) := w(Φ(x)) + 1

2 log Jac(Φ).

S2 P w wΦλ S2 λ ≫ 1

All the volume associated to w, e2w, 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(S2) in this way: w(x) → wΦ(x) := w(Φ(x)) + 1

2 log Jac(Φ).

S2 P w wΦλ S2 λ ≫ 1

All the volume associated to w, e2w, concentrates at one point for λ large. The energy IK stays bounded despite this loss of compactness.

IK K > 0 λ w wΦλ

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 + Kg = 4πδP

• n 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 + Kg = 4πδP

• n 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 + Kg = 4πδP

• n 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}, e2wg) 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 + Kg = 4πδP

• n 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}, e2wg) is simply connected, complete and with zero Gaussian curvature. A classical theorem in differential geome- try implies that (Σ \ {P}, e2wg) 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 + Kg = 4πδP

• n 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}, e2wg) is simply connected, complete and with zero Gaussian curvature. A classical theorem in differential geome- try implies that (Σ \ {P}, e2wg) 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

• f 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

• f 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

• f 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

• f 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 θ1, . . . , θm at given points p1, . . . , pm.

p1 p2 pi

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 θ1, . . . , θm at given points p1, . . . , pm.

p1 p2 pi

If ˜ g is conformal to g, in normal coordinates z near each pi it must be ˜ g(z) ≃ |z|2αidz ⊗ dz; θi = 2π(1 + αi), αi > −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 θ1, . . . , θm at given points p1, . . . , pm.

p1 p2 pi

If ˜ g is conformal to g, in normal coordinates z near each pi it must be ˜ g(z) ≃ |z|2αidz ⊗ dz; θi = 2π(1 + αi), αi > −1. With this notation, the singular structure is encoded in a divisor α =

m

• i=1

αj pj.

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 = 2π(1 + α) the curvature is as follows:

K = 0 p K = −2παδp

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

Singular Liouville equations

For a standard cone of angle θ = 2π(1 + α) the curvature is as follows:

K = 0 p K = −2παδp

Looking for constant K˜

g ≡ ρ ∈ R, we need to solve for the following

singular Liouville equation (Eρ,α) −∆w + Kg = ρ e 2w − 2π

m

• j=1

αjδpj.

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

Singular Liouville equations

For a standard cone of angle θ = 2π(1 + α) the curvature is as follows:

K = 0 p K = −2παδp

Looking for constant K˜

g ≡ ρ ∈ R, we need to solve for the following

singular Liouville equation (Eρ,α) −∆w + Kg = ρ e 2w − 2π

m

• j=1

αjδpj. By the Gauss-Bonnet formula ρ must satisfy the constraint ρ = 2π

m

• j=1

αj + 2πχ(Σ) (w.l.o.g. assume V olg(Σ) = 1).

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

Variational structure of (Eρ,α)

Andrea Malchiodi (SNS, Pisa) BMC 2016 14 / 30

Variational structure of (Eρ,α)

(Eρ,α) can be desingularized by a substitution.

Andrea Malchiodi (SNS, Pisa) BMC 2016 14 / 30

Variational structure of (Eρ,α)

(Eρ,α) can be desingularized by a substitution. For p ∈ Σ, let wp solve − ∆wp = δp − 1

• n Σ.

Andrea Malchiodi (SNS, Pisa) BMC 2016 14 / 30

Variational structure of (Eρ,α)

(Eρ,α) can be desingularized by a substitution. For p ∈ Σ, let wp solve − ∆wp = δp − 1

• n Σ.

Again, wp(x) ∼ − 1

2π log |x − p| for x close to p.

Andrea Malchiodi (SNS, Pisa) BMC 2016 14 / 30

Variational structure of (Eρ,α)

(Eρ,α) can be desingularized by a substitution. For p ∈ Σ, let wp solve − ∆wp = δp − 1

• n Σ.

Again, wp(x) ∼ − 1

2π log |x − p| for x close to p. Changing variables as

u → u + 2π

j αwpj, we obtain the equivalent problem:

− ∆u = ρ

• h(x)e2u − 1
• ;

h(x) ∼ dist(x, pj)2αj. ( ˜ Eρ,α)

Andrea Malchiodi (SNS, Pisa) BMC 2016 14 / 30

Variational structure of (Eρ,α)

(Eρ,α) can be desingularized by a substitution. For p ∈ Σ, let wp solve − ∆wp = δp − 1

• n Σ.

Again, wp(x) ∼ − 1

2π log |x − p| for x close to p. Changing variables as

u → u + 2π

j αwpj, we obtain the equivalent problem:

− ∆u = ρ

• h(x)e2u − 1
• ;

h(x) ∼ dist(x, pj)2αj. ( ˜ Eρ,α) ( ˜ Eρ,α) is the Euler-Lagrange eq. for the functional Iρ,α : H1(Σ) → R Iρ,α(u) =

• Σ

|∇u|2 + 2ρ

• Σ

u − ρ log

• Σ

h(x)e2u,

Andrea Malchiodi (SNS, Pisa) BMC 2016 14 / 30

Variational structure of (Eρ,α)

(Eρ,α) can be desingularized by a substitution. For p ∈ Σ, let wp solve − ∆wp = δp − 1

• n Σ.

