Laplace eigenvalues and minimal surfaces in spheres Mikhail - - PowerPoint PPT Presentation

Laplace eigenvalues and minimal surfaces in spheres Mikhail Karpukhin

(UC Irvine) Partially based on a joint work with Nikolai Nadirashvili, Alexei Penskoi and Iosif Polterovich

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Laplace-Beltrami operator

Let (M, g) be a closed Riemannian surface.

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Laplace-Beltrami operator

Let (M, g) be a closed Riemannian surface. The Laplace-Beltrami

• perator is defined by

∆f = − 1

• |g|

∂ ∂xi

• |g|gij ∂f

∂xj

• ,

where gij is the Riemannian metric, gij are the components of the matrix inverse to gij and |g| = det g.

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Eigenvalues of the Laplacian

Consider the eigenvalue problem: ∆f = λf

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Eigenvalues of the Laplacian

Consider the eigenvalue problem: ∆f = λf The spectrum is discrete, 0 = λ0(M, g) < λ1(M, g) λ2(M, g) · · · ր +∞

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Eigenvalues of the Laplacian

Consider the eigenvalue problem: ∆f = λf The spectrum is discrete, 0 = λ0(M, g) < λ1(M, g) λ2(M, g) · · · ր +∞ Set ¯ λk(M, g) = λk(M, g)Area(M, g).

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Geometric optimization of eigenvalues

Consider ¯ λk(M, g) as a functional on the space R of Riemannian metrics on M. g − → ¯ λk(M, g)

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Geometric optimization of eigenvalues

Consider ¯ λk(M, g) as a functional on the space R of Riemannian metrics on M. g − → ¯ λk(M, g) We are primarily interested in the following quantity Λk(M) = sup

g

¯ λk(M, g).

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Geometric optimization of eigenvalues

Consider ¯ λk(M, g) as a functional on the space R of Riemannian metrics on M. g − → ¯ λk(M, g) We are primarily interested in the following quantity Λk(M) = sup

g

¯ λk(M, g). Korevaar (1993): Λk(M) < ∞

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Maximal metrics for λ1: first examples

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Maximal metrics for λ1: first examples

• Hersch (1970): Λ1(S2) = 8π and the maximum is achieved on

the standard metric on S2.

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Maximal metrics for λ1: first examples

• Hersch (1970): Λ1(S2) = 8π and the maximum is achieved on

the standard metric on S2.

• Li, Yau (1982): Λ1(RP2) = 12π and the maximum is achieved
• n the standard metric on RP2.

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Maximal metrics for λ1: first examples

• Hersch (1970): Λ1(S2) = 8π and the maximum is achieved on

the standard metric on S2.

• Li, Yau (1982): Λ1(RP2) = 12π and the maximum is achieved
• n the standard metric on RP2.

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• Nadirashvili (1996): Λ1(T2) = 8π2

√ 3 and the maximum is achieved on the flat equilateral torus.

Extremality conditions

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Extremality conditions

Theorem (Nadirashvili 1996; El Soufi, Ilias 2008)

Suppose that (M, g) is extremal for the functional ¯ λk(M, g) and λk(M, g) = 2.

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Extremality conditions

Theorem (Nadirashvili 1996; El Soufi, Ilias 2008)

Suppose that (M, g) is extremal for the functional ¯ λk(M, g) and λk(M, g) = 2. Let Ek be the corresponding eigenspace.

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Extremality conditions

Theorem (Nadirashvili 1996; El Soufi, Ilias 2008)

Suppose that (M, g) is extremal for the functional ¯ λk(M, g) and λk(M, g) = 2. Let Ek be the corresponding eigenspace. Then there exists a collection Φ = (φ1, . . . , φn+1), ui ∈ Ek such that

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Extremality conditions

Theorem (Nadirashvili 1996; El Soufi, Ilias 2008)

Suppose that (M, g) is extremal for the functional ¯ λk(M, g) and λk(M, g) = 2. Let Ek be the corresponding eigenspace. Then there exists a collection Φ = (φ1, . . . , φn+1), ui ∈ Ek such that

• Φ: M → Rn+1 is a map to the unit sphere Sn ⊂ Rn+1;

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Extremality conditions

Theorem (Nadirashvili 1996; El Soufi, Ilias 2008)

Suppose that (M, g) is extremal for the functional ¯ λk(M, g) and λk(M, g) = 2. Let Ek be the corresponding eigenspace. Then there exists a collection Φ = (φ1, . . . , φn+1), ui ∈ Ek such that

• Φ: M → Rn+1 is a map to the unit sphere Sn ⊂ Rn+1;
• Φ: M → Sn ⊂ Rn+1 is an isometric (branched) immersion to

the unit sphere.

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Extremality conditions

Theorem (Nadirashvili 1996; El Soufi, Ilias 2008)

Suppose that (M, g) is extremal for the functional ¯ λk(M, g) and λk(M, g) = 2. Let Ek be the corresponding eigenspace. Then there exists a collection Φ = (φ1, . . . , φn+1), ui ∈ Ek such that

• Φ: M → Rn+1 is a map to the unit sphere Sn ⊂ Rn+1;
• Φ: M → Sn ⊂ Rn+1 is an isometric (branched) immersion to

the unit sphere. Such map is automatically minimal.

