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 1 / 20

Laplace-Beltrami operator Let ( M , g ) be a closed Riemannian surface. 2 / 20

Laplace-Beltrami operator Let ( M , g ) be a closed Riemannian surface. The Laplace-Beltrami operator is defined by �� � 1 | g | g ij ∂ f ∂ ∆ f = − � , ∂ x i ∂ x j | g | where g ij is the Riemannian metric, g ij are the components of the matrix inverse to g ij and | g | = det g . 2 / 20

Eigenvalues of the Laplacian Consider the eigenvalue problem: ∆ f = λ f 3 / 20

Eigenvalues of the Laplacian Consider the eigenvalue problem: ∆ f = λ f The spectrum is discrete, 0 = λ 0 ( M , g ) < λ 1 ( M , g ) � λ 2 ( M , g ) � · · · ր + ∞ 3 / 20

Eigenvalues of the Laplacian Consider the eigenvalue problem: ∆ f = λ f The spectrum is discrete, 0 = λ 0 ( M , g ) < λ 1 ( M , g ) � λ 2 ( M , g ) � · · · ր + ∞ Set ¯ λ k ( M , g ) = λ k ( M , g ) Area ( M , g ) . 3 / 20

Geometric optimization of eigenvalues Consider ¯ λ k ( M , g ) as a functional on the space R of Riemannian metrics on M . → ¯ g �− λ k ( M , g ) 4 / 20

Geometric optimization of eigenvalues Consider ¯ λ k ( M , g ) as a functional on the space R of Riemannian metrics on M . → ¯ g �− λ k ( M , g ) We are primarily interested in the following quantity ¯ Λ k ( M ) = sup λ k ( M , g ) . g 4 / 20

Geometric optimization of eigenvalues Consider ¯ λ k ( M , g ) as a functional on the space R of Riemannian metrics on M . → ¯ g �− λ k ( M , g ) We are primarily interested in the following quantity ¯ Λ k ( M ) = sup λ k ( M , g ) . g Korevaar (1993): Λ k ( M ) < ∞ 4 / 20

Maximal metrics for λ 1 : first examples 5 / 20

Maximal metrics for λ 1 : first examples • Hersch (1970): Λ 1 ( S 2 ) = 8 π and the maximum is achieved on the standard metric on S 2 . 5 / 20

Maximal metrics for λ 1 : first examples • Hersch (1970): Λ 1 ( S 2 ) = 8 π and the maximum is achieved on the standard metric on S 2 . • Li, Yau (1982): Λ 1 ( RP 2 ) = 12 π and the maximum is achieved on the standard metric on RP 2 . 5 / 20

Maximal metrics for λ 1 : first examples • Hersch (1970): Λ 1 ( S 2 ) = 8 π and the maximum is achieved on the standard metric on S 2 . • Li, Yau (1982): Λ 1 ( RP 2 ) = 12 π and the maximum is achieved on the standard metric on RP 2 . • Nadirashvili (1996): Λ 1 ( T 2 ) = 8 π 2 √ 3 and the maximum is achieved on the flat equilateral torus . 5 / 20

Extremality conditions 6 / 20

Extremality conditions Theorem (Nadirashvili 1996; El Soufi, Ilias 2008) Suppose that ( M , g ) is extremal for the functional ¯ λ k ( M , g ) and λ k ( M , g ) = 2 . 6 / 20

Extremality conditions Theorem (Nadirashvili 1996; El Soufi, Ilias 2008) Suppose that ( M , g ) is extremal for the functional ¯ λ k ( M , g ) and λ k ( M , g ) = 2 . Let E k be the corresponding eigenspace. 6 / 20

Extremality conditions Theorem (Nadirashvili 1996; El Soufi, Ilias 2008) Suppose that ( M , g ) is extremal for the functional ¯ λ k ( M , g ) and λ k ( M , g ) = 2 . Let E k be the corresponding eigenspace. Then there exists a collection Φ = ( φ 1 , . . . , φ n +1 ) , u i ∈ E k such that 6 / 20

