Part 3 Gauss Curvature flow Panagiota Daskalopoulos Columbia - PowerPoint PPT Presentation

Part 3 Gauss Curvature flow Panagiota Daskalopoulos Columbia University Summer School on Extrinsic flows ICTP - Trieste June 4-8, 2018 Panagiota Daskalopoulos Part 3 Gauss Curvature flow

The Gauss Curvature flow - Introduction Consider the evolution of a hypersurface M t in R n +1 by the α -Gauss Curvature flow ∂ P ∂ t = K α ν ( ∗ k ) with speed K α = ( λ 1 , · · · , λ n ) α , α > 0. This is a well known example of fully-nolinear degenerate diffusion of Monge-Amp´ ere type It was introduced by W. Firey in 1974 and has been widely studied especially in the compact case. We note important geometric works in the compact case by: K. Tso, B. Chow, R. Hamilton, J. Urbas, B. Andrews, K. Lee, X. Chen, P. Guan, L. Ni, S. Brendle, K. Choi among many others. Panagiota Daskalopoulos Part 3 Gauss Curvature flow

The Gauss Curvature Flow on compact surfaces Firey 1974: The GCF ( α = 1) models the wearing process of tumbling stones subjected to collisions from all directions with uniform frequency. Firey: The GCF shrinks strictly convex compact and centrally symmetric surfaces to round points. Firey’s conjecture: The GCF shrinks any strictly convex compact hypersurface to spherical points. Tso 1985: Existence and uniqueness for compact strictly convex and smooth initial data up. Andrews 1999: Firey’s Conjecture for strictly convex surfaces in dim n = 2. Brendle, Choi and D., 2017: Firey’s Conjecture for the GCF α , 1 α > n +2 , flow in any dimension n ≥ 2. Based on previous work by Andrews, Guan and Ni on convergence to self-similar solutions. Other works: Andrews, Guan-Ni, Kim-Lee. Panagiota Daskalopoulos Part 3 Gauss Curvature flow

Outline We will discuss the following topics on GCF: GCF on complete non-compact convex hypersurfaces Optimal regularity of solutions Surfaces with Flat sides Firey’s Conjecture Panagiota Daskalopoulos Part 3 Gauss Curvature flow

Gauss Curvature flow - the PDE If x n +1 = u ( x , t ) defines M n locally, then the GCF becomes equivalent to the Monge-Amp´ ere type of eq. det D 2 u u t = . n +1 (1 + | Du | 2 ) 2 To understand the nature of the PDE let us look at the case n = 2: u t = u xx u yy − u 2 xy . 3 (1 + | Du | 2 ) 2 The linearized equation at u is h t = u yy h xx + u xx h yy − 2 u xy h xy + lower order 3 (1 + | Du | 2 ) 2 One see that this equation becomes degenerate what points where u is not strictly convex . Panagiota Daskalopoulos Part 3 Gauss Curvature flow

The Regularity of solutions to GCF -Known Results Hamilton: Convex surfaces with at most one vanishing principal curvature, will instantly become strictly convex and hence smooth. Chopp, Evans and Ishii: If M n is C 3 , 1 at a point P 0 and two or more principal curvatures vanish at P 0 , then P 0 will not move for some time τ > 0. Andrews: A surface M 2 in R 3 evolving by the GCF is always C 1 , 1 on 0 < t < T and smooth on t 0 ≤ t < T , for some t 0 > 0. This is the optimal regularity in dimension n = 2. Remark: The regularity of solutions M n in dimensions n ≥ 3 poses a much harder question. Hamilton: If a surface M 2 in R 3 has flat sides, then the flat sides will persist for some time. Panagiota Daskalopoulos Part 3 Gauss Curvature flow

Basic equations under GCF ∂ t g ij = 2 K α h ij ∂ t g ij = − 2 K α h ij , ∂ t ν = −∇ K α ∂ t h ij = L h ij + α K α A klmn ∇ i h mn ∇ j h kl + α K α Hh ij − (1 + n α ) K α h ik h k j ∂ t K α = L K α + α K 2 α H ∂ t b pq = L b pq − α K α b ip b jq B klmn ∇ i h kl ∇ j h mn − α K α Hb pq +(1 + n α ) K α g pq ∂ t v = L v − 2 v − 1 �∇ v � 2 L − α K α H Panagiota Daskalopoulos Part 3 Gauss Curvature flow

Basic equations under GCF The function ψ β ( p , t ) = ( M − β t − ¯ u ( p , t )) + satisfies ∂ t ψ β = L ψ β + ( n α − 1) v − 1 K α − β ψ ( p , t ) = ( R 2 − | F ( p , t ) − ¯ The function ¯ x 0 | 2 ) + satisfies ∂ t ¯ ψ ≤ L ¯ n α + 1)( λ − 1 min + R ) K α � ψ + 2 Panagiota Daskalopoulos Part 3 Gauss Curvature flow

The GCF-flow on complete non-compact graphs Jointly with K. Choi, L. Kim and K. Lee we studied the evolution of complete non-compact graphs M t in R n +1 by the α -Gauss Curvature flow ∂ P ∂ t = K α ν ( ∗ k ) with speed K α = ( λ 1 , · · · , λ n ) α , α > 0. Here ν is the inner normal. We assume that M 0 is a complete non-compact strictly convex graph over a domain Ω ⊂ R n . The domain Ω may be bounded or unbounded (e.g. Ω = R n ). H. Wu (1974): a complete non-compact smooth and strictly convex hypersurface M 0 in R n +1 is the graph of a function u 0 defined on a domain Ω ⊂ R n . Panagiota Daskalopoulos Part 3 Gauss Curvature flow

