The Willmore Functional: a perturbative approach

Transcription

1 The Willmore Functional: a perturbative approach p. 1/21 The Willmore Functional: a perturbative approach Andrea Mondino Carthage, May 2010

2 The Willmore Functional: a perturbative approach p. 2/21 Basic definitions -(M, g) 3-dim Riemannian Manifold

3 The Willmore Functional: a perturbative approach p. 2/21 Basic definitions -(M, g) 3-dim Riemannian Manifold -(N, g) M isometrically immersed closed oriented surface

4 The Willmore Functional: a perturbative approach p. 2/21 Basic definitions -(M, g) 3-dim Riemannian Manifold -(N, g) M isometrically immersed closed oriented surface - H mean curvature, H := A ij g ij = k 1 + k 2

5 The Willmore Functional: a perturbative approach p. 2/21 Basic definitions -(M, g) 3-dim Riemannian Manifold -(N, g) M isometrically immersed closed oriented surface - H mean curvature, H := A ij g ij = k 1 + k 2 - The Willmore functional I is defined as I(N) := H 2 dσ

6 The Willmore Functional: a perturbative approach p. 3/21 Applications - Reilly Theorem: λ 1 (N) 2 Area(N) N H 2 dσ

7 The Willmore Functional: a perturbative approach p. 3/21 Applications - Reilly Theorem: λ 1 (N) 2 Area(N) N H 2 dσ - General Relativity: Hawking mass Area(N) Mass(N) := (1 1 16π 16π N H 2 dσ)

8 The Willmore Functional: a perturbative approach p. 3/21 Applications - Reilly Theorem: λ 1 (N) 2 Area(N) N H 2 dσ - General Relativity: Hawking mass Area(N) Mass(N) := (1 1 16π 16π - Biology: Hellfrich Energy N H 2 dσ)

9 The Willmore Functional: a perturbative approach p. 3/21 Applications - Reilly Theorem: λ 1 (N) 2 Area(N) N H 2 dσ - General Relativity: Hawking mass Area(N) Mass(N) := (1 1 16π 16π - Biology: Hellfrich Energy - String Theory: Polyakov extrinsic action N H 2 dσ)

10 The Willmore Functional: a perturbative approach p. 4/21 Problem: find critical points - (N, g) (M, g) is a critical point of I if for all φ C (N), called ν the unit normal vector to N and N(t) := exp N (tφν) = N + tφν one has d dt I(N(t)) = 0

11 The Willmore Functional: a perturbative approach p. 4/21 Problem: find critical points - (N, g) (M, g) is a critical point of I if for all φ C (N), called ν the unit normal vector to N and N(t) := exp N (tφν) = N + tφν one has d dt I(N(t)) = 0 - Critical points: called Willmore Surfaces"

12 The Willmore Functional: a perturbative approach p. 4/21 Problem: find critical points - (N, g) (M, g) is a critical point of I if for all φ C (N), called ν the unit normal vector to N and N(t) := exp N (tφν) = N + tφν one has d dt I(N(t)) = 0 - Critical points: called Willmore Surfaces" - N is a Willmore surface if and only if satisfies 2 N H + H ( H 2 4k 1 k 2 + 2Ric M (ν, ν) ) = 0

13 The Willmore Functional: a perturbative approach p. 5/21 Known Results in Euclidean Space - Conformal Invariance (Blaschke and Thomsen 30)

14 The Willmore Functional: a perturbative approach p. 5/21 Known Results in Euclidean Space - Conformal Invariance (Blaschke and Thomsen 30) - Strict global minimum on standard spheres S ρ p (Willmore 60): I(N) 16π and I(N) = 16π N = S ρ p

15 The Willmore Functional: a perturbative approach p. 5/21 Known Results in Euclidean Space - Conformal Invariance (Blaschke and Thomsen 30) - Strict global minimum on standard spheres S ρ p (Willmore 60): I(N) 16π and I(N) = 16π N = S ρ p - For each genus the infimum ( 16π) is reached: L.Simon (1993) and M.Bauer, E.Kuwert (2003)

16 The Willmore Functional: a perturbative approach p. 5/21 Known Results in Euclidean Space - Conformal Invariance (Blaschke and Thomsen 30) - Strict global minimum on standard spheres S ρ p (Willmore 60): I(N) 16π and I(N) = 16π N = S ρ p - For each genus the infimum ( 16π) is reached: L.Simon (1993) and M.Bauer, E.Kuwert (2003) - Divergence structure of the PDE: Riviére (2008)

17 The Willmore Functional: a perturbative approach p. 6/21 In Manifold? - Closed minimal surfaces (H = 0), the only general result in general manifold

18 The Willmore Functional: a perturbative approach p. 6/21 In Manifold? - Closed minimal surfaces (H = 0), the only general result in general manifold - In Space forms: Chen (1974),Weiner (1978), Li (2004), Guo(2007)