Extremality conditions Theorem (Nadirashvili 1996; El Soufi, Ilias 2008) Suppose that ( M , g ) is extremal for the functional ¯ λ k ( M , g ) and λ k ( M , g ) = 2 . Let E k be the corresponding eigenspace. Then there exists a collection Φ = ( φ 1 , . . . , φ n +1 ) , u i ∈ E k such that • Φ: M → R n +1 is a map to the unit sphere S n ⊂ R n +1 ; 6 / 20

Extremality conditions Theorem (Nadirashvili 1996; El Soufi, Ilias 2008) Suppose that ( M , g ) is extremal for the functional ¯ λ k ( M , g ) and λ k ( M , g ) = 2 . Let E k be the corresponding eigenspace. Then there exists a collection Φ = ( φ 1 , . . . , φ n +1 ) , u i ∈ E k such that • Φ: M → R n +1 is a map to the unit sphere S n ⊂ R n +1 ; • Φ: M → S n ⊂ R n +1 is an isometric (branched) immersion to the unit sphere. 6 / 20

Extremality conditions Theorem (Nadirashvili 1996; El Soufi, Ilias 2008) Suppose that ( M , g ) is extremal for the functional ¯ λ k ( M , g ) and λ k ( M , g ) = 2 . Let E k be the corresponding eigenspace. Then there exists a collection Φ = ( φ 1 , . . . , φ n +1 ) , u i ∈ E k such that • Φ: M → R n +1 is a map to the unit sphere S n ⊂ R n +1 ; • Φ: M → S n ⊂ R n +1 is an isometric (branched) immersion to the unit sphere. Such map is automatically minimal. 6 / 20

Maximal metrics for λ 1 : first examples 7 / 20

Maximal metrics for λ 1 : first examples • Hersch (1970): Λ 1 ( S 2 ) = 8 π and the maximum is achieved on the standard metric on S 2 . • Li, Yau (1982): Λ 1 ( RP 2 ) = 12 π and the maximum is achieved on the standard metric on RP 2 . • Nadirashvili (1996): Λ 1 ( T 2 ) = 8 π 2 3 and the maximum is √ achieved on the flat equilateral torus . 7 / 20

Maximal metrics: S 2 and RP 2 revisited 8 / 20

Maximal metrics: S 2 and RP 2 revisited • The eigenfunctions of S 2 ⊂ R 3 are the restrictions of harmonic polynomials p on R 3 . 8 / 20

Maximal metrics: S 2 and RP 2 revisited • The eigenfunctions of S 2 ⊂ R 3 are the restrictions of harmonic polynomials p on R 3 . Eigenvalue is deg p (deg p + 1) 8 / 20

Maximal metrics: S 2 and RP 2 revisited • The eigenfunctions of S 2 ⊂ R 3 are the restrictions of harmonic polynomials p on R 3 . Eigenvalue is deg p (deg p + 1) degree 1: x , y , z degree 2: xy , yz , xz , x 2 − y 2 , x 2 − z 2 8 / 20

Maximal metrics: S 2 and RP 2 revisited • The eigenfunctions of S 2 ⊂ R 3 are the restrictions of harmonic polynomials p on R 3 . Eigenvalue is deg p (deg p + 1) degree 1: x , y , z degree 2: xy , yz , xz , x 2 − y 2 , x 2 − z 2 • S 2 : the identity map S 2 → S 2 is an isometric minimal immersion. 8 / 20

Maximal metrics: S 2 and RP 2 revisited • The eigenfunctions of S 2 ⊂ R 3 are the restrictions of harmonic polynomials p on R 3 . Eigenvalue is deg p (deg p + 1) degree 1: x , y , z degree 2: xy , yz , xz , x 2 − y 2 , x 2 − z 2 • S 2 : the identity map S 2 → S 2 is an isometric minimal immersion. • RP 2 : Veronese immersion v : RP 2 → S 4 � � √ 2 ( x 2 − y 2 ) , 1 3 2( x 2 + y 2 ) − z 2 v ( x , y , z ) = xy , xz , yz , 8 / 20