Again, wp(x) ∼ − 1

2π log |x − p| for x close to p. Changing variables as

u → u + 2π

j αwpj, we obtain the equivalent problem:

− ∆u = ρ

• h(x)e2u − 1
• ;

h(x) ∼ dist(x, pj)2αj. ( ˜ Eρ,α) ( ˜ Eρ,α) is the Euler-Lagrange eq. for the functional Iρ,α : H1(Σ) → R Iρ,α(u) =

• Σ

|∇u|2 + 2ρ

• Σ

u − ρ log

• Σ

h(x)e2u, 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 (MT) log

• Σ

e2(u−u) ≤ 1 4π

• Σ

|∇u|2 + C; u ∈ H1(Σ).

Andrea Malchiodi (SNS, Pisa) BMC 2016 15 / 30

Moser-Trudinger inequality

Recall the classical Moser-Trudinger inequality (MT) log

• Σ

e2(u−u) ≤ 1 4π

• Σ

|∇u|2 + C; u ∈ H1(Σ). With power-type weights, it turns out that ([Chen,’90], [Troyanov,’91]) (CT) log

• Σ

h(x)e2(u−u) ≤ 1 4π min{1, 1 + minj αj}

• Σ

|∇u|2 + C.

Andrea Malchiodi (SNS, Pisa) BMC 2016 15 / 30

Moser-Trudinger inequality

Recall the classical Moser-Trudinger inequality (MT) log

• Σ

e2(u−u) ≤ 1 4π

• Σ

|∇u|2 + C; u ∈ H1(Σ). With power-type weights, it turns out that ([Chen,’90], [Troyanov,’91]) (CT) log

• Σ

h(x)e2(u−u) ≤ 1 4π min{1, 1 + minj αj}

• Σ

|∇u|2 + C.

• 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 = 2π m

j=1 αj + 2πχ(Σ)

Andrea Malchiodi (SNS, Pisa) BMC 2016 16 / 30

Three geometric cases: ρ = 2π m

j=1 αj + 2πχ(Σ)

Sub-critical case ρ < 4π min{1, 1 + minj αj}.

Andrea Malchiodi (SNS, Pisa) BMC 2016 16 / 30

Three geometric cases: ρ = 2π m

j=1 αj + 2πχ(Σ)

Sub-critical case ρ < 4π min{1, 1 + minj αj}. Iρ,α coercive, and one can solve by minimization ([Troyanov, ’91], [Thurston, ’98]).

Andrea Malchiodi (SNS, Pisa) BMC 2016 16 / 30

Three geometric cases: ρ = 2π m

j=1 αj + 2πχ(Σ)

Sub-critical case ρ < 4π min{1, 1 + minj αj}. Iρ,α coercive, and one can solve by minimization ([Troyanov, ’91], [Thurston, ’98]). Critical caseρ = 4π min{1, 1+minj αj}.

Andrea Malchiodi (SNS, Pisa) BMC 2016 16 / 30

Three geometric cases: ρ = 2π m

j=1 αj + 2πχ(Σ)

Sub-critical case ρ < 4π min{1, 1 + minj αj}. Iρ,α coercive, and one can solve by minimization ([Troyanov, ’91], [Thurston, ’98]). Critical caseρ = 4π min{1, 1+minj αj}.Iρ,α bd. below but not coercive.

Andrea Malchiodi (SNS, Pisa) BMC 2016 16 / 30

Three geometric cases: ρ = 2π m

j=1 αj + 2πχ(Σ)

Sub-critical case ρ < 4π min{1, 1 + minj αj}. Iρ,α coercive, and one can solve by minimization ([Troyanov, ’91], [Thurston, ’98]). Critical caseρ = 4π min{1, 1+minj αj}.Iρ,α bd. below but not coercive. Supercritical case ρ > 4π min{1, 1 + minj αj}.

Andrea Malchiodi (SNS, Pisa) BMC 2016 16 / 30

Three geometric cases: ρ = 2π m

j=1 αj + 2πχ(Σ)

Sub-critical case ρ < 4π min{1, 1 + minj αj}. Iρ,α coercive, and one can solve by minimization ([Troyanov, ’91], [Thurston, ’98]). Critical caseρ = 4π min{1, 1+minj αj}.Iρ,α bd. below but not coercive. Supercritical case ρ > 4π min{1, 1 + minj αj}. Iρ,α unbounded below.

Andrea Malchiodi (SNS, Pisa) BMC 2016 16 / 30

Three geometric cases: ρ = 2π m

j=1 αj + 2πχ(Σ)

Sub-critical case ρ < 4π min{1, 1 + minj αj}. Iρ,α coercive, and one can solve by minimization ([Troyanov, ’91], [Thurston, ’98]). Critical caseρ = 4π min{1, 1+minj αj}.Iρ,α bd. below but not coercive. Supercritical case ρ > 4π min{1, 1 + minj αj}. Iρ,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 = 2π m

j=1 αj + 2πχ(Σ)

Sub-critical case ρ < 4π min{1, 1 + minj αj}. Iρ,α coercive, and one can solve by minimization ([Troyanov, ’91], [Thurston, ’98]). Critical caseρ = 4π min{1, 1+minj αj}.Iρ,α bd. below but not coercive. Supercritical case ρ > 4π min{1, 1 + minj αj}. Iρ,α unbounded below. In some supercritical cases the problem is not solvable, as for example the tear-drop: S2 with one singular point. p S2

