Ancient solutions to Geometric Flows Lecture No 2

Transcription

1 Ancient solutions to Geometric Flows Lecture No 2 Panagiota Daskalopoulos Columbia University Frontiers of Mathematics and Applications IV UIMP 2015 July 20-24, 2015

2 Topics to be discussed In this lecture we will discuss: Liouville s theorem for the heat equation on manifolds Eternal solutions to the semi linear diffusion. Classification of ancient solutions to the curve shortening flow Ancient compact solutions to the mean curvature flow.

3 Ancient and Eternal solutions Definition: A solution u(, t) to a parabolic equation is called ancient if it is defined for all time < t < T, T < +. Ancient solutions typically arise as blow up limits at a type I singularity. Definition: A solution u(, t) to a parabolic equation is called eternal if it is defined for all < t < +. Eternal solutions as blow up limits at a type II singularity.

4 Solitons Solitons (self-similar solutions) are typical examples of ancient or eternal solutions and often models of singularities. A remarkable technique introduced by R. Hamilton (1995) has been widely used to characterize as solitons the eternal solutions to geometric flows which attain a space-time curvature maximum. Such solutions typically appear as carefully chosen blow up limits near type II singularities. Its proof relies on a clever combination of the strong maximum principle and Li-Yau type differentiable Harnack estimates.

5 Other Ancient and eternal solutions However, there exist other ancient or eternal solutions which are not solitons. These, often may be visualized as obtained from the gluing as t of two or more solitons. Obtaining more information about such solutions, often leads to the better understanding of the singularities. Objective: How to construct such solutions and how to characterize them among all ancient or eternal solutions.

6 Liouville s theorem for the heat equation on manifolds Let M n be a complete non-compact Riemannian manifold of dimension n 2 with Ricci (M n ) 0. Yau (1975): be constant. Any positive harmonic function u on M n must This is the analogue of Liouville s Theorem for harmonic functions on R n. Question: Does the analogue of Yau s theorem hold for positive solutions of the heat equation u t = u on M n? Answer: No. Example u(x, t) = e x 1+t, x = (x 1,, x n ) on M n := R n.

7 A Liouville type theorem for the heat equation Souplet - Zhang (2006): Let M n be a complete non-compact Riemannian manifold of dimension n 2 with Ricci (M n ) 0. (a) If u be a positive ancient solution to the heat equation on M n (, T ) such that u(p, t) = e o(d(p)+ t ) as d(p) then u is a constant. (b) If u be an ancient solution to the heat equation on M n (, T ) such that u(p, t) = o(d(p) + t ) as d(p) then u is a constant. Proof: By using a local gradient estimate of Li-Yau type on large appropriately scaled parabolic cylinders.

8 The sub-critical semi-linear heat equation Consider nonnegative solutions u(t) H 1 (R n ) of equation u t = u + u p on R n (0, T ), 1 < p < n + 2 n 2. u blows up in finite time T if lim t T u(t) H 1 (R n ) = +. Recall: u H 1 (R n ) := ( R n u 2 + Du 2 dx ) 1/2. Fillipas, Giga, Herrero, Kohn, Merle, Velazquez, Zaag etc: analyzed the blow up behavior behavior of the solution u(t) near a singular point (a, T ). Self-similar scaling: w(y, τ) = (T t) 1 p 1 u(x, t), y = x a, τ = log(t t). T t Giga, Kohn : Showed w(, τ) L (R n ) C, τ > log T.

9 The rescaled semi-linear heat equation The rescaled solution satisfies the equation ( ) w τ = w 1 2 y w w p 1 + w p. Objective: To analyze the blow up behavior of u we need to understand the long time behavior of w as τ +. If τ k + then (passing to a subsequence) w k (y, τ) := w(y, τ + τ k ) converges to an eternal solution ŵ of equation ( ). To understand the blow up behavior of u we need to classify the bounded eternal solutions of ( ).

10 Eternal solutions of the semi-linear heat equation Consider a bounded nonnegative eternal solutions of equation ( ) w τ = w 1 2 y w w p 1 + w p on R n R. Steady states: w = 0 or w = κ, with κ := (p 1) 1 (p 1). lim τ ± w(, τ) = steady state. Solutions independent of y: φ(τ) = κ(1 + e τ ) 1 (p 1). Theorem (Giga-Kohn 87 and Merle-Zaag 98) If w is bounded nonnegative eternal solution of ( ) then w = 0 or w = κ or w(τ) = φ(τ τ 0 ).