Examples of the initial hypersurface M 0 M 0 M 0 (a) Ω = R n (b) Ω = B R (0) M 0 (c) Ω = R n − 1 × R + Figure: Examples of the initial hypersurface M 0 Panagiota Daskalopoulos Part 3 Gauss Curvature flow

The Main Results Theorem 1. Let M 0 = { ( x , u 0 ( x )) : x ∈ Ω } be a locally uniformly convex graph given by u 0 : Ω → R defined on a convex domain Ω ⊂ R n . Then, for any α > 0, there exists a smooth strictly convex solution u : Ω × (0 , + ∞ ) → R of the α -Gauss curvature flow (det D 2 u ) α ( ∗∗ α ) u t = ( n +2) α − 1 (1 + | Du | 2 ) 2 such that lim t → 0 u ( x , t ) = u 0 ( x ). Theorem 2. Let M 0 be a smooth complete non-compact and strictly convex hypersurface embedded in R n +1 . Then, for any α > 0, there exists a smooth complete non-compact and strictly convex solution M t of the α -Gauss curvature flow defined for all time 0 < t < + ∞ and having initial data M 0 . Panagiota Daskalopoulos Part 3 Gauss Curvature flow

Proof of Theorem 1 - Main steps for α = 1 For simplicity assume that α = 1. Assume that M 0 is a convex graph in the direction of ω := e n +1 . Then, M t will remain a convex graph. Define the heignt function ¯ u := � F , e n +1 � . The proof of Theorem 1 replies on local a’priori geometric bounds which are shown by the maximum principle. Local gradient estimate on v:= � e n +1 , ν � − 1 = � 1 + | Du | 2 . 1 Local lower bound for the principal curvatures, i.e. 2 on λ min . Local upper speed bound, i.e. on K . 3 Linearized operator: L = Kb ij ∇ i ∇ j , where b ij = ( h ij ) − 1 . Remark: It is easier to use geometric bounds, rather than pure PDE bounds on the evolution of u . Panagiota Daskalopoulos Part 3 Gauss Curvature flow

The Gradient Estimate u ) t = n − 1 L ¯ Height function: ¯ u := � F , e n +1 � . It satisfies (¯ u . Cut off function: ψ β ( p , t ) = ( M − β t − ¯ u ( p , t )) + . It satisfies ∂ t ψ β = L ψ β + ( n − 1) v − 1 K − β. Gradient: v = � e n +1 , ν � − 1 = 1 + | Du | 2 satisfies the � equation ∂ t v = L v − 2 v − 1 �∇ v � 2 L − K H v Gradient Estimate: Given β > 0 and M ≥ β : 1 v ( p , 0) , β − 1 n n +1 ( n − 1) � � v ( p , t ) ψ β ( p , t ) ≤ M max sup u ≤ M ¯ Panagiota Daskalopoulos Part 3 Gauss Curvature flow

Local lower bound on λ min Recall that ψ β ( p , t ) = ( M − β t − ¯ u ( p , t )) + . The most crucial estimate is the following lower curvature bound: � � ( p , t ) ≥ M − 2 n min ψ − 2 n � � λ min u ≤ M λ min ( p , 0) , B n ,β inf β ¯ where B n ,β constant depending on parameters. Proof: By a rather involved Pogorelov type computation to bound from above ψ 2 n β λ − 1 min . Panagiota Daskalopoulos Part 3 Gauss Curvature flow

Local upper bound on K Let ψ := ( M − ¯ u ) + , where ¯ u := � F , e n +1 � is the height function. Recall that v = � e n +1 , ν � − 1 = � 1 + | Du | 2 . We show the following local upper bound for the speed K . t � � ( ψ 2 K 1 1 n )( p , t ) ≤ (4 n α + 1) 2 (2 θ ) 1+ 2 n α ( θ Λ + M 2 ) 1 + t where θ and Λ are constants given by θ = sup { v 2 ( p , s ) : ¯ u ( p , s ) < M , s ∈ [0 , t ] } , Λ = sup { λ − 1 min ( p , s ) : ¯ u ( p , s ) < M , s ∈ [0 , t ] } . v 2 Proof: Following the CGN trick we set ϕ ( v ) := 2 θ − v 2 and apply the maximum principle on K 2 ϕ ( v ). Remark: The upper bound on K at time t > 0 does not depend on an upper bound on K at time t = 0. Panagiota Daskalopoulos Part 3 Gauss Curvature flow

The Proof of Long time existence We obtain a solution M t := { ( x , u ( · , t )) : x ∈ Ω t ⊂ R n } as j → + ∞ Γ j M t := lim t where Γ j t is a strictly convex closed solution symmetric with respect to the hyperplane x n +1 = j . To pass to the limit we show that our a’priori estimates imply a uniform local C 2 ,α bound for Γ j t . Finally we construct barriers to show that Ω t = Ω for all 0 < t < + ∞ . This is expected since K ( x , u ( x , t )) → 0, as x → ∂ Ω t . Panagiota Daskalopoulos Part 3 Gauss Curvature flow

The Regularity of solutions to GCF If x n +1 = u ( x , t ) defines M n locally, then u evolves by the PDE det D 2 u u t = . n +1 (1 + | Du | 2 ) 2 A strictly convex surface evolving by the GCF remains strictly convex and hence smooth up to its collapsing time T . The problem of the regularity of solutions in the weakly convex case is a difficult question. It is related to the regularity of solutions of the evolution Monge-Amp´ ere equation u t = det D 2 u . Question: What is the optimal regularity of weakly convex solutions to the Gauss Curvature flow ? Panagiota Daskalopoulos Part 3 Gauss Curvature flow