Part 2 Nonlinear extrinsic flows on entire graphs MCF and IMCF Panagiota Daskalopoulos Columbia University Summer School on Extrinsic flows ICTP - Trieste June 4-8, 2018 Panagiota Daskalopoulos Part 2 Nonlinear extrinsic flows on entire graphs MCF and IMCF
Future Lectures We have seen in Lecture 1 some basic properties for degenerate and fast diffusion. Also classical results the solvability for the Cauchy problem for these equations on R n . In our future lectures we will see how these properties and results relate to more recent works on extrinsic geometric flows on complete non-compact graphs. Panagiota Daskalopoulos Part 2 Nonlinear extrinsic flows on entire graphs MCF and IMCF
The evolution of complete graphs Assume that M t is a complete non-compact graph over a domain Ω ⊂ R n . Σ 0 Σ n 0 Let F t : N n → R n +1 be a family of immersions of our graph M t := F t ( N n ) in R n +1 evolving by the flow ∂ p ∈ N n ∂ t F ( p , t ) = σ ( p , t ) ν ( p , t ) , where ν ( p , t ) is a choice of normal vector and σ ( λ 1 · · · λ n ) is the speed. Panagiota Daskalopoulos Part 2 Nonlinear extrinsic flows on entire graphs MCF and IMCF
Examples of such flows Examples of nonlinear extrinsic geometric flows Mean curvature flow: σ = H = λ 1 + · · · + λ n σ = − 1 1 Inverse mean curvature flow: H = − λ 1 + ··· + λ n Gauss curvature flow: σ = K = λ 1 · · · λ n σ = K α = ( λ 1 · · · λ n ) α , α -Gauss curvature flow:: 0 < α < ∞ . Panagiota Daskalopoulos Part 2 Nonlinear extrinsic flows on entire graphs MCF and IMCF
Outline We will discuss: The Mean curvature flow where on entire graphs. An example of quasilinear diffusion which resembles the heat equation. The Inverse mean curvature flow on entire convex graphs. An example for fully-nonlinear ultra-fast diffusion. The α -Gauss curvature flow on complete non-compact graphs. An example of fully-nonlinear slow diffusion. Panagiota Daskalopoulos Part 2 Nonlinear extrinsic flows on entire graphs MCF and IMCF
The evolution of complete graphs - Graph Parametrization Let x n +1 = ¯ u ( x , t ), ¯ u : Ω × [0 , T ) → R , be a graph over Ω ⊂ R n which evolves by a non-linear extrinsic flow. ¯ u ( x )), x ∈ R n . Graph Parametrization: F ( x ) := ( x , ¯ This is not the same as the geometric parametrization F ( · , t ) : N n → R n +1 under which ∂ p ∈ N n ∂ t F ( p , t ) = σ ( p , t ) ν ( p , t ) , In the graph parametrization the equation is: � ∂ � ⊥ ¯ F ( x , t ) = σ ν ∂ t Then ¯ u satisfies the equation � u , D 2 ¯ u | ) 2 ¯ u t = ¯ 1 + | D ¯ σ ( x , ¯ u , D ¯ u ) u , D 2 ¯ where ¯ σ is the speed as a function of x , ¯ u , D ¯ u . Panagiota Daskalopoulos Part 2 Nonlinear extrinsic flows on entire graphs MCF and IMCF