19 The Willmore Functional: a perturbative approach p. 6/21 In Manifold? - Closed minimal surfaces (H = 0), the only general result in general manifold - In Space forms: Chen (1974),Weiner (1978), Li (2004), Guo(2007) - GOAL: Say something about existence of Willmore surfaces in (non constantly) curved manifold

20 The Willmore Functional: a perturbative approach p. 7/21 Which tecnic? Geometrical properties of the functional:

21 The Willmore Functional: a perturbative approach p. 7/21 Which tecnic? Geometrical properties of the functional: - Invariance under isometries of ambient manifold (useful for multiplicity results)

22 The Willmore Functional: a perturbative approach p. 7/21 Which tecnic? Geometrical properties of the functional: - Invariance under isometries of ambient manifold (useful for multiplicity results) - Invariance under dilations: g λg, λ > 0

23 The Willmore Functional: a perturbative approach p. 7/21 Which tecnic? Geometrical properties of the functional: - Invariance under isometries of ambient manifold (useful for multiplicity results) - Invariance under dilations: g λg, λ > 0 loss of compactness

24 The Willmore Functional: a perturbative approach p. 7/21 Which tecnic? Geometrical properties of the functional: - Invariance under isometries of ambient manifold (useful for multiplicity results) - Invariance under dilations: g λg, λ > 0 loss of compactness USE A PERTURBATIVE METHOD

25 The Willmore Functional: a perturbative approach p. 8/21 The perturbative method: general setting H Hilbert space

26 The Willmore Functional: a perturbative approach p. 8/21 The perturbative method: general setting H Hilbert space I ǫ = I 0 + ǫg with I 0, G : H R of class C 2 such that

27 The Willmore Functional: a perturbative approach p. 8/21 The perturbative method: general setting H Hilbert space I ǫ = I 0 + ǫg with I 0, G : H R of class C 2 such that I 0 possesses a finite dimensional manifold Z of critical points: Z = {z H : I 0(z) = 0}. With

28 The Willmore Functional: a perturbative approach p. 8/21 The perturbative method: general setting H Hilbert space I ǫ = I 0 + ǫg with I 0, G : H R of class C 2 such that I 0 possesses a finite dimensional manifold Z of critical points: Z = {z H : I 0(z) = 0}. With (ND) T z Z = Ker[I 0(z)], z Z

29 The Willmore Functional: a perturbative approach p. 8/21 The perturbative method: general setting H Hilbert space I ǫ = I 0 + ǫg with I 0, G : H R of class C 2 such that I 0 possesses a finite dimensional manifold Z of critical points: Z = {z H : I 0(z) = 0}. With (ND) T z Z = Ker[I 0(z)], z Z (Fr) z Z, I 0(z) is Fredholm map of index 0

30 The Willmore Functional: a perturbative approach p. 8/21 The perturbative method: general setting H Hilbert space I ǫ = I 0 + ǫg with I 0, G : H R of class C 2 such that I 0 possesses a finite dimensional manifold Z of critical points: Z = {z H : I 0(z) = 0}. With (ND) T z Z = Ker[I 0(z)], z Z (Fr) z Z, I 0(z) is Fredholm map of index 0 AIM: find critical points of I ǫ i.e. find u H s.t I ǫ(u) = 0.

31 The Willmore Functional: a perturbative approach p. 9/21 The reduced functional CRUCIAL FACT: there exist a function in finitely many variables called reduced fuctional such that Φ ǫ : Z R

32 The Willmore Functional: a perturbative approach p. 9/21 The reduced functional CRUCIAL FACT: there exist a function in finitely many variables called reduced fuctional Φ ǫ : Z R such that -fixed a compact subset Z c Z

33 The Willmore Functional: a perturbative approach p. 9/21 The reduced functional CRUCIAL FACT: there exist a function in finitely many variables called reduced fuctional Φ ǫ : Z R such that -fixed a compact subset Z c Z -there exist a neighborhood V c of Z c such that for ǫ > 0 small enough

34 The Willmore Functional: a perturbative approach p. 9/21 The reduced functional CRUCIAL FACT: there exist a function in finitely many variables called reduced fuctional Φ ǫ : Z R such that -fixed a compact subset Z c Z -there exist a neighborhood V c of Z c such that for ǫ > 0 small enough {critical points of I ǫ Vc } {critical points of Φ ǫ Z c } bijection

35 The Willmore Functional: a perturbative approach p. 10/21 Who is Φ ǫ? With a Taylor expansion Φ ǫ (z) = I ǫ (z) + o(ǫ)

36 The Willmore Functional: a perturbative approach p. 10/21 Who is Φ ǫ? With a Taylor expansion Φ ǫ (z) = I ǫ (z) + o(ǫ) to study the reduced functional Φ ǫ is enough to know the I ǫ Z

37 The Willmore Functional: a perturbative approach p. 11/21 The Willmore functional in metric g ǫ -Consider R 3 with the metric g ǫ = δ + ǫh, h a symmetric bilinear form

38 The Willmore Functional: a perturbative approach p. 11/21 The Willmore functional in metric g ǫ -Consider R 3 with the metric g ǫ = δ + ǫh, h a symmetric bilinear form - I ǫ = Willmore functional in ambient manifold (R 3, g ǫ ): I ǫ (N) := N H2 ǫ dσ ǫ.