Andrea Malchiodi (SNS, Pisa) BMC 2016 16 / 30

Three geometric cases: ρ = 2π m

j=1 αj + 2πχ(Σ)

Sub-critical case ρ < 4π min{1, 1 + minj αj}. Iρ,α coercive, and one can solve by minimization ([Troyanov, ’91], [Thurston, ’98]). Critical caseρ = 4π min{1, 1+minj αj}.Iρ,α bd. below but not coercive. Supercritical case ρ > 4π min{1, 1 + minj αj}. Iρ,α unbounded below. In some supercritical cases the problem is not solvable, as for example the tear-drop: S2 with one singular point. p S2 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

K = 4π

1) Sphere

K = 4π(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 un to ( ˜ Eρ,α) blows-up.

Andrea Malchiodi (SNS, Pisa) BMC 2016 18 / 30

Blow-up implies quantization

Theorem ([Bartolucci-Tarantello, ’02]) Suppose a sequence of solutions un to ( ˜ Eρ,α) blows-up. Then un concentrates at finitely-many points, developing k spheres, k ≥ 0, plus possibly American Footballs at some

• f the singular points pi.

Andrea Malchiodi (SNS, Pisa) BMC 2016 18 / 30

Blow-up implies quantization

Theorem ([Bartolucci-Tarantello, ’02]) Suppose a sequence of solutions un to ( ˜ Eρ,α) blows-up. Then un concentrates at finitely-many points, developing k spheres, k ≥ 0, plus possibly American Footballs at some

• f the singular points pi. Therefore ρ ∈ Λα, where

Λα :=

• 4kπ +
• j∈J

4π(1 + αj) : k ∈ N∗, J ⊆ {1, . . . , m}

• .

Andrea Malchiodi (SNS, Pisa) BMC 2016 18 / 30

Blow-up implies quantization

Theorem ([Bartolucci-Tarantello, ’02]) Suppose a sequence of solutions un to ( ˜ Eρ,α) blows-up. Then un concentrates at finitely-many points, developing k spheres, k ≥ 0, plus possibly American Footballs at some

• f the singular points pi. Therefore ρ ∈ Λα, where

Λα :=

• 4kπ +
• j∈J

4π(1 + αj) : k ∈ N∗, J ⊆ {1, . . . , m}

• .
• Notice that Λα is discrete, for example Λα = 4πN when α = 0.

Andrea Malchiodi (SNS, Pisa) BMC 2016 18 / 30

Blow-up implies quantization

Theorem ([Bartolucci-Tarantello, ’02]) Suppose a sequence of solutions un to ( ˜ Eρ,α) blows-up. Then un concentrates at finitely-many points, developing k spheres, k ≥ 0, plus possibly American Footballs at some

• f the singular points pi. Therefore ρ ∈ Λα, where

Λα :=

• 4kπ +
• j∈J

4π(1 + αj) : k ∈ N∗, J ⊆ {1, . . . , m}

• .
• Notice that Λα is discrete, for example Λα = 4πN when α = 0. This

discreteness implies that compactness holds for most ρ’s, which is good.

Andrea Malchiodi (SNS, Pisa) BMC 2016 18 / 30

Blow-up implies quantization

Theorem ([Bartolucci-Tarantello, ’02]) Suppose a sequence of solutions un to ( ˜ Eρ,α) blows-up. Then un concentrates at finitely-many points, developing k spheres, k ≥ 0, plus possibly American Footballs at some

• f the singular points pi. Therefore ρ ∈ Λα, where

Λα :=

• 4kπ +
• j∈J

4π(1 + αj) : k ∈ N∗, J ⊆ {1, . . . , m}

• .
• Notice that Λα is discrete, for example Λα = 4πN when α = 0. This

discreteness implies that compactness holds for most ρ’s, which is good. The structure of Iρ,α changes each time ρ crosses an element of Λα.

Andrea Malchiodi (SNS, Pisa) BMC 2016 18 / 30

Concentration of volume for low energy

Andrea Malchiodi (SNS, Pisa) BMC 2016 19 / 30

Concentration of volume for low energy

The supercritical case is when inf Iρ,α = −∞.

Andrea Malchiodi (SNS, Pisa) BMC 2016 19 / 30

Concentration of volume for low energy

The supercritical case is when inf Iρ,α = −∞. It turns out that concen- tration occurs only assuming low energy (no use of the equation).

Andrea Malchiodi (SNS, Pisa) BMC 2016 19 / 30

Concentration of volume for low energy

The supercritical case is when inf Iρ,α = −∞. It turns out that concen- tration occurs only assuming low energy (no use of the equation).

h(x)e2u p2,u p1,u · · · pk,u Iρ,α(u) low = ⇒

Andrea Malchiodi (SNS, Pisa) BMC 2016 19 / 30

Concentration of volume for low energy

The supercritical case is when inf Iρ,α = −∞. It turns out that concen- tration occurs only assuming low energy (no use of the equation).

h(x)e2u p2,u p1,u · · · pk,u Iρ,α(u) low = ⇒

The number of limit points and their character (regular or singular) de- pends on the topology of Σ and on the singular structure α.