Examples: continued • Jakobson–Nadirashvili–Polterovich (2006), El Soufi–Giacomini–Jazar (2006) (see also Cianci-K.-Medvedev (2019)): Λ 1 ( K ) = ¯ λ 1 ( K , g ˜ τ 3 , 1 ), 9 / 20

Examples: continued • Jakobson–Nadirashvili–Polterovich (2006), El Soufi–Giacomini–Jazar (2006) (see also Cianci-K.-Medvedev (2019)): τ 3 , 1 : K → S 4 is a Lawson bipolar Λ 1 ( K ) = ¯ λ 1 ( K , g ˜ τ 3 , 1 ), where ˜ surface and is a unique minimal immersion of K into S n by first eigenfunctions. 9 / 20

Examples: continued • Jakobson–Levitin– Nadirashvili–Nigam– Polterovich (2005), Nayatani–Shoda (2017): Λ 1 (Σ 2 ) = 16 π . Bolza surface w 2 = z 5 − z 10 / 20

Higher eigenvalues: ”bubbling” phenomenon 11 / 20

Higher eigenvalues: ”bubbling” phenomenon • Nadirashvili (2002), Petrides (2014): Λ 2 ( S 2 ) = 16 π. 11 / 20

Higher eigenvalues: ”bubbling” phenomenon • Nadirashvili (2002), Petrides (2014): Λ 2 ( S 2 ) = 16 π. • Nadirashvili–Sire (2017): Λ 3 ( S 2 ) = 24 π. 11 / 20

Higher eigenvalues: ”bubbling” phenomenon • Nadirashvili (2002), Petrides (2014): Λ 2 ( S 2 ) = 16 π. • Nadirashvili–Sire (2017): Λ 3 ( S 2 ) = 24 π. • Nadirashvili–Penskoi (2018): Λ 2 ( RP 2 ) = 20 π. 11 / 20

Higher eigenvalues on S 2 12 / 20

Higher eigenvalues on S 2 Theorem (K.–Nadirashvili–Penskoi–Polterovich, to appear in JDG ) Let ( S 2 , g ) be the sphere endowed with a metric g which is smooth outside a finite number of conical singularities. Then λ k ( S 2 , g ) < 8 π k , k � 2 . ¯ 12 / 20

Higher eigenvalues on S 2 Theorem (K.–Nadirashvili–Penskoi–Polterovich, to appear in JDG ) Let ( S 2 , g ) be the sphere endowed with a metric g which is smooth outside a finite number of conical singularities. Then λ k ( S 2 , g ) < 8 π k , k � 2 . ¯ In particular, Λ k ( S 2 ) = 8 π k 12 / 20

Higher eigenvalues on RP 2 13 / 20

Higher eigenvalues on RP 2 Theorem (K., Preprint 2019) Let ( RP 2 , g ) be the projective plane endowed with a metric g which is smooth outside a finite number of conical singularities. Then ¯ λ k ( RP 2 , g ) < 4 π (2 k + 1) , k � 2 . 13 / 20

Higher eigenvalues on RP 2 Theorem (K., Preprint 2019) Let ( RP 2 , g ) be the projective plane endowed with a metric g which is smooth outside a finite number of conical singularities. Then ¯ λ k ( RP 2 , g ) < 4 π (2 k + 1) , k � 2 . In particular, Λ k ( RP 2 ) = 4 π (2 k + 1) 13 / 20

Maximal metrics: existence condition 14 / 20

Maximal metrics: existence condition Theorem (Petrides, 2017) Let k ≥ 2 and suppose that Λ k ( RP 2 ) > Λ k − 1 ( RP 2 ) + 8 π 14 / 20