39 The Willmore Functional: a perturbative approach p. 11/21 The Willmore functional in metric g ǫ -Consider R 3 with the metric g ǫ = δ + ǫh, h a symmetric bilinear form - I ǫ = Willmore functional in ambient manifold (R 3, g ǫ ): I ǫ (N) := N H2 ǫ dσ ǫ. -I 0 = functional in euclidean metric

40 The Willmore Functional: a perturbative approach p. 11/21 The Willmore functional in metric g ǫ -Consider R 3 with the metric g ǫ = δ + ǫh, h a symmetric bilinear form - I ǫ = Willmore functional in ambient manifold (R 3, g ǫ ): I ǫ (N) := N H2 ǫ dσ ǫ. -I 0 = functional in euclidean metric -Critical manifold Z = {S ρ p : p R 3, ρ > 0} R 3 R +

41 The Willmore Functional: a perturbative approach p. 12/21 Perturbed spheres -look for critical points of the form S ρ p(w), parametrized by with w C 4,α (S 2 ). Θ p + ρ(1 w(θ))θ

42 The Willmore Functional: a perturbative approach p. 12/21 Perturbed spheres -look for critical points of the form S ρ p(w), parametrized by Θ p + ρ(1 w(θ))θ with w C 4,α (S 2 ). -I 0(S ρ p)[w] = 2 ρ 3 S 2( S 2 + 2)w

43 The Willmore Functional: a perturbative approach p. 12/21 Perturbed spheres -look for critical points of the form S ρ p(w), parametrized by Θ p + ρ(1 w(θ))θ with w C 4,α (S 2 ). -I 0(S p)[w] ρ = 2 ρ 3 S 2( S 2 + 2)w -Ker[I 0(S p)] ρ = costants + affine functions = T S ρ p Z

44 The Willmore Functional: a perturbative approach p. 12/21 Perturbed spheres -look for critical points of the form S ρ p(w), parametrized by Θ p + ρ(1 w(θ))θ with w C 4,α (S 2 ). -I 0(S p)[w] ρ = 2 ρ 3 S 2( S 2 + 2)w -Ker[I 0(S p)] ρ = costants + affine functions = T S ρ p Z -Fredholm condition

45 The Willmore Functional: a perturbative approach p. 12/21 Perturbed spheres -look for critical points of the form S ρ p(w), parametrized by Θ p + ρ(1 w(θ))θ with w C 4,α (S 2 ). -I 0(S p)[w] ρ = 2 ρ 3 S 2( S 2 + 2)w -Ker[I 0(S p)] ρ = costants + affine functions = T S ρ p Z -Fredholm condition I 0 satisfies the assumptions of the abstract method

46 The Willmore Functional: a perturbative approach p. 12/21 Perturbed spheres -look for critical points of the form S ρ p(w), parametrized by Θ p + ρ(1 w(θ))θ with w C 4,α (S 2 ). -I 0(S p)[w] ρ = 2 ρ 3 S 2( S 2 + 2)w -Ker[I 0(S p)] ρ = costants + affine functions = T S ρ p Z -Fredholm condition I 0 satisfies the assumptions of the abstract method AIM: give hypothesys on h to get that Sp(w) ρ critical point.

47 The Willmore Functional: a perturbative approach p. 13/21 Expansions on standard spheres -recall Φ ǫ (S ρ p) = I ǫ (S ρ p) + o(ǫ) do computations on standard spheres.

48 The Willmore Functional: a perturbative approach p. 13/21 Expansions on standard spheres -recall Φ ǫ (Sp) ρ = I ǫ (Sp) ρ + o(ǫ) do computations on standard spheres. Lemma The following expansion for Φ ǫ in ǫ holds Φ ǫ (S ρ p) = 16π + 2ǫ S 2 [ Trh 3h(Θ, Θ) +ρa µµλ Θ λ ρa µνλ Θ µ Θ ν Θ λ ]dσ 0 + o(ǫ) with A µνλ := [D µ h λν + D λ h νµ D ν h µλ ].