Andrea Malchiodi (SNS, Pisa) BMC 2016 19 / 30

Concentration of volume for low energy

The supercritical case is when inf Iρ,α = −∞. It turns out that concen- tration occurs only assuming low energy (no use of the equation).

h(x)e2u p2,u p1,u · · · pk,u Iρ,α(u) low = ⇒

The number of limit points and their character (regular or singular) de- pends on the topology of Σ and on the singular structure α. Multiple points can only occur in supercritical regimes.

Andrea Malchiodi (SNS, Pisa) BMC 2016 19 / 30

Weigthted counting and admissible measures

Andrea Malchiodi (SNS, Pisa) BMC 2016 20 / 30

Weigthted counting and admissible measures

We introduce a weighted cardinality as follows

Andrea Malchiodi (SNS, Pisa) BMC 2016 20 / 30

Weigthted counting and admissible measures

We introduce a weighted cardinality as follows

• set ω(q) = 4π if q is a regular point

Andrea Malchiodi (SNS, Pisa) BMC 2016 20 / 30

Weigthted counting and admissible measures

We introduce a weighted cardinality as follows

• set ω(q) = 4π if q is a regular point;
• set ω(q) = 4π(1 + αi) if q = pi for some i = 1, . . . , m

Andrea Malchiodi (SNS, Pisa) BMC 2016 20 / 30

Weigthted counting and admissible measures

We introduce a weighted cardinality as follows

• set ω(q) = 4π if q is a regular point;
• set ω(q) = 4π(1 + αi) if q = pi for some i = 1, . . . , m;
• extend ω to finite sets by additivity.

Andrea Malchiodi (SNS, Pisa) BMC 2016 20 / 30

Weigthted counting and admissible measures

We introduce a weighted cardinality as follows

• set ω(q) = 4π if q is a regular point;
• set ω(q) = 4π(1 + αi) if q = pi for some i = 1, . . . , m;
• extend ω to finite sets by additivity.

We claim that the admissible limit measures (for low energy) are Σρ,α := {unit measures supported on finite sets A ⊆ Σ : ω(A) < ρ} .

Andrea Malchiodi (SNS, Pisa) BMC 2016 20 / 30

Weigthted counting and admissible measures

We introduce a weighted cardinality as follows

• set ω(q) = 4π if q is a regular point;
• set ω(q) = 4π(1 + αi) if q = pi for some i = 1, . . . , m;
• extend ω to finite sets by additivity.

We claim that the admissible limit measures (for low energy) are Σρ,α := {unit measures supported on finite sets A ⊆ Σ : ω(A) < ρ} . Notice that all these measures are finite combinations of Dirac masses.

Andrea Malchiodi (SNS, Pisa) BMC 2016 20 / 30

Weigthted counting and admissible measures

We introduce a weighted cardinality as follows

• set ω(q) = 4π if q is a regular point;
• set ω(q) = 4π(1 + αi) if q = pi for some i = 1, . . . , m;
• extend ω to finite sets by additivity.

We claim that the admissible limit measures (for low energy) are Σρ,α := {unit measures supported on finite sets A ⊆ Σ : ω(A) < ρ} . Notice that all these measures are finite combinations of Dirac masses.

• Example. If α = 0 and if ρ ∈ (4kπ, 4(k + 1)π), Σρ,α are the measures

supported on at most k points of Σ.

Andrea Malchiodi (SNS, Pisa) BMC 2016 20 / 30

Structure of Σρ,α

Andrea Malchiodi (SNS, Pisa) BMC 2016 21 / 30

Structure of Σρ,α

Elements of Σρ,α are measures, acting on smooth functions.

Andrea Malchiodi (SNS, Pisa) BMC 2016 21 / 30

Structure of Σρ,α

Elements of Σρ,α are measures, acting on smooth functions. It is natural to use the weak topology of distributions

≃

x1 x2 x3 t1 t2 t3 x1 x2 t1 t2 + t3

Andrea Malchiodi (SNS, Pisa) BMC 2016 21 / 30

Structure of Σρ,α

Elements of Σρ,α are measures, acting on smooth functions. It is natural to use the weak topology of distributions

≃

x1 x2 x3 t1 t2 t3 x1 x2 t1 t2 + t3

Σρ,α is finite-dimensional, but not smooth.

Andrea Malchiodi (SNS, Pisa) BMC 2016 21 / 30

Structure of Σρ,α

Elements of Σρ,α are measures, acting on smooth functions. It is natural to use the weak topology of distributions

≃

x1 x2 x3 t1 t2 t3 x1 x2 t1 t2 + t3

Σρ,α is finite-dimensional, but not smooth. It is indeed a stratified set, union of open manifolds of different dimensions, as the following one

Andrea Malchiodi (SNS, Pisa) BMC 2016 21 / 30

Structure of Σρ,α

Elements of Σρ,α are measures, acting on smooth functions. It is natural to use the weak topology of distributions

≃

x1 x2 x3 t1 t2 t3 x1 x2 t1 t2 + t3

Σρ,α is finite-dimensional, but not smooth. It is indeed a stratified set, union of open manifolds of different dimensions, as the following one Such objects can be complicated.