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Maximal metrics for λ1: first examples

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Maximal metrics for λ1: first examples

• Hersch (1970): Λ1(S2) = 8π and the maximum is achieved on

the standard metric on S2.

• Li, Yau (1982): Λ1(RP2) = 12π and the maximum is achieved
• n the standard metric on RP2.
• Nadirashvili (1996): Λ1(T2) = 8π2

√ 3 and the maximum is

achieved on the flat equilateral torus.

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Maximal metrics: S2 and RP2 revisited

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Maximal metrics: S2 and RP2 revisited

• The eigenfunctions of S2 ⊂ R3 are the restrictions of

harmonic polynomials p on R3.

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Maximal metrics: S2 and RP2 revisited

• The eigenfunctions of S2 ⊂ R3 are the restrictions of

harmonic polynomials p on R3. Eigenvalue is deg p(deg p + 1)

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Maximal metrics: S2 and RP2 revisited

• The eigenfunctions of S2 ⊂ R3 are the restrictions of

harmonic polynomials p on R3. Eigenvalue is deg p(deg p + 1) degree 1: x, y, z degree 2: xy, yz, xz, x2 − y2, x2 − z2

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Maximal metrics: S2 and RP2 revisited

• The eigenfunctions of S2 ⊂ R3 are the restrictions of

harmonic polynomials p on R3. Eigenvalue is deg p(deg p + 1) degree 1: x, y, z degree 2: xy, yz, xz, x2 − y2, x2 − z2

• S2: the identity map S2 → S2 is an isometric minimal

immersion.

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Maximal metrics: S2 and RP2 revisited

• The eigenfunctions of S2 ⊂ R3 are the restrictions of

harmonic polynomials p on R3. Eigenvalue is deg p(deg p + 1) degree 1: x, y, z degree 2: xy, yz, xz, x2 − y2, x2 − z2

• S2: the identity map S2 → S2 is an isometric minimal

immersion.

• RP2: Veronese immersion v : RP2 → S4

v(x, y, z) =

• xy, xz, yz,

√ 3 2 (x2 − y2), 1 2(x2 + y2) − z2

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Examples: continued

• Jakobson–Nadirashvili–Polterovich (2006),

El Soufi–Giacomini–Jazar (2006) (see also Cianci-K.-Medvedev (2019)): Λ1(K) = ¯ λ1(K, g˜

τ3,1),

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Examples: continued

• Jakobson–Nadirashvili–Polterovich (2006),

El Soufi–Giacomini–Jazar (2006) (see also Cianci-K.-Medvedev (2019)): Λ1(K) = ¯ λ1(K, g˜

τ3,1), where ˜

τ3,1 : K → S4 is a Lawson bipolar surface and is a unique minimal immersion of K into Sn by first eigenfunctions.

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Examples: continued

• Jakobson–Levitin–

Nadirashvili–Nigam– Polterovich (2005), Nayatani–Shoda (2017): Λ1(Σ2) = 16π. Bolza surface w2 = z5 − z

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Higher eigenvalues: ”bubbling” phenomenon

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Higher eigenvalues: ”bubbling” phenomenon

• Nadirashvili (2002), Petrides (2014): Λ2(S2) = 16π.

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Higher eigenvalues: ”bubbling” phenomenon

• Nadirashvili (2002), Petrides (2014): Λ2(S2) = 16π.
• Nadirashvili–Sire (2017): Λ3(S2) = 24π.

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Higher eigenvalues: ”bubbling” phenomenon

• Nadirashvili (2002), Petrides (2014): Λ2(S2) = 16π.
• Nadirashvili–Sire (2017): Λ3(S2) = 24π.
• Nadirashvili–Penskoi (2018): Λ2(RP2) = 20π.

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Higher eigenvalues on S2

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Higher eigenvalues on S2

Theorem (K.–Nadirashvili–Penskoi–Polterovich, to appear in JDG) Let (S2, g) be the sphere endowed with a metric g which is smooth outside a finite number of conical singularities. Then ¯ λk(S2, g) < 8πk, k 2.

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Higher eigenvalues on S2

Theorem (K.–Nadirashvili–Penskoi–Polterovich, to appear in JDG) Let (S2, g) be the sphere endowed with a metric g which is smooth outside a finite number of conical singularities. Then ¯ λk(S2, g) < 8πk, k 2. In particular, Λk(S2) = 8πk

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Higher eigenvalues on RP2

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Higher eigenvalues on RP2

Theorem (K., Preprint 2019) Let (RP2, g) be the projective plane endowed with a metric g which is smooth outside a finite number

• f conical singularities. Then

¯ λk(RP2, g) < 4π(2k + 1), k 2.