49 The Willmore Functional: a perturbative approach p. 14/21 Expansion for small radius Lemma For small radius spheres Φ ǫ (Sp) ρ = 16π 8π 3 R 1(p)ρ 2 ǫ + o(ρ 2 )ǫ + o(ǫ) where R 1 := µν D µνh µν Trh.

50 The Willmore Functional: a perturbative approach p. 14/21 Expansion for small radius Lemma For small radius spheres Φ ǫ (S ρ p) = 16π 8π 3 R 1(p)ρ 2 ǫ + o(ρ 2 )ǫ + o(ǫ) where R 1 := µν D µνh µν Trh. Remark The scalar curvature R gǫ of g ǫ is R gǫ = ǫr 1 + o(ǫ)

51 The Willmore Functional: a perturbative approach p. 15/21 Existence Theorem Theorem Assume - p R 3 such that R 1 ( p) 0, - Said h(p) := sup v =1 h p (v, v) i) lim p h(p) = 0. ii) C > 0 and α > 2 s.t. D λ h µν (p) < C p α λ, µ, ν =

52 The Willmore Functional: a perturbative approach p. 15/21 Existence Theorem Theorem Assume - p R 3 such that R 1 ( p) 0, - Said h(p) := sup v =1 h p (v, v) i) lim p h(p) = 0. ii) C > 0 and α > 2 s.t. D λ h µν (p) < C p λ, µ, ν = α Then, for ǫ small enough, there exist (p ǫ, ρ ǫ ) R 3 R + + and w ǫ (p ǫ, ρ ǫ ) C 4,α (S 2 ) s.t the perturbed sphere S ρ ǫ p ǫ (w ǫ (p ǫ, ρ ǫ )) is critical point of the Willmore functional I ǫ.

53 The Willmore Functional: a perturbative approach p. 16/21 Perturbed geodesic sphere -(M, g) 3-dim ambient Riemannian manifold.

54 The Willmore Functional: a perturbative approach p. 16/21 Perturbed geodesic sphere -(M, g) 3-dim ambient Riemannian manifold. -fixed p M, the geodesic sphere S p,ρ of radius ρ is the image of Θ S 2 T p M Exp p (ρθ)

55 The Willmore Functional: a perturbative approach p. 16/21 Perturbed geodesic sphere -(M, g) 3-dim ambient Riemannian manifold. -fixed p M, the geodesic sphere S p,ρ of radius ρ is the image of Θ S 2 T p M Exp p (ρθ) - fix a function w C 4,α (S 2 ), the perturbed geodesic sphere S p,ρ (w) is defined as the image of Θ S 2 T p M Exp p ( ρ(1 w(θ))θ )

56 The Willmore Functional: a perturbative approach p. 17/21 Expansion of the Willmore functional The Willmore functional H 2 dσ on S p,ρ (w) can be expanded as I(S p,ρ (w)) = 16π 8π 3 R(p)ρ2 + (Q (2) p (w) + ρ 2 L p (w))dθ + O p (ρ 3 ), S 2 where R is the scalar curvature of the ambient manifold (M, g).

57 The Willmore Functional: a perturbative approach p. 18/21 w is constricted Lemma For small ρ and w, if S p,ρ (w) is a critical point then

58 The Willmore Functional: a perturbative approach p. 18/21 w is constricted Lemma For small ρ and w, if S p,ρ (w) is a critical point then i) w is a C 1 function in (p, ρ),

59 The Willmore Functional: a perturbative approach p. 18/21 w is constricted Lemma For small ρ and w, if S p,ρ (w) is a critical point then i) w is a C 1 function in (p, ρ), ii) w = O(ρ 2 ) for ρ 0,

60 The Willmore Functional: a perturbative approach p. 18/21 w is constricted Lemma For small ρ and w, if S p,ρ (w) is a critical point then i) w is a C 1 function in (p, ρ), ii) w = O(ρ 2 ) for ρ 0, iii) ρw(p, ρ) is bounded as ρ 0 = ρ I[S ( ) 16π p,ρ w(p, ρ) ] = 3 R( p)ρ + O(ρ2 ) 0 if R( p) 0 and ρ << 1

61 The Willmore Functional: a perturbative approach p. 19/21 Non Existence THEOREM: Assume that at the point p M the scalar curvature is non null: R( p) 0. Then, for radius ρ and perturbation w C 4,α (S 2 ) small enough, the surfaces S p,ρ (w) are not critical points of the Willmore functional I.

62 The Willmore Functional: a perturbative approach p. 20/21 References -About the method: A. Ambrosetti, A. Malchiodi, Perturbation methods and semilinear elliptic problems in R n ", Progress in mathematics, Birkhauser (2006). -The results: A. Mondino, Some results about the existence of critical points for the Willmore functional", Math. Zeit. (2009, in press).

63 Thank you The Willmore Functional: a perturbative approach p. 21/21