On Mean curvature flow of Singular Non compact cases Riemannian - PowerPoint PPT Presentation

On Mean curvature flow of Singular Riemannian foliations: On Mean curvature flow of Singular Non compact cases Riemannian foliations: Non compact cases Marcos M. Alexandrino (IME-USP) In honor of Professor Marcos M. Alexandrino (IME-USP) Jürgen Berndt’s In honor of Professor Jürgen Berndt’s 60 th birthday. 60 th birthday. Definitions ThmA [ACG19] Marcos M. Alexandrino, Leonardo F. Cavenaghi Basins of and Icaro Gonçalves, Mean curvature flow of singular attraction Riemannian foliations: Non compact cases , MCF of non-closed arXiv:1909.04201 (2019) regular leaf Cylinder structure Type 1

On Mean curvature flow of Singular Riemannian foliations: Non compact Definition cases Marcos M. Given a Riemannian manifold M and an immersion Alexandrino (IME-USP) ϕ : L 0 → M , a smooth family of immersions ϕ t : L 0 → M , In honor of t ∈ [ 0 , T ) is called a solution of the mean curvature flow Professor Jürgen (MCF for short) if ϕ t satisfies the evolution equation Berndt’s 60 th birthday. d dt ϕ t ( x ) = � H ( t , x ) , Definitions ThmA where � H ( t , x ) is the mean curvature of L ( t ) := ϕ t ( L 0 ) . Basins of attraction MCF of non-closed regular leaf Cylinder structure Type 1

On Mean curvature flow of Definition Singular Riemannian foliations: A submanifold L of a space form M ( k ) is called isoparametric Non compact if its normal bundle is flat and the principal curvatures along cases any parallel normal vector field are constant. Marcos M. Alexandrino (IME-USP) In honor of An isoparametric foliation F on M ( k ) is a partition of M ( k ) Professor Jürgen by submanifolds parallel to a given isoparametric submanifold L . Berndt’s 60 th birthday. Jurgen Berndt, Sergio Console, Carlos Enrique Olmos Definitions Submanifolds and Holonomy Chapman & Hall/CRC ThmA Monographs and Research Notes in Mathematics(2003) Basins of attraction MCF of G. Thorbergsson, Singular Riemannian Foliations and non-closed regular leaf Isoparametric Submanifolds Milan J. Math. Vol. 78 (2010) Cylinder 355–370 structure Type 1

On Mean curvature flow of Singular Definition Riemannian foliations: A singular foliation F = { L } is called a generalized Non compact isoparametric if cases Marcos M. 1 F is Riemannian , i.e., every geodesic perpendicular to one Alexandrino (IME-USP) leaf is perpendicular to every leaf it meets. In honor of Professor Jürgen Berndt’s 60 th birthday. Definitions ThmA Basins of attraction MCF of non-closed regular leaf Cylinder structure Type 1

On Mean curvature flow of Singular Definition Riemannian foliations: A singular foliation F = { L } is called a generalized Non compact isoparametric if cases Marcos M. 1 F is Riemannian , i.e., every geodesic perpendicular to one Alexandrino (IME-USP) leaf is perpendicular to every leaf it meets. In honor of Professor 2 the mean curvature field � H is basic in the principal Jürgen Berndt’s stratum M 0 60 th birthday. Definitions ThmA Basins of attraction MCF of non-closed regular leaf Cylinder structure Type 1

On Mean curvature flow of Singular Definition Riemannian foliations: A singular foliation F = { L } is called a generalized Non compact isoparametric if cases Marcos M. 1 F is Riemannian , i.e., every geodesic perpendicular to one Alexandrino (IME-USP) leaf is perpendicular to every leaf it meets. In honor of Professor 2 the mean curvature field � H is basic in the principal Jürgen Berndt’s stratum M 0 60 th birthday. Examples: Definitions 1 F = { G ( x ) } x ∈ M , where G is Lie subgroup of Iso ( M ) ThmA Basins of 2 isoparametric foliations, attraction 3 Singular Riemannian foliations with compact leaves on R n , MCF of S n and projective spaces (see Clifford foliations for non non-closed regular leaf homogenous examples). Cylinder structure Type 1

On Mean curvature flow of Singular Example (Holonomy foliations) Riemannian foliations: Non compact • L is a Riemannian manifold , cases • E is a Euclidean vector bundle over L (i.e., with an inner Marcos M. Alexandrino product � , � p on each fiber E p ) (IME-USP) In honor of • ∇ E is a metric connection on E , i.e. Professor Jürgen Berndt’s 60 th X � ξ, η � = �∇ E X ξ, η � + � ξ, ∇ E X η � . birthday. • the connection (Sasaki) metric g E on E Definitions ThmA Define the holonomy foliation F h on E , by declaring two Basins of attraction vectors ξ, η ∈ E in the same leaf if they can be connected to MCF of one another via a composition of parallel transports (with non-closed regular leaf respect to ∇ E ). Cylinder structure Type 1