11 Classification of eternal solutions - Sketch of Proof Lyapunov functional: E(w) = 1 w 2 1 dµ+ 2 2(p 1) w 2 dµ 1 p + 1 w p+1 dµ where dµ = (4π) n 2 e y 2 4 dy. d Monotonicity: dτ E(w(τ)) = wτ 2 dµ. Limits as τ ± : The limits w ± := lim s ± w(, τ) are steady states of ( ). Hence w ± = 0, κ. Since E(w(τ)) is decreasing, there are only three possibilities: w ± = 0 or w ± = κ or w = κ and w + = 0. Easy case - Giga-Kohn 89: If w ± = 0 or w ± = κ, then by the monotonicity of the energy E(w(s)) one concludes that w 0 or w κ.

12 Classification of eternal solutions - Sketch of Proof Difficult case - Marle-Zaag 98 : If w = κ and w + = 0, then w is independent of y, hence the solution of the ODE w τ = w p 1 + w p s R connecting the two steady states w = κ and w + = 0. It follows that w(τ) = φ(τ τ 0 ) with φ(τ) = κ(1 + e τ ) 1 (p 1). Idea of the Proof: They consider the linearized operator L at w = κ and carefully analyze the projections of w near τ = into the positive, null and negative eigenspaces of L. They conclude that the bounded solution w satisfies w(y, τ) a(τ) θ(y) as τ where θ is the eigenfunction corresponding to the largest (positive) eigenvalue of L which is independent of y.

13 The Curve shortening flow Let Γ t be a family of closed curves which is an embedded solution to the Curve shortening flow, i.e. the embedding F : Γ t R 2 satisfies F t = κ ν with κ the curvature of the curve and ν the outer normal. Gage (1984); Gage and Hamilton (1996): if Γ 0 is convex, then the CSF shrinks Γ t to a round point. Grayson (1987): Γ t does not develop singularities before it becomes strictly convex.

14 The evolution of the curvature The curvature κ of Γ t evolves, in terms of its arc-length s, by κ t = κ ss + κ 3. If θ is the angle between the tangent vector of Γ t and the x-axis, then on convex curves one can express κ as a function of θ and compute its evolution κ t = κ 2 κ θθ + κ 3. We introduce the pressure function p = κ 2 which evolves by p t = p p θθ 1 2 p2 θ + 2 p2.

15 Examples of Ancient solutions We say that Γ t is type I if sup t p(θ, t) <. Γ t (, 1] Otherwise we say that Γ t is of type II. Example of a type I solution (contracting circles): p(θ, t) = 1 2( t), t < 0. Example of a type II solution (Angenent ovals): 1 p(θ, t) = λ( 1 e 2λt sin2 (θ + γ)), t < 0 with parameters λ > 0 and γ.

16 The Classification of Ancient Convex solutions to the CSF The Angenent ovals (paper clips) as t may be visualized as two grim reaper solutions glued together. Theorem (D., Hamilton, Sesum ) The only ancient convex solutions to the CSF are the contracting spheres or the Angenent ovals.

17 Sketch of proof - Monotone functional We will introduce a monotone functional I (t) which depends on our solution and analyze its behavior as t and as t 0 (vanishing time). Set α(θ, t) := p θ (θ, t). Then, α satisfies: We introduce the functional We have I (α) = α t = p (α θθ + 4 α). 2π 0 (α 2 θ 4α2 ) dθ. d 2π αt 2 I (α(t)) = 2 dt 0 p dθ 0. In particular: lim t I (α)(t) exists or it is +.

18 Sketch of proof - Classification of limits We show that Since lim I (α(t)) = 0 and lim I (α(t)) = 0. t 0 t d 2π αt 2 I (α(t)) = 2 dt 0 p dθ 0 we conclude that I (α(t)) 0. This implies that α t 0. Hence α θθ + 4α = 0. Solving in θ gives: α := p θ = a(t) cos 2θ + b(t) sin 2θ plugging back to the equation we conclude p(θ, t) = 1 2t and 1 or p(θ, t) = λ ( 1 e 2λt sin2 (θ + γ)).