Andrea Malchiodi (SNS, Pisa) BMC 2016 21 / 30

Structure of Σρ,α

Elements of Σρ,α are measures, acting on smooth functions. It is natural to use the weak topology of distributions

≃

x1 x2 x3 t1 t2 t3 x1 x2 t1 t2 + t3

Σρ,α is finite-dimensional, but not smooth. It is indeed a stratified set, union of open manifolds of different dimensions, as the following one Such objects can be complicated. However the above PDE can be redu- ced to the study of this explicit finite-dimensional topological space.

Andrea Malchiodi (SNS, Pisa) BMC 2016 21 / 30

A general existence result

Andrea Malchiodi (SNS, Pisa) BMC 2016 22 / 30

A general existence result

Theorem ([Carlotto-M., w.i.p.]) Suppose ρ ∈ Λα and that Σρ,α is not contractible.

Andrea Malchiodi (SNS, Pisa) BMC 2016 22 / 30

A general existence result

Theorem ([Carlotto-M., w.i.p.]) Suppose ρ ∈ Λα and that Σρ,α is not

• contractible. Then (Eρ,α) is solvable.

Andrea Malchiodi (SNS, Pisa) BMC 2016 22 / 30

A general existence result

Theorem ([Carlotto-M., w.i.p.]) Suppose ρ ∈ Λα and that Σρ,α is not

• contractible. Then (Eρ,α) is solvable.

The assumption ρ ∈ Λα is used for compactness reasons.

Andrea Malchiodi (SNS, Pisa) BMC 2016 22 / 30

A general existence result

Theorem ([Carlotto-M., w.i.p.]) Suppose ρ ∈ Λα and that Σρ,α is not

• contractible. Then (Eρ,α) is solvable.

The assumption ρ ∈ Λα is used for compactness reasons. The topology

• f Σρ,α allows to employ min-max theory.

Andrea Malchiodi (SNS, Pisa) BMC 2016 22 / 30

A general existence result

Theorem ([Carlotto-M., w.i.p.]) Suppose ρ ∈ Λα and that Σρ,α is not

• contractible. Then (Eρ,α) is solvable.

The assumption ρ ∈ Λα is used for compactness reasons. The topology

• f Σρ,α allows to employ min-max theory. Example of a saddle

disconnected sublevel

Andrea Malchiodi (SNS, Pisa) BMC 2016 22 / 30

A general existence result

Theorem ([Carlotto-M., w.i.p.]) Suppose ρ ∈ Λα and that Σρ,α is not

• contractible. Then (Eρ,α) is solvable.

The assumption ρ ∈ Λα is used for compactness reasons. The topology

• f Σρ,α allows to employ min-max theory. Example of a saddle

disconnected sublevel

• For the tear-drop Σρ,α ≃ {p} or Σρ,α ≃ S2 \ {p}, both contractible.

Andrea Malchiodi (SNS, Pisa) BMC 2016 22 / 30

A general existence result

Theorem ([Carlotto-M., w.i.p.]) Suppose ρ ∈ Λα and that Σρ,α is not

• contractible. Then (Eρ,α) is solvable.

The assumption ρ ∈ Λα is used for compactness reasons. The topology

• f Σρ,α allows to employ min-max theory. Example of a saddle

disconnected sublevel

• For the tear-drop Σρ,α ≃ {p} or Σρ,α ≃ S2 \ {p}, both contractible.

Other results in [Ding-Jost-Li-Wang, ’99], [Bartolucci-De Marchis-M., ’11], [M.- Ruiz, ’11], [Carlotto-M.,’12], [Bartolucci-M.,’13], [Chen-Lin, ’15] + al..

Andrea Malchiodi (SNS, Pisa) BMC 2016 22 / 30

Trying to persuade you that Σρ,α is a natural object

Andrea Malchiodi (SNS, Pisa) BMC 2016 23 / 30

Trying to persuade you that Σρ,α is a natural object

Reason 1: new counterexamples when Σρ,α is very contractible.

Andrea Malchiodi (SNS, Pisa) BMC 2016 23 / 30

Trying to persuade you that Σρ,α is a natural object

Reason 1: new counterexamples when Σρ,α is very contractible. Theorem ([Carlotto, ’12]) Given any surface (Σ, g) and points p1, . . . , pm in Σ, there exists weights α1, . . . , αm such that (Eρ,α) is not solvable.

Andrea Malchiodi (SNS, Pisa) BMC 2016 23 / 30

Trying to persuade you that Σρ,α is a natural object

Reason 1: new counterexamples when Σρ,α is very contractible. Theorem ([Carlotto, ’12]) Given any surface (Σ, g) and points p1, . . . , pm in Σ, there exists weights α1, . . . , αm such that (Eρ,α) is not solvable. Other non existence results only available for S2 via monodromy methods ([Luo-Tian, ’92], [Eremenko et al., ’04, ’13, ’15], [Mondello-Panov, ’15]).