MCF on entire graphs Let M t := F t ( N n ) ⊂ R n +1 , where F t : N n → R n +1 immersions evolving by the Mean curvature flow ∂ p ∈ N n . ( ⋆ MCF ) ∂ t F ( p , t ) = H ( p , t ) ν ( p , t ) , We assume that M t is an entire graph over R n ; i.e. there exists a unit vector ω ∈ R n +1 , such that � ω, ν � > 0 , on M t . From now on we take ω := e n +1 . Graph parametrization: We may write M t as x n +1 = ¯ u ( x , t ), u : R n × [0 , T ) → R . Then the (MCF) is equivalent to: for ¯ D ¯ u � u | 2 div � � ( ⋆ MCF ¯ u ) u t = ¯ 1 + | D ¯ . � u | 2 1 + | D ¯ Remark: We will use the geometric parametrization and not the graph parametrization. Panagiota Daskalopoulos Part 2 Nonlinear extrinsic flows on entire graphs MCF and IMCF
Geometry on a graph Consider the graph F = ( x , u ( x )), x ∈ R n . Metric: � ∂ F , ∂ F � g ij = = δ ij + D i ¯ uD j ¯ u . ∂ x i ∂ x j Second fundamental form: ν ( x ) , ∂ 2 F � � h ij = − ∂ x i ∂ x j Mean curvature H : � � � δ ij − D i ¯ uD j ¯ � D ij ¯ D i ¯ u u u H = g ij h ij = u | 2 = D i u | 2 1 + | D ¯ � � u | 2 1 + | D ¯ 1 + | D ¯ Panagiota Daskalopoulos Part 2 Nonlinear extrinsic flows on entire graphs MCF and IMCF
MCF on entire graphs - Long time existence The following is work by K. Ecker and G. Huisken (1989-1991): Theorem (Ecker-Huisken) Let M 0 be a locally Lipschitz entire graph over R n .Then, the (MCF) admits a C ∞ solution M t with initial data M 0 , for all t > 0. Moreover, M t remains an entire graph over R n . Idea of the Proof: (i) Show a local bound on the gradient function v := −� e n +1 , ν � − 1 = � 1 + | Du | 2 ; (ii) Use this bound to obtain a local bound on the second fundamental form | A | 2 which is independent of the initial data. Remark 1: Note that no growth assumptions or smoothness need to be imposed on the initial graph. Remark 2: No uniqueness was shown. Panagiota Daskalopoulos Part 2 Nonlinear extrinsic flows on entire graphs MCF and IMCF
MCF on entire graphs - Local gradient Estimate Assume that M t is an entire graph over R n ; i.e. � e n +1 , ν � > 0 , on M t for a choice of unit normal ν . Let M t be given by x n +1 = u ( x , t ), for u : R n × [0 , T ) → R . Then v := � e n +1 , ν � − 1 = 1 + | Du | 2 satisfies: � ∂ ∂ t v = ∆ v − 2 v − 1 |∇ v | 2 − | A | 2 v . For any R > 0, let η ( F , t ) = ( R 2 − | F | 2 − 2 nt ) + a cut off function, where F ( · , t ) ∈ M t denotes the position vector. Local gradient estimate: We have v ( F , t ) η ( F , t ) ≤ sup ( v η ) . M 0 Panagiota Daskalopoulos Part 2 Nonlinear extrinsic flows on entire graphs MCF and IMCF
MCF on entire graphs - Bound on | A | 2 The second fundamental form A = { h ij } satisfies: ∂ ∂ t | A | 2 = ∆ | A | 2 − 2 |∇ A | 2 + 2 | A | 4 . The gradient v := � e n +1 , ν � − 1 = 1 + | Du | 2 satisfies: � ∂ ∂ t v = ∆ v − 2 v − 1 |∇ v | 2 − | A | 2 v . Combine the evolutions of v and | A | 2 to obtain a local bound on | A | 2 which is independent of the initial data. Crucial bound on | A | 2 : For ρ > 0, let B ρ ( y 0 ) ⊂ R n . Then, for any θ ∈ (0 , 1) and 0 ≤ t ≤ 1: ρ − 2 + t − 1 � | A | 2 ( · , t ) ≤ C n (1 − θ 2 ) − 2 � v 4 . sup sup B θρ ( y 0 ) B ρ ( y 0 ) × [0 , t ] Panagiota Daskalopoulos Part 2 Nonlinear extrinsic flows on entire graphs MCF and IMCF