19 Sketch of proof - lim t I (α(t)) 0 We recall that the pressure p satisfies p t = p p θθ 1 2 p2 θ + 2 p2 and also that p t 0 (Harnack estimate for ancient solutions). A direct calculation shows that ( p 2 ) θ p θ (p θ ) θθ + 4pθ 2 2p Integration by parts gives d dt 2π 0 p 2 θ 2p dθ = t 2π 0 ( pθθ 2 + 4p2 θ ) dθ = I (α(t)). On the other hand, from the inequality p t 0 we obtain 2π 0 p 2 θ 2p dθ 2 Combining the above gives 2π lim I (α(t)) 0. t 0 p dθ C

20 Sketch of proof - lim t 0 I (α(t)) = 0 Theorem (Gage - Hamilton) If Γ 0 is a closed convex curve embedded in the plane R 2, the curve shortening flow shrinks Γ t to a point in a circular manner. Recalling that α := p θ = (κ 2 ) θ we show that lim I (t) = lim t 0 t 0 2π by refining the above convergence result. 0 (α 2 θ 4α2 ) dθ = 0

21 Conclusion We set α(θ, t) := p θ (θ, t) and showed that α satisfies: α t = p (α θθ + 4 α). We introduced the functional I (α) = 2π 0 (α2 θ 4α2 ) dθ and showed that d 2π αt 2 I (α(t)) = 2 dθ 0. dt 0 p We showed that lim I (α(t)) = 0 and lim I (α(t)) = 0. t 0 t Hence α t 0, implying α θθ + 4α = 0. Recalling that α = p θ and using the equation we conclude p(θ, t) = 1 2t 1 or p(θ, t) = λ ( 1 e 2λt sin2 (θ + γ)).

22 Non-Convex ancient solutions Question: Do they exist non convex compact embedded solutions to the curve shortening flow? Angenent (2011): Presents a YouTube video of an ancient solution to the CSF built out from one Yin-Yang spiral and one Grim Reaper. S. Angenent: is currently working on a rigorous construction of these solutions.

23 The Mean curvature flow Let F : N n (, T ) R n+1 be a smooth ancient compact and convex solution of the Mean curvature flow (MCF) F t = H ν where H(p, t) denotes the Mean curvature of the surface M t := F (, t)(n n ) at the point F (p, t). The only ancient compact and convex self-similar solutions to the MCF the are the contracting spheres. Problem: Understand ancient compact solutions M t of the Mean curvature flow.

24 Ancient compact non-collapsed solutions to the MCF α-andrews flow: Solutions that satisfy an α-noncollapsed condition. B=B α H(p) Definition. We call an ancient oval to be any ancient compact and α-noncollapsed solution to MCF that is not self-similar. Haslhofer & Kleiner (2013): Ancient ovals are convex at each instant t, i.e ancient + α-noncollapsed convex. One may construct various examples of ancient compact and collapsed solutions of the MCF.

25 Ancient MCF ovals White (2003); Haslhofer & Hershkovits (2013): Establish the existence of ancient ovals with O(k) O(l) symmetry. We will refer to them as White ancient ovals. Angenent (2012): Establishes the formal matched asymptotics of the White ancient ovals as t. He shows that are small perturbations of ellipsoids. Problem: Show the uniqueness of White ovals, i.e. show: ancient oval White oval.

26 Unique asymptotics of Ancient MCF ovals S. Angenent, D., and N. Sesum (2015): All ancient ovals with O(1) O(n) symmetry have unique asymptotics as t. In addition we establish their precise asymptotic description at t = as formally predicted by Angenent. It follows from our analysis that as t : diam(t) 8 t log t and H max (t) 1 2 t log t. The proof involves: the analysis of the linearized operator at the cylinder, Huisken s monotonicity formula and carefully constructed barriers at the intermediate region.

27 Uniqueness of Ancient MCF ovals Work in progress: to establish such asymptotics in the non-symmetric case. Next Step: Establish the uniqueness of the Ancient ovals. Conjecture 1: The Ancient ovals with O(l) O(k) symmetry are uniquely determined by their asymptotics at t. Hence: they are unique (up to dilation and translation in rescaled time). Conjecture 2: All Ancient ovals are O(l) O(k) symmetric. Similarities with ancient compact solutions of the 3-dim Ricci flow.