On Mean curvature Example (Model) flow of Singular Consider a Euclidean vector bundle R n → E → L , with a metric Riemannian foliations: connection ∇ E and a the Sasaki metric g E . Let F 0 Non p = { L 0 ξ } ξ ∈ E p compact cases be a SRF with compact leaves on the fiber E p . Assume F 0 is Marcos M. invariant by the the holonomy group H p at p i.e., the group Alexandrino (IME-USP) sends leaves to leaves. In honor of Professor • F = { L ξ } ξ ∈ E p with leaves L ξ = H ( L 0 Jürgen ξ ) where H is the Berndt’s 60 th holonomy groupoid associate to the connection ∇ E . birthday. Definitions ThmA Basins of attraction MCF of non-closed regular leaf Cylinder structure Type 1

On Mean curvature Example (Model) flow of Singular Consider a Euclidean vector bundle R n → E → L , with a metric Riemannian foliations: connection ∇ E and a the Sasaki metric g E . Let F 0 Non p = { L 0 ξ } ξ ∈ E p compact cases be a SRF with compact leaves on the fiber E p . Assume F 0 is Marcos M. invariant by the the holonomy group H p at p i.e., the group Alexandrino (IME-USP) sends leaves to leaves. In honor of Professor • F = { L ξ } ξ ∈ E p with leaves L ξ = H ( L 0 Jürgen ξ ) where H is the Berndt’s 60 th holonomy groupoid associate to the connection ∇ E . birthday. ACG19 + Alexandrino, Inagaki, Struchiner(18) imply Definitions ThmA Lemma (Semi-local Model) Basins of attraction Let F be a SRF with closed leaves. Then F| Tub ǫ ( L q ) is foliated MCF of non-closed diffeomorphic to the foliation defined in Model. Therefore regular leaf Tub ǫ ( L q ) admits a metric so that F| Tub ǫ ( L q ) is a generalized Cylinder structure isoparametric foliation. Type 1

On Mean curvature Theorem A (ACG19) flow of Singular Riemannian Let F := { L } be a generalized isoparametric foliation with foliations: Non closed leaves on a complete manifold M so that M / F is compact cases compact. Let L 0 ∈ F be a regular leaf of M and let L ( t ) denote Marcos M. the MCF evolution of L 0 . Assume that T < ∞ . Then: Alexandrino (IME-USP) In honor of Professor Jürgen Berndt’s 60 th birthday. Definitions ThmA Basins of attraction MCF of non-closed regular leaf Cylinder structure Type 1

On Mean curvature Theorem A (ACG19) flow of Singular Riemannian Let F := { L } be a generalized isoparametric foliation with foliations: Non closed leaves on a complete manifold M so that M / F is compact cases compact. Let L 0 ∈ F be a regular leaf of M and let L ( t ) denote Marcos M. the MCF evolution of L 0 . Assume that T < ∞ . Then: Alexandrino (IME-USP) In honor of (a) L ( t ) converges to a singular leaf L T of F . Professor Jürgen Berndt’s 60 th birthday. Definitions ThmA Basins of attraction MCF of non-closed regular leaf Cylinder structure Type 1

On Mean curvature Theorem A (ACG19) flow of Singular Riemannian Let F := { L } be a generalized isoparametric foliation with foliations: Non closed leaves on a complete manifold M so that M / F is compact cases compact. Let L 0 ∈ F be a regular leaf of M and let L ( t ) denote Marcos M. the MCF evolution of L 0 . Assume that T < ∞ . Then: Alexandrino (IME-USP) In honor of (a) L ( t ) converges to a singular leaf L T of F . Professor Jürgen (b) If the curvature of M is bounded and the shape operator Berndt’s 60 th along each leaf is bounded, then ϕ t ( p ) converges to a birthday. point of L T , for each p ∈ L ( 0 ) . In addition Definitions the singularity is of type I, i.e., ThmA Basins of t → T − � A t � 2 lim sup ∞ ( T − t ) < ∞ , attraction MCF of non-closed regular leaf where � A t � ∞ is the sup norm of the second fundamental Cylinder form of L ( t ) . structure Type 1

On Mean curvature flow of Singular Lemma (basins of attraction) Riemannian foliations: Non Let L q be a singular leaf. Then there exists an ǫ = ǫ ( L q ) such compact cases that if L ( t 0 ) lies in Tub ǫ ( L q ) we have: Marcos M. (a) For any t > t 0 the distance r ( t ) = dist ( L ( t ) , L q ) satisfies Alexandrino (IME-USP) In honor of Professor C 2 1 ( t − t 0 ) ≤ r 2 ( t 0 ) − r 2 ( t ) ≤ C 2 2 ( t − t 0 ) Jürgen Berndt’s 60 th where C 1 and C 2 are positive constants that depend only birthday. on Tub ǫ ( L q ) . Definitions ThmA Basins of attraction MCF of non-closed regular leaf Cylinder structure Type 1