Large curvature on open manifolds

Transcription

1 Large curvature on open manifolds Nadine Große Universität Leipzig (joint with: Marc Nardmann (Hamburg)) Leipzig,

2 Motivation Surface M - e.g. plane, sphere, torus

3 Motivation Surface M - e.g. plane, sphere, torus

4 Motivation Surface M - e.g. plane, sphere, torus Gauss curvature K: ( ) vol(b ɛ (p) M) = vol(b ɛ (0) R 2 ) 1 K(p) 12 ɛ2 + O(ɛ 3 )

5 Motivation Surface M - e.g. plane K = 0, sphere K = const > 0, torus Gauss curvature K: ( ) vol(b ɛ (p) M) = vol(b ɛ (0) R 2 ) 1 K(p) 12 ɛ2 + O(ɛ 3 )

6 Motivation Higher dimension n: (M n, g) several notions of curvature

7 Motivation Higher dimension n: (M n, g) several notions of curvature Scalar curvature: vol(b ɛ (p) M) = vol(b ɛ (0) R n ) ( 1 scal ) g (p) 6(n + 2) ɛ2 + O(ɛ 3 )

8 Motivation Higher dimension n: (M n, g) several notions of curvature Scalar curvature: scal g : M R ( vol(b ɛ (p) M) = vol(b ɛ (0) R n ) 1 scal ) g (p) 6(n + 2) ɛ2 + O(ɛ 3 )

9 Motivation Higher dimension n: (M n, g) several notions of curvature Scalar curvature: scal g : M R ( vol(b ɛ (p) M) = vol(b ɛ (0) R n ) 1 scal ) g (p) 6(n + 2) ɛ2 + O(ɛ 3 ) Sectional curvature: plane σ T p M M B ɛ (p) 2D-surface whose tangent space in p is σ

10 Motivation Higher dimension n: (M n, g) several notions of curvature Scalar curvature: scal g : M R ( vol(b ɛ (p) M) = vol(b ɛ (0) R n ) 1 scal ) g (p) 6(n + 2) ɛ2 + O(ɛ 3 ) Sectional curvature: plane σ T p M M B ɛ (p) 2D-surface whose tangent space in p is σ sec g (σ) = K M (p) = Gauss curvature of M in p

11 Motivation Higher dimension n: (M n, g) several notions of curvature Scalar curvature: scal g : M R ( vol(b ɛ (p) M) = vol(b ɛ (0) R n ) 1 scal ) g (p) 6(n + 2) ɛ2 + O(ɛ 3 ) Ricci curvature: Ric g : T p M T p M R Sectional curvature: plane σ T p M M B ɛ (p) 2D-surface whose tangent space in p is σ sec g (σ) = K M (p) = Gauss curvature of M in p

12 Motivation Given a manifold M. Can one prescribe the curvature? local: global:

13 Motivation Given a manifold M. Can one prescribe the scalar curvature? local: no obstruction global:

14 Motivation Given a manifold M. Can one prescribe the scalar curvature? local: no obstruction global: compact M: obstructions e.g. M compact surface: Gauss-Bonnet M KdA = 2π(2 2#holes)

15 Motivation Given a manifold M. Can one prescribe the scalar curvature? local: no obstruction global: compact M: obstructions e.g. M compact surface: Gauss-Bonnet M KdA = 2π(2 2#holes) noncompact connected M: On each M there is a metric whose scal is everywhere positive (negative).

16 Gromov s relative h-principle - Black Box M noncompact and connected h-principle implies: For constants c 1 < c 2 each of the relations scal > c 1, scal < c 2, c 1 < scal < c 2 can be fulfilled. Analog for Ric, sec.

17 Gromov s relative h-principle - Black Box M noncompact and connected h-principle implies: For constants c 1 < c 2 each of the relations scal > c 1, scal < c 2, c 1 < scal < c 2 can be fulfilled. Analog for Ric, sec. relative h-principle implies: Let B M be a closed subset of M such that M \ B has exits to infinity. Let there be a metric g on M that on B fulfills a relation as above. Then there is a metric g fulfilling the same relation on all of M and g B = g B. (M \ B has exits to infinity = each connected component of M \ B is not relatively compact in M)

18 Question Do there exist obstructions for curvature on noncompact connected manifolds?

19 Question Do there exist obstructions for curvature on noncompact connected manifolds? Today: Can curvature grow in an arbitrary way?

20 Enlarging scal g - First version Theorem (G.-Nardmann 12) Let M be a noncompact connected manifold of dimension n 2. Let f C (M, R). Then there is a metric g on M with scal g > f on M.

21 Enlarging scal g - First version Theorem (G.-Nardmann 12) Let M be a noncompact connected manifold of dimension n 2. Let f C (M, R). Then there is a metric g on M with scal g > f on M. Basic idea: Mixture out of explicit constructions on cylinder and Gromov s relative h-principle cylinders: to enlarge the curvature h-principle: to save curvature inequalities when topology changes

22 Enlarging scal g - First version Basic idea: Mixture out of explicit constructions on cylinder and Gromov s relative h-principle cylinders: to enlarge the curvature h-principle: to save curvature inequalities when topology changes (M i ) i compact exhaustion ( exits to infinity )

23 Enlarging scal g - First version Basic idea: Mixture out of explicit constructions on cylinder and Gromov s relative h-principle cylinders: to enlarge the curvature h-principle: to save curvature inequalities when topology changes (M i ) i compact exhaustion ( exits to infinity ) near N i := M i cyl: N i I

24 Enlarging scal g - First version Basic idea: c i := max M i :=M i (N i I ) f

25 Enlarging scal g - First version Cylinders:

26 Enlarging scal g - First version Cylinders:

27 Enlarging scal g - First version Cylinders:

28 Enlarging scal g - First version Cylinders:

29 Enlarging scal g - First version Cylinders: ḡ = e 2h(t) g: scalḡ = e 2h (scal g 2(n 1)g tt h + a 1 h + a 2 (h ) 2 )

30 Enlarging scal g - First version Cylinders: ḡ = e 2h(t) g: scalḡ = e 2h (scal g 2(n 1)g tt h + a 1 h + a 2 (h ) 2 )

31 Enlarging scal g - Second Version Theorem (G.-Nardmann 12) Let (M, g 0 ) be a noncompact connected manifold of dim n 2. Let A be a closed codimension-0 submanifold-with-boundary, such that M \ A has exits to infinity. Let f C (M, R) with scal g 0 = f on A. Then there is a metric g on M with g = g 0 on A and scal g > f on M \ A.

32 Enlarging ric g ric g (X ) := Ric g (X, X ) X 2, X T x M \ {0} g

33 Enlarging ric g ric g ([X ]) := Ric g (X, X ) X 2, X T x M\{0}, [X ] := {λx λ R 0 } g

34 Enlarging ric g ric g ([X ]) := Ric g (X, X ) X 2, X T x M\{0}, [X ] := {λx λ R 0 } g Theorem (G.-Nardmann 12) Let (M, g 0 ) be a noncompact connected manifold of dim n 2. Let A be a closed codimension-0 submanifold-with-boundary, such that M \ A has exits to infinity. Let F C (S x M, R) with ric g 0 = F on A. Then there is a metric g on M with g = g 0 on A and ric g > F on M \ A.