Andrea Malchiodi (SNS, Pisa) BMC 2016 23 / 30

Trying to persuade you that Σρ,α is a natural object

Reason 1: new counterexamples when Σρ,α is very contractible. Theorem ([Carlotto, ’12]) Given any surface (Σ, g) and points p1, . . . , pm in Σ, there exists weights α1, . . . , αm such that (Eρ,α) is not solvable. Other non existence results only available for S2 via monodromy methods ([Luo-Tian, ’92], [Eremenko et al., ’04, ’13, ’15], [Mondello-Panov, ’15]). Reason 2: construction of natural test functions.

Andrea Malchiodi (SNS, Pisa) BMC 2016 23 / 30

Trying to persuade you that Σρ,α is a natural object

Reason 1: new counterexamples when Σρ,α is very contractible. Theorem ([Carlotto, ’12]) Given any surface (Σ, g) and points p1, . . . , pm in Σ, there exists weights α1, . . . , αm such that (Eρ,α) is not solvable. Other non existence results only available for S2 via monodromy methods ([Luo-Tian, ’92], [Eremenko et al., ’04, ’13, ’15], [Mondello-Panov, ’15]). Reason 2: construction of natural test functions. Proposition There exist test functions ϕλ,σ, modelled on σ ∈ Σρ,α, s.t. h(x)e2ϕλ,σ ⇀ 4π

• i

tiδxi = 4πσ as λ → +∞

Andrea Malchiodi (SNS, Pisa) BMC 2016 23 / 30

Trying to persuade you that Σρ,α is a natural object

Reason 1: new counterexamples when Σρ,α is very contractible. Theorem ([Carlotto, ’12]) Given any surface (Σ, g) and points p1, . . . , pm in Σ, there exists weights α1, . . . , αm such that (Eρ,α) is not solvable. Other non existence results only available for S2 via monodromy methods ([Luo-Tian, ’92], [Eremenko et al., ’04, ’13, ’15], [Mondello-Panov, ’15]). Reason 2: construction of natural test functions. Proposition There exist test functions ϕλ,σ, modelled on σ ∈ Σρ,α, s.t. h(x)e2ϕλ,σ ⇀ 4π

• i

tiδxi = 4πσ as λ → +∞, and for which Iρ,α(ϕλ,σ) → −∞ uniformly in σ ∈ Σρ,α.

Andrea Malchiodi (SNS, Pisa) BMC 2016 23 / 30

Characterization of low energy levels

Andrea Malchiodi (SNS, Pisa) BMC 2016 24 / 30

Characterization of low energy levels

We claimed previously a converse statement

Andrea Malchiodi (SNS, Pisa) BMC 2016 24 / 30

Characterization of low energy levels

We claimed previously a converse statement, namely that ∃ L large s.t. Iρ,α(u) ≤ −L

Andrea Malchiodi (SNS, Pisa) BMC 2016 24 / 30

Characterization of low energy levels

We claimed previously a converse statement, namely that ∃ L large s.t. Iρ,α(u) ≤ −L = ⇒ h(x)e2u is close to Σρ,α distributionally.

Andrea Malchiodi (SNS, Pisa) BMC 2016 24 / 30

Characterization of low energy levels

We claimed previously a converse statement, namely that ∃ L large s.t. Iρ,α(u) ≤ −L = ⇒ h(x)e2u is close to Σρ,α distributionally.

• Fact. The Moser-Trudinger constant can be improved for functions that

are macroscopically spread over Σ, using localization via cut-off functions.

Andrea Malchiodi (SNS, Pisa) BMC 2016 24 / 30

Characterization of low energy levels

We claimed previously a converse statement, namely that ∃ L large s.t. Iρ,α(u) ≤ −L = ⇒ h(x)e2u is close to Σρ,α distributionally.

• Fact. The Moser-Trudinger constant can be improved for functions that

are macroscopically spread over Σ, using localization via cut-off functions. Then one uses a contradiction argument as follows spreading of h e2u ⇒ better const. in (MT) ⇒ lower bd. on Iρ,α

Andrea Malchiodi (SNS, Pisa) BMC 2016 24 / 30

Characterization of low energy levels

We claimed previously a converse statement, namely that ∃ L large s.t. Iρ,α(u) ≤ −L = ⇒ h(x)e2u is close to Σρ,α distributionally.

• Fact. The Moser-Trudinger constant can be improved for functions that

are macroscopically spread over Σ, using localization via cut-off functions. Then one uses a contradiction argument as follows spreading of h e2u ⇒ better const. in (MT) ⇒ lower bd. on Iρ,α Iρ,α low ⇒ no better const. in (MT) ⇒ concentration of h e2u

Andrea Malchiodi (SNS, Pisa) BMC 2016 24 / 30

Improved Moser-Trudinger inequalities

Andrea Malchiodi (SNS, Pisa) BMC 2016 25 / 30

Improved Moser-Trudinger inequalities

The constant

1 4π in (MT) is sharp for u arbitrary, but under extra

assumptions it can be reduced.

Andrea Malchiodi (SNS, Pisa) BMC 2016 25 / 30

Improved Moser-Trudinger inequalities

The constant

1 4π in (MT) is sharp for u arbitrary, but under extra

assumptions it can be reduced. Well known examples are 1) ([Moser, ’73]) For (Σ, g) = (S2, g0) one can take

1 8π in (MT) provided

u is even (u(x) = u(−x) for all x).

