1 M = (M, g) m Riemann N = (N, h) n Riemann M N C f : M N f df : T M T N M T M f N T N M f 1 T N T M f 1 T N C X, Y Γ(T M) M C T M f 1 T N M Levi-Civi

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1 1 Surveys in Geometry , 7 Harmonic Map Plateau Eells-Sampson [5] Siu [19, 20] Kähler 6 Reports on Global Analysis [15] Sacks- Uhlenbeck [18] Siu-Yau [21] Frankel Siu Yau Frankel [13] 1 Surveys in Geometry Finsler 4 Hartshorne Frankel 5 Kähler Finsler Surveys in Geometry, Special Edition ( ), , 1 Sacks-Uhlenbeck Frankel Yau

2 1 M = (M, g) m Riemann N = (N, h) n Riemann M N C f : M N f df : T M T N M T M f N T N M f 1 T N T M f 1 T N C X, Y Γ(T M) M C T M f 1 T N M Levi-Civita N Levi-Civita f C df Γ(T M f 1 T N) df Γ(T M T M f 1 T N) (1) df(x, Y ) = f 1 T N X df(y ) df( T X M Y ) f df Riemann f(m) N 2 Riemann f (tension field) f τ(f) Γ(f 1 T M) {E i } M τ(f) = Trace g df = m df(e i, E i ) i=1 τ(f) f M N (harmonic map) M (x i ) N (y α ) M N Riemann g h m n g = g ij dx i dx j, h = h αβ dy α dy β i,j=1 f α,β=1 f(x) = ( f 1 (x 1,..., x m ),..., f n (x 1,..., x m ) ) = ( f α (x i ) ) m n e(f) = df 2 = Trace g f h = g ij h αβ (f) f α f β x i x j i,j=1 α,β=1 C e(f) : M R f (energy density) M e(f) E(f) = 1 2 M e(f)dµ g 2

3 f (energy) (g ij ) (g ij ) µ g Riemann g M M N C C (M, N) E(f) C (M, N) E : C (M, N) R f C (M, N) {f t } t I, I = ( ɛ, ɛ) (ɛ > 0) f = f 0 C V = d (2) dt f t Γ(f 1 T N) t=0 f E(f t ) 1 d dt E(f (3) t) = τ(f), V dµ g t=0 M, Riemann h f 1 T N n ( ) τ(f) = τ(f) α = 0 y α α=1 E Euler-Lagrange (1) M N m n τ(f) α = f α + g ij Γ α βγ(f) f β f γ (4) x i x = 0, 1 α n j i,j=1 β,γ=1 M Laplace-Beltrami Γ α βγ N Christoffel Γ α βγ = 1 n ( h αδ hγδ 2 y + h βδ β y h ) βγ γ y δ δ=1 f (4) M 2 C 2 f : M N (4) C N = R f : M R Dirichlet (4) Laplace f α = 0 f M = R f : R N (4) d 2 f α n + Γ α dt βγ(f) df β df γ 2 dt dt = 0 β,γ=1 3

4 N f : M N M N f : M N M N Kähler M N Eells-Lemaire [2, 3, 4] [16] 2 Riemann 1964 Eells-Sampson 1 ([5]) (M, g) (N, h) Riemann N C f C (M, N) f f : M N N 1 f f N C (M, N) ([9]) M N Riemann 1 Hamilton [8] Eells-Sampson Morse (heat flow method) 1. M = C (M, N ), E : C (M, N) R M. f {f t } t I M (2) V Γ(f 1 T N) t = 0 E(f t ) 1 M E f M V T f M de f (V ) W 1, W 2 Γ(f 1 T N) W 1, W 2 = W 1, W 2 dµ g, M, T f M 1 (3) M E de f (V ) = τ(f), V 4

5 f τ(f) M E grad E Morse E τ(f). τ(f) f 0 C (M, N) f t u t t = τ(f t). f t f 0 E τ(f ) = 0 f : M [0, T ) N. f (x, t) = τ(f(x, t)), (x, t) M (0, T ) (5) t f(x, 0) = f 0 (x) T > 0 f 0 C (M, N) f M [0, T ) M (0, T ) C 1 (1) f 0 (5) f : M [0, ) N (2) f 0 f t f t (x) = f(x, t) t f tn f t f : M N f 0 f (1), (2) (5) f(x, t) t. (2) Riemann N. N N f : M [0, ) N f(m [0, )) N (2) [5] [16, 15] 3 Finsler M m π : E M M E M z M z E E z = π 1 (z) 5