MCF on entire graphs - Proof of the bound on | A | 2 Proof: It uses the Caffarelli, Nirenberg and Spruck trick: v 2 Let ϕ ( v 2 ) = 1 − kv 2 . You derive that g := | A | 2 φ ( v 2 ) satisfies: 2 k g ≤ − 2 kg 2 − (1 − kv 2 ) 2 |∇ v | 2 g − 2 ϕ v − 3 ∇ v ∇ g . � � ( ∗ ) ∂ t − ∆ Let m ( t ) := sup M t g (if it is finite !). Then, formally if you could apply the maximum principle, ( ∗ ) would give d dt m ( t ) ≤ − 2 k m ( t ) 2 . 1 Comparing with the solution of the ODE gives m ( t ) ≤ 2 kt , i.e. 1 | A | 2 φ ( v 2 ) ≤ sup 2 kt . M t 1 + | Du | 2 ≫ 1, as � However this is not possible if v := | F | → + ∞ . Panagiota Daskalopoulos Part 2 Nonlinear extrinsic flows on entire graphs MCF and IMCF
MCF on entire graphs - Proof of the bound on | A | 2 To over come this difficulty we need to localize the equation of g by multiplying with the cut-off function η ( F , t ) = ( R 2 − r ) + , r := | F | 2 + 2 nt . Then, using ( ∗ ) we obtain that G := g η t satisfies 1 G ≤ A · ∇ G − 2 kg 2 η t + C kv 2 ) r + R 2 � � � � ∂ t − ∆ (1 + gt + g η. Applying the maximum principle to m ( t ) := max r < R 2 G , we conclude for any θ ∈ (0 , 1) the bound ρ − 2 + t − 1 � | A | 2 ( · , t ) ≤ C n (1 − θ 2 ) − 2 � v 4 . sup sup B θρ ( y 0 ) B ρ ( y 0 ) × [0 , t ] Panagiota Daskalopoulos Part 2 Nonlinear extrinsic flows on entire graphs MCF and IMCF
Conclusion Theorem. If M 0 is a locally Lipschitz entire graph over R n , then the (MCF) admits a C ∞ -smooth solution M t with initial data M 0 , for all t > 0. M t is an entire graph over R n . At a maximal existence time T , if T < + ∞ the graph M T is locally Lipschitz and also | A | 2 is locally bounded. Hence, the flow may be continued to show that T = + ∞ . In addition, since the evolution equation is uniformly parabolic on compact sets, the solution M t is C ∞ smooth by standard parabolic regularity. Panagiota Daskalopoulos Part 2 Nonlinear extrinsic flows on entire graphs MCF and IMCF
Inverse Mean curvature flow - Introduction Let F : N n × [0 , T ] → R n +1 be a smooth family of closed hypersurfaces in R n +1 . F defines a classical solution to the Inverse mean curvature flow in R n +1 if it satisfies ∂ 1 p ∈ N n ∂ t F ( p , t ) = H ( p , t ) ν ( p , t ) , where H ( · , t ) > 0 and ν ( p , t ) denote the mean curvature and exterior unit normal of the surface M t at the point F ( p , t ). M t t compact in R n +1 or a graph over R n Figure: Hypersurface M n Panagiota Daskalopoulos Part 2 Nonlinear extrinsic flows on entire graphs MCF and IMCF
IMCF- An ultra-fast diffusion Under (IMCF) the Mean curvature satisfies the ultra-fast diffusion: H 3 |∇ H | 2 − | A | 2 ∂ t H = 1 ∂ H 2 ∆ H − 2 H . The above equation can also be written as ∂ t H = − ∆ H − 1 − | A | 2 ∂ H . This resembles the ultra fast diffusion equation on R n : u t = − ∆ u m , m < 0 . Panagiota Daskalopoulos Part 2 Nonlinear extrinsic flows on entire graphs MCF and IMCF
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