Andrea Malchiodi (SNS, Pisa) BMC 2016 25 / 30

Improved Moser-Trudinger inequalities

The constant

1 4π in (MT) is sharp for u arbitrary, but under extra

assumptions it can be reduced. Well known examples are 1) ([Moser, ’73]) For (Σ, g) = (S2, g0) one can take

1 8π in (MT) provided

u is even (u(x) = u(−x) for all x). 2) ([Aubin, ’76]) For (Σ, g) = (S2, g0) one can take 1+ε

8π in (MT) provided

u is balanced.

Andrea Malchiodi (SNS, Pisa) BMC 2016 25 / 30

Improved Moser-Trudinger inequalities

The constant

1 4π in (MT) is sharp for u arbitrary, but under extra

assumptions it can be reduced. Well known examples are 1) ([Moser, ’73]) For (Σ, g) = (S2, g0) one can take

1 8π in (MT) provided

u is even (u(x) = u(−x) for all x). 2) ([Aubin, ’76]) For (Σ, g) = (S2, g0) one can take 1+ε

8π in (MT) provided

u is balanced. The balancing condition means that, viewing S2 as embedded in R3

• S2 xi eu dV = 0,

i = 1, 2, 3.

Andrea Malchiodi (SNS, Pisa) BMC 2016 25 / 30

The role of singularities

Andrea Malchiodi (SNS, Pisa) BMC 2016 26 / 30

The role of singularities

Recall Chen-Troyanov’s inequality, and that h(x) ≃ d(x, pi)2αi near pi (CT) log

• Σ

h(x)e2(u−u) ≤ 1 4π min{1, 1 + minj αj}

• Σ

|∇u|2 + C.

Andrea Malchiodi (SNS, Pisa) BMC 2016 26 / 30

The role of singularities

Recall Chen-Troyanov’s inequality, and that h(x) ≃ d(x, pi)2αi near pi (CT) log

• Σ

h(x)e2(u−u) ≤ 1 4π min{1, 1 + minj αj}

• Σ

|∇u|2 + C. When localizing it, one sees either 4π or 4π(1 + α) when α is negative.

Andrea Malchiodi (SNS, Pisa) BMC 2016 26 / 30

The role of singularities

Recall Chen-Troyanov’s inequality, and that h(x) ≃ d(x, pi)2αi near pi (CT) log

• Σ

h(x)e2(u−u) ≤ 1 4π min{1, 1 + minj αj}

• Σ

|∇u|2 + C. When localizing it, one sees either 4π or 4π(1 + α) when α is negative. Problem: for positive α’s there is a mismatch between the local Moser- Trudinger constant and the weighted counting 4π(1 + α).

Andrea Malchiodi (SNS, Pisa) BMC 2016 26 / 30

The role of singularities

Recall Chen-Troyanov’s inequality, and that h(x) ≃ d(x, pi)2αi near pi (CT) log

• Σ

h(x)e2(u−u) ≤ 1 4π min{1, 1 + minj αj}

• Σ

|∇u|2 + C. When localizing it, one sees either 4π or 4π(1 + α) when α is negative. Problem: for positive α’s there is a mismatch between the local Moser- Trudinger constant and the weighted counting 4π(1 + α). The hope is that if e2u concentrates near a singular point, the (CT) constant should improve due to the vanishing of h(x).

Andrea Malchiodi (SNS, Pisa) BMC 2016 26 / 30

The role of singularities

Recall Chen-Troyanov’s inequality, and that h(x) ≃ d(x, pi)2αi near pi (CT) log

• Σ

h(x)e2(u−u) ≤ 1 4π min{1, 1 + minj αj}

• Σ

|∇u|2 + C. When localizing it, one sees either 4π or 4π(1 + α) when α is negative. Problem: for positive α’s there is a mismatch between the local Moser- Trudinger constant and the weighted counting 4π(1 + α). The hope is that if e2u concentrates near a singular point, the (CT) constant should improve due to the vanishing of h(x). To see it, one should look at microscopic structure of the conformal volume h(x)e2u.

Andrea Malchiodi (SNS, Pisa) BMC 2016 26 / 30

A new improved (MT) inequality via angular moments

Andrea Malchiodi (SNS, Pisa) BMC 2016 27 / 30

A new improved (MT) inequality via angular moments

For simplicity, consider the unit disk D of R2 with p = 0 and α > 0.

Andrea Malchiodi (SNS, Pisa) BMC 2016 27 / 30

A new improved (MT) inequality via angular moments

For simplicity, consider the unit disk D of R2 with p = 0 and α > 0. Then one can improve the constant via angular spreading only.

Andrea Malchiodi (SNS, Pisa) BMC 2016 27 / 30

A new improved (MT) inequality via angular moments

For simplicity, consider the unit disk D of R2 with p = 0 and α > 0. Then one can improve the constant via angular spreading only. Proposition ([M.-Ruiz, ’11, Bartolucci-M., ’13]) Let α > 0 and fix ε > 0.