6 ζ E z (z, ζ) E E E = E \ {0} E F : E R E Finsler (complex Finsler metric) (1) F (z, ζ) 0 F (z, ζ) = 0 ζ = 0 (2) F C 2 (E ) F E C 2 (3) F (z, λζ) = λ F (z, ζ) λ C (z, ζ) E z = (z 1,..., z m ) M ζ = (ζ 1,..., ζ r ) E s = (s 1,..., s r ) E (z, ζ) = (z 1,..., z m, ζ 1,..., ζ r ) = (z µ, ζ i ) E (z µ, ζ i ) G(z, ζ) = F 2 (z, ζ) Hesse ( ) ( Gi j) 2 G = ζ i ζ j E F (strongly pseudoconvex) Finsler E Hermite r g(z, ζ) = g i j(z)ζ i ζj i,j=1 F (z, ζ) = g(z, ζ) F E Finsler Finsler F (3) E G(z, ζ) r G(z, ζ) = G i j(z, ζ)ζ i ζj i,j=1 λ C = C \ {0} G i j(z, λζ) = G i j(z, ζ), (z, ζ) E F G(z, ζ) E p : P (E) = E /C M E Ẽ = p 1 E Hermite L(E) P (E) (tautological line bundle) L(E) (z, [ζ]) P (E) [ζ] E z 1 Ẽ L(E) Ẽ π P (E) 6 p E π p M

7 L(E) L(E) p : L(E) E L(E) E 6 L(E) Hermite h E Finsler F 1 1 (z, ζ) E (6) F 2 (z, ζ) = h( p 1 (z, ζ), p 1 (z, ζ)) E L(E) 1 Chern c 1 (L(E)) L(E) Hermite h (6) L(E) ĥ L(E) 1Φ = 1 log ĥ P (E) E (negative) E < 0 L(E) 1 Chern c 1 (L(E)) E (positive) (ample) E > 0 E E E 2 ([11, 12]) M E h L(E) Hermite (6) h E Finsler F E < 0 1Φ P (E) F Finsler F R i jµ ν = 2 G i j z µ z ν + k,l G kl G i k z µ G l j z ν, ( ) ( ) 1 G kl = Gi j F Hermite ( i,j R i jµ νζ ) i ζj (z, ζ) E E Ẽ Ẽ Finsler 4 Hartshorne Frankel 1 Surveys in Geometry 1979 [14] Siu-Yau [21] 7

8 Hartshorne ([10], [14]) m 2 M m P m (C) Frankel ([6], [21]) m Kähler M m P m (C) [13] P m (C) m M 1 Chern c 1 (M) M L c 1 (M) (m + 1)c 1 (L) M P m (C) M P 1 (C) ([7]) M P 1 (C) M Siu Yau Siu Yau Sacks-Uhlenbeck [18] S 2 M S 2 = P 1 (C) M 3 M Kähler 3 ([21]) M Kähler f : P 1 (C) M Hermite M M Frankel Hartshorne p > 0 Hartshorne Frankel E E P (E) L(E) Hermite Finsler Finsler Hartshorne 2 Hartshorne C k p > 0 8

9 5 Kähler Finsler M m π : T 1,0 M M M z = (z 1,..., z m ) M T 1,0 M v = µ vµ ( / z µ ) T 1,0 M (z, v) = (z 1,..., z m, v 1,..., v m ) = (z µ, v α ) F : T 1,0 M R T 1,0 M Finsler M = T 1,0 M \ {0} M G = F 2 m G(z, v) = G α β(z, v)v α v β, G α β = 2 G/ v α v β α,β=1 N Riemann w N f : N M N M C f(w) = (f 1 (w),..., f m (w)) = (f µ (w)) f N T 0,1 N M T 1,0 M f : T 0,1 N T 1,0 M ( ) m f f µ = w w z µ f f - ( -energy) ( ( )) 1 (7) E (f) = F 2 f(w), f dw d w w 2 α=1 N µ=1 (7) N w Finsler (3) π : T 1,0 M M M T 1,0 M (vertical bundle) V = Ker dπ T 1,0 M Hermite m ( ) m ( ) W 1 = W1 α, W v α 2 = W β 2 (z,v) v β (z,v) β=1 m W 1, W 2 (z,v) = G α β(z, v)w1 α W β 2, α,β=1 (z, v) M Hermite V Hermite D D T 1,0 M (horizontal bundle) H T 1,0 M V H T 1,0 M = V H Hermite, Hermite D : Γ(T 1,0 M) Γ(T C M T 1,0 M) 9