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Higher eigenvalues on RP2

Theorem (K., Preprint 2019) Let (RP2, g) be the projective plane endowed with a metric g which is smooth outside a finite number

• f conical singularities. Then

¯ λk(RP2, g) < 4π(2k + 1), k 2. In particular, Λk(RP2) = 4π(2k + 1)

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Maximal metrics: existence condition

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Maximal metrics: existence condition

Theorem (Petrides, 2017) Let k ≥ 2 and suppose that Λk(RP2) > Λk−1(RP2) + 8π

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Maximal metrics: existence condition

Theorem (Petrides, 2017) Let k ≥ 2 and suppose that Λk(RP2) > Λk−1(RP2) + 8π Then there exists a maximal metric g, which is smooth except possibly at a finite number of conical singularities, such that Λk(RP2) = ¯ λk(RP2, g).

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Existence of maximal metrics on RP2

Assuming the existence of a maximal metric for ¯ λk we obtain a minimal immersion Φ: RP2 → Sn such that

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Existence of maximal metrics on RP2

Assuming the existence of a maximal metric for ¯ λk we obtain a minimal immersion Φ: RP2 → Sn such that 1) The components of Φ are λk-eigenfunctions;

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Existence of maximal metrics on RP2

Assuming the existence of a maximal metric for ¯ λk we obtain a minimal immersion Φ: RP2 → Sn such that 1) The components of Φ are λk-eigenfunctions; 2) Area(RP2, Φ∗gSn) is large (recall that λk(RP2, Φ∗gSn) = 2).

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Existence of maximal metrics on RP2

Assuming the existence of a maximal metric for ¯ λk we obtain a minimal immersion Φ: RP2 → Sn such that 1) The components of Φ are λk-eigenfunctions; 2) Area(RP2, Φ∗gSn) is large (recall that λk(RP2, Φ∗gSn) = 2). Thus we are looking for an inequality of the form k F(Area)

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Quadratic forms

k = ind(QS)

QS =

• M

|∇u|2

g − |∇Φ|2 gu2 dvg

indE(Φ) = ind(QE)

QE =

• M

|∇V |2

g −|∇Φ|2 g|V |2 dvg

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Quadratic forms

k = ind(QS)

QS =

• M

|∇u|2

g − |∇Φ|2 gu2 dvg

u is a function.

indE(Φ) = ind(QE)

QE =

• M

|∇V |2

g −|∇Φ|2 g|V |2 dvg

V is a vector-function V ⊥ Φ.

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Quadratic forms

k = ind(QS)

QS =

• M

|∇u|2

g − |∇Φ|2 gu2 dvg

u is a function.

indE(Φ) = ind(QE)

QE =

• M

|∇V |2

g −|∇Φ|2 g|V |2 dvg

V is a vector-function V ⊥ Φ. QE(V ) =

n+1

• i=1

QS(V i)

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Index inequality

• Proposition(K. 2019) If Φ: (M, g) → Sn is a harmonic map,

then k indE(Φ) n + 1

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Index inequality

• Proposition(K. 2019) If Φ: (M, g) → Sn is a harmonic map,

then k indE(Φ) n + 1 Proof: Let FE be a negative subspace of QE, dim FE = indE(Φ)

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Index inequality

• Proposition(K. 2019) If Φ: (M, g) → Sn is a harmonic map,

then k indE(Φ) n + 1 Proof: Let FE be a negative subspace of QE, dim FE = indE(Φ) Assuming the contrary, there is V ∈ FE, such that all of its components are orthogonal to the negative subspace of QS.

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Index inequality

• Proposition(K. 2019) If Φ: (M, g) → Sn is a harmonic map,

then k indE(Φ) n + 1 Proof: Let FE be a negative subspace of QE, dim FE = indE(Φ) Assuming the contrary, there is V ∈ FE, such that all of its components are orthogonal to the negative subspace of QS. 0 > QE(V ) =

n+1

• i=1

QS(V i) 0.

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Index inequality II

Theorem (K. 2019) If Φ: S2 → Sn is a harmonic map, then indE(Φ) (n − 2)k.

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Thank you for your attention!

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Ideas of the proof for RP2: area index

Area index ind( Φ) is the Morse index of Φ as a critical point of area.

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Ideas of the proof for RP2: area index

Area index ind( Φ) is the Morse index of Φ as a critical point of area. Propositon (K.) If Φ: RP2 → Sn is branched minimal, then k ind( Φ) n + 1

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Ideas of the proof for RP2: area index

Area index ind( Φ) is the Morse index of Φ as a critical point of area. Propositon (K.) If Φ: RP2 → Sn is branched minimal, then k ind( Φ) n + 1 Proposition (K.) Let Φ: S2 → Sn be a lift of Φ, then ind(Φ) = 2 ind( Φ).

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Ideas of the proof for RP2: area index

Area index ind( Φ) is the Morse index of Φ as a critical point of area. Propositon (K.) If Φ: RP2 → Sn is branched minimal, then k ind( Φ) n + 1 Proposition (K.) Let Φ: S2 → Sn be a lift of Φ, then ind(Φ) = 2 ind( Φ). Start of the proof: k ind( Φ) n + 1 = ind(Φ) 2(n + 1)

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