Andrea Malchiodi (SNS, Pisa) BMC 2016 27 / 30

A new improved (MT) inequality via angular moments

For simplicity, consider the unit disk D of R2 with p = 0 and α > 0. Then one can improve the constant via angular spreading only. Proposition ([M.-Ruiz, ’11, Bartolucci-M., ’13]) Let α > 0 and fix ε > 0. If u ∈ H1

0(D) satisfies the condition (vanishing of the first k moments)

• D

|x|2αe2u eijθ r dr dθ = 0 in C; j = 1, . . . k

Andrea Malchiodi (SNS, Pisa) BMC 2016 27 / 30

A new improved (MT) inequality via angular moments

For simplicity, consider the unit disk D of R2 with p = 0 and α > 0. Then one can improve the constant via angular spreading only. Proposition ([M.-Ruiz, ’11, Bartolucci-M., ’13]) Let α > 0 and fix ε > 0. If u ∈ H1

0(D) satisfies the condition (vanishing of the first k moments)

• D

|x|2αe2u eijθ r dr dθ = 0 in C; j = 1, . . . k, then one has the inequality log

• D

|x|2αe2udx ≤ 1 + ε 4π min{1 + α, 1 + k}

• D

|∇u|2dx + Cε.

Andrea Malchiodi (SNS, Pisa) BMC 2016 27 / 30

Some comments

Andrea Malchiodi (SNS, Pisa) BMC 2016 28 / 30

Some comments

The main new feature of the last inequality are the scaling invariant assumptions on u.

Andrea Malchiodi (SNS, Pisa) BMC 2016 28 / 30

Some comments

The main new feature of the last inequality are the scaling invariant assumptions on u. These allow volume concentration at any pace.

Andrea Malchiodi (SNS, Pisa) BMC 2016 28 / 30

Some comments

The main new feature of the last inequality are the scaling invariant assumptions on u. These allow volume concentration at any pace. The best constant for radial functions is known to be

1 4π(1+α), so one

cannot improve the multiplicative constant indefinitely in k.

Andrea Malchiodi (SNS, Pisa) BMC 2016 28 / 30

Some comments

The main new feature of the last inequality are the scaling invariant assumptions on u. These allow volume concentration at any pace. The best constant for radial functions is known to be

1 4π(1+α), so one

cannot improve the multiplicative constant indefinitely in k. The above inequality is employed to tear-apart ambiguous volume distri- butions, and determine whether it is better to regard them as regular or singular Dirac masses

p h(x)e2u

Andrea Malchiodi (SNS, Pisa) BMC 2016 28 / 30

Some open problems

Andrea Malchiodi (SNS, Pisa) BMC 2016 29 / 30

Some open problems

It would be nice to have a clean algebraic condition in terms of α and g(Σ) for guaranteeing topological non-triviality of Σρ,α (⇒ existence).

Andrea Malchiodi (SNS, Pisa) BMC 2016 29 / 30

Some open problems

It would be nice to have a clean algebraic condition in terms of α and g(Σ) for guaranteeing topological non-triviality of Σρ,α (⇒ existence). When ρ belongs to the critical set Λα, compactness might fail.

Andrea Malchiodi (SNS, Pisa) BMC 2016 29 / 30

Some open problems

It would be nice to have a clean algebraic condition in terms of α and g(Σ) for guaranteeing topological non-triviality of Σρ,α (⇒ existence). When ρ belongs to the critical set Λα, compactness might fail. Possibly some geometric insight as for the uniformization problem might help.

Andrea Malchiodi (SNS, Pisa) BMC 2016 29 / 30

Some open problems

It would be nice to have a clean algebraic condition in terms of α and g(Σ) for guaranteeing topological non-triviality of Σρ,α (⇒ existence). When ρ belongs to the critical set Λα, compactness might fail. Possibly some geometric insight as for the uniformization problem might help. In dimension d ≥ 3 one has also the singular Yamabe problem (deforming the scalar curvature to a constant). This is mostly unexplored.

Andrea Malchiodi (SNS, Pisa) BMC 2016 29 / 30

Some open problems

It would be nice to have a clean algebraic condition in terms of α and g(Σ) for guaranteeing topological non-triviality of Σρ,α (⇒ existence). When ρ belongs to the critical set Λα, compactness might fail. Possibly some geometric insight as for the uniformization problem might help. In dimension d ≥ 3 one has also the singular Yamabe problem (deforming the scalar curvature to a constant). This is mostly unexplored. Liouville equations also play a role in studying Mean field equations in stationary flows (Onsager), Electroweak theory (Glashow-Salam-Weinberg), Chern-Simons theory in a self-dual regime (Jackiw-Winberg).

Andrea Malchiodi (SNS, Pisa) BMC 2016 29 / 30

Some open problems

It would be nice to have a clean algebraic condition in terms of α and g(Σ) for guaranteeing topological non-triviality of Σρ,α (⇒ existence). When ρ belongs to the critical set Λα, compactness might fail. Possibly some geometric insight as for the uniformization problem might help. In dimension d ≥ 3 one has also the singular Yamabe problem (deforming the scalar curvature to a constant). This is mostly unexplored. Liouville equations also play a role in studying Mean field equations in stationary flows (Onsager), Electroweak theory (Glashow-Salam-Weinberg), Chern-Simons theory in a self-dual regime (Jackiw-Winberg). Non-abelian and non self-dual models are particularly challenging.

Andrea Malchiodi (SNS, Pisa) BMC 2016 29 / 30

Thanks for the attention!

Andrea Malchiodi (SNS, Pisa) BMC 2016 30 / 30