10 ([1]) T 1,0 M Finsler F T 1,0 M Hermite D Hermite, D X Y, Z = X Y, Z + Y, X Z, X, Y, Z Γ(T 1,0 M) (torsion) X Y X Y T 1,0 M M (2, 0) θ X Y Y X = [X, Y ] + θ(x, Y ), X, Y Γ(T 1,0 M) θ (8) θ(h, χ), χ = 0 H H χ : M H F Kähler (weakly Kähler) Finsler (8) M G α [Γ α ν;µ Γ α µ;ν]v µ = 0, (z, v) M α,µ G α = G/ v α Γ α ν;µ = τ,γ,ρ G τα ( Gβ τ z µ G ργ G ν τ v γ 2 G v ρ z µ ), ( G γδ ) = ( G α β) 1 3 D F : T 1,0 M R Kähler Finsler f C (N, M) {f s } s I, I = {s C s < ɛ} f = f 0 C - E (f s ) 1 E Euler- Lagrange (9) 2 f σ w w + m µ,ν=1 Γ σ f µ f ν ν;µ w w = 0, 1 σ m 2 f σ w w (w) + m µ,ν=1 Γ σ ν;µ ( ( )) f µ f(w), f ν w w (w) f w (w) = 0 3 θ 0 Kähler 10

11 f ( / w ) = 0 Γ σ ν;µ ( f(w), f ( / w) ) = 0 Riemann N Kähler F M C f : N M (9) f N M 3 4 ([17]) M Kähler F : T 1,0 M R F - f : P 1 (C) M E (f s ) 2 2 s s E (f s ) ( 0) s=0 F [1] M. Abate and G. Patrizio, Finsler Metrics A Global Approach, Lecture Notes in Math. Vol. 1591, Springer-Verlag, [2] J. Eells and L. Lemaire, A report on harmonic maps, Bull. London Math. Soc. 10 (1978), [3] J. Eells and L. Lemaire, Selected Topics in Harmonic Maps, C. B. M. S. Regional Conference Series No. 50, Amer Math Soc., [4] J. Eells and L. Lemaire, Another report on harmonic maps, Bull. London Math. Soc. 20 (1988), [5] J. Eells and J. H. Sampson, Harmonic mappings of Riemannian manifolds, Amer. J. Math. 86 (1964), [6] T. Frankel, Manifolds with positive curvature, Pacific J. Math. 11 (1961), [7] A. Grothendieck, Sur la classification des fibrés holomorphes sur la sphère de Riemann, Amer. J. Math. 79 (1957),

12 [8] R. S. Hamilton, Harmonic Maps of Manifolds with Boundary, Lecture Notes in Math. Vol. 471, Springer-Verlag, [9] P. Hartman, On homotopic harmonic maps, Canad J. Math. 19 (1967), [10] R. Hartshorne, Ample subvarieties of algebraic varieties, Lecture Notes in Math. Vol. 156, Springer-Verlag, [11] S. Kobayashi, Negative vector bundles and complex Finsler structures, Nagoya Math. J. 57 (1975), [12] S. Kobayashi, Complex Finsler bundles, Contemp. Math. 196 (1996), [13] S. Kobayashi and T. Ochiai, Characterizations of complex projective spaces and hyperquadrics, J. Math. Kyoto Univ. 13 (1973), [14] S. Mori, Projective manifolds with ample tangent bundles, Ann. of Math. 110 (1979), [15], Harmonic Maps, Reports on Global Analysis II, [16],,, [17] S. Nishikawa, Harmonic maps into complex Finsler manifolds, in preparation. [18] J. Sacks and K. Uhlenbeck, The existence of minimal immersions of 2-spheres, Ann. of Math. 113 (1981), [19] Y.-T. Siu, The complex-analyticity of harmonic maps and the strong rigidity of compact Kähler manifolds, Ann. of Math. 112 (1980), [20] Y.-T. Siu, Strong rigidity of compact quotients of exceptional bounded symmetric domains, Duke Math J. 48 (1981), [21] Y.-T. Siu and S.-T. Yau, Compact Kähler manifolds of positive bisectional curvature, Invent. Math. 59 (1980),

(2) Fisher α (α) α Fisher α ( α) 0 Levi Civita (1) ( 1) e m (e) (m) ([1], [2], [13]) Poincaré e m Poincaré e m Kähler-like 2 Kähler-like

(2) Fisher α (α) α Fisher α ( α) 0 Levi Civita (1) ( 1) e m (e) (m) ([1], [2], [13]) Poincaré e m Poincaré e m Kähler-like 2 Kähler-like () 10 9 30 1 Fisher α (α) α Fisher α ( α) 0 Levi Civita (1) ( 1) e m (e) (m) ([1], [], [13]) Poincaré e m Poincaré e m Kähler-like Kähler-like Kähler M g M X, Y, Z (.1) Xg(Y, Z) = g( X Y, Z) + g(